BIOMATHEMATICS TRAINING PROGRAM
•
UNIFORMITY STUDIES WITH SOYBEA.NS AT THE
NORTH CAROLINA STATE UNIVERSITY PHYTOTRON
by
Chong Soon Lee
Institute of Statistics
Mimeograph Series No. 1153
Raleigh - December 1977
(iv)
TABLE OF CONTENTS
Page
1
INTRODUCTION . . • • .
1
2
REVIEW OF LITERATURE
4
3
MATERIALS AND METHODS
7
4
STUDIES IN THE WALK-IN 'A' CHAMBERS
4.1
4.2
·e
4.3
11
Materials and Methods . • . . .
4.1.1 Studies on juvenile plants
4.1.2 Studies on mature plants
11
11
Results
•...•.•.•
4.2.1 Preliminary data analyses. .
. ...
4.2.1.1 Mean, variance (within experiment),
range and coefficient of variation.
4.2.1.2 Correlations among variables
4.2.2 Within experiment variation with 20 day
old plants
•....
•...
4.2.2.1 Position effects
••••
4.2.2.2 Characterization and description of
growth pattern
. • . . •
4.2.2.3 Consistency of pattern over
experiments . . . . . . . .
4.2.2.4 Temperature and light distribution
effects . . . . • . . .
4.2.2.5 Use of prior information as a
covariate
.
4.2.2.6 Within chamber designs .
4.2.3 Between experiment variation with 20 day
old plants
.....•....
4.2.3.1 Between chamber designs . . . . .
4.2.3.2 Relative sizes of the between to
the within experiment error mean
squares . . . . . . • .
4.2.4 Experiment with mature plants • . .
4.2.4.1 Within chamber variation . . •
4.2.4.2 Correlations between variables at
the two growth stages . . . .
4.2.4.3 Correlations between variables in
maturity study with variables in
juvenile studies in A4
15
15
Discussion and Conclusions . . .
14
15
16
17
17
18
30
36
39
41
44
44
48
50
50
51
52
57
(v)
TABLE OF CONTENTS (continued).
Page
5
61
STUDIES IN THE WALK-IN 'B' CHAMBERS ••
5.1
5.2
Materials and Methods • • • • • •
5.1.1 Studies on juvenile plants
5.1.2 Studies on mature plants.
Results . . . . . . . . . . .
......
61
61
62
.. . . . . . .
5.2.1
Preliminary data analyses
••••
5.2.1.1 Mean, variance (within experiment),
range and coefficient of variation •
5.2.1.2 Correlations among variables
5.2.2 Within experiment variation with 20 day old
plants
. . . . . . . . . . . . . . . . .
5.2.2.1
5.2.2.2
Position effects • • • • • • • • • •
Characterization and description of
pattern of growth • • • •
•• • • •
5.2.2.3 Consistency of pattern over
experiments • • • • • • •
5.2.2.4 Light and temperature effects ••
5.2.2.5 Use of prior information as a
covariate • • • • • • • • • •
5.2.2.6 Within chamber designs • • • •
5.2.3 Between experiment variation with 20 day old
plants
.... . . . . . . . . ..
.. . .
5.2.3.1
5.2.3.2
Between chamber designs • • •
Relative sizes of the between to within
experiment error variances
5.2.4 Experiment with mature plants
5.2.4.1 Within chamber variation
5.2.4.2 Correlations between measurements at
two growth stages • • • • • • •
5.2.4.3 Correlations between variables in
maturity study with variables in
juvenile studies in chamber B8 • • • • •
5.3
6
Discussion and Conclusions
STUDIES IN THE REACH-IN
6.1
6.2
'c'
CHAMBERS
Materials and Methods •
Results.
• • • • •
• • • • • •
6.2.1 Preliminary data analyses
• • • • • • • • •
6.2.1.1 Mean, variance (within experiment)
range and coefficient of variation •
65
65
65
66
67
67
68
76
79
82
85
87
87
88
91
91
92
95
97
103
103
106
106
106
(vi)
TABLE OF CONTENTS (continued).
Page
6.2.2
6.2.1.2 Correlations among variables . . . .
Within experiment variation with 20 day old
planes
. . . . . . ..
.
6.2.2.1
6.2.2.2
Position effects . . . . . • . .
Characterization and description of
pattern of growth • . . . . • •
6.2.2.3 Consistency of pattern of growth
6.2.2.4 Light effects . . . . .
6.2.2.5 Use of prior information as a
covariate . . . . . . .
6.2.2.6 Within chamber designs .
6.2.3 Between experiment variation with 20 day
old plants . . . . . . . . . . . .
6.2.3.1 Between chamber designs
....
6.2.3.2 Relative sizes of the between to
the within experiment error mean
squares . . . .
e.
6.3
Discussion and Conclusions.
107
108
108
109
119
124
125
127
128
128
130
132
7
GENERAL CONSIDERATIONS FOR EXPERIMENTS IN PHYTOTRONS
135
8
APPENDICES
139
9
LIST OF
REFER&~CES
.
144
1
INTRODUCTION
Individually controlled growth chambers and greenhouses
located in one building are collectively referred to as a phytotron.
Downs et. al. (1972) called phytotrons "laboratories of precisely
controlled environmental chambers used to study the effects of
temperature, humidity, light and other factors on plant growth."
Since the establishment of the first phytotron at the Earhart
Laboratory in California Institute of Technology in America in 1949,
other phytotrons have been established here and in other parts of
the world.
The phytotron at the North Carolina State University
was one of the two units established in 1968 under the Southeastern
Plant Environment Laboratories (SEPEL) in North Carolina.
the four general engineering
de~ign
One of
criteria considered in the
planning of the phytotrons f0r SEPEL was uniformity of· environmental
conditions within chambers, (Downs et. al., 1972).
horizontal plane one meter
fr~m
Temperatures in a
the light barrier of not more than
+ .50 degrees Celsius with lights on and + .25 degrees Celsius with
lights off, at any given temperature setting, were the measure of
uniformity within chambers.
Lang (1963) recognized that an engineer
could construct a chamber in which temperature and light distribution
approached ideal uniformity but with plants placed in a chamber, this
uniformity will be reduced.
He believed that the biologist who
works with the plants and the engineer who designs the chambers, must
come to a reasonable compromise on the uniformity that could be
attained for plants.
Kramer (1963), in a discussion on Lang's paper
2
felt that the basic objectives of controlled environment equipment
was that pla~t growth, under the same environmental condition· should
be reproducible (presumably in different chambers set under the same
conditions), and that all plants within a chamber should be subjected
to essentially uniform environment.
With the capacity of precise control over several
environmental factors influencing plant growth in chambers, which had
been impossible under natural conditions, biologists can now study
several factors simultaneously.
A chamber is a physical unit so
constructed that it cannot be partitioned to accomodate several
levels of certain treatment factors applied to a chamber.
For
example, in a study of four day temperatures at least four separate
chambers are necessary.
The experiment may take a fraction or a
full chamber space but the experimental unit to which a treatment is
randomly assigned is a chamber.
The number of chambers required
will be large in a factorial experiment where several factors are
studied simultaneously.
Replications of treatments are essential
to obtain a valid estimate of the experimental error (the between
chamber variance) to test between chamber treatment factors.
Limitations in space and time in the phytotron have often led to
experiments involving environmental factors being conducted without
replication.
Without an estimate of the between chamber variance,
tests of the between chamber factors are often erroneously made using
a within chamber estimate of experimental error.
Cooke (1968)
recognized that for greenhouse experiments, the within-house estimate
3
of error is not appropriate for testing the between-house factors,
as he believed that an additional component of error at the wholehouse stage may be substantial.
However, he believed that in
experiments utilizing growth cabinets (or chambers), this additional
component of error may be small.
In addition to treatment factors applied to whole chambers,
several within chamber treatments mayor may not be included.
When
within chamber treatments are included, the experiment is analogous
to the split-plot, with whole chamber treatment factors as the main
plot treatments and the within chamber factors as the sub plot
treatments.
The main plot experimental unit is a chamber within the
period of time specified by the experimenter.
The sub plot
experimental unit is usually a single pot containing one or more
plants.
Chamber to chamber variation is often assumed to be
negligible.
With this view, many experimenters working under
controlled environments in the phytotron have ignored the need for
appropriate between chamber and within chamber experimental designs.
4
2 REVIEW OF LITERATURE
Many research workers conducting experiments in greenhouses and growth chambers have reported variation in plant
performance within chambers, usually in qualitative rather than
quantitative terms.
In addition, several research workers have
tried to explain the possible sources of variation within chambers in
terms of some physical measurements.
Hruschka and Koch (1964) reported that 'strong gradients
in sprout weight of potatoes were found front to back, top to bottom,
and left to right in storage room experiments where temperature and
humidity were controlled.
They concluded that a completely
homogeneous environment would never be realized and recommended
stratification to increase precision of experiments.
Collip and
Acock (1967) reported significant variation in relative growth rates
in lettuce in growth cabinets (1.25 m x 1.25 m) with outside columns
and rows having higher variability and also higher growth rates
compared to the other regions within cabinets.
They recommended that
the outside columns and rows be treated as guard rows to reduce the
within cabinets variation.
Rigorous plant selection for uniform
plant size at the start of the study was given as another way of
reducing the variation.
In terms of a within chamber design, they
recommended the randomized complete block over the completely random
design.
Went (1955) also believed that as much as 50 to 70 percent
of the within chamber variation could be eliminated by exercising
vigorous phenotype selection of planting materials, even though the
5
materials were of the same genotype.
Hammer and Langhans (1972) at
Cornell measured physical and biological variation within a walk-in
chamber, similar in dimensions to an 'A' chamber at N.C.S.U. (2.44m
x 3.66m x 2.l3m).
Light pattern was described as a dome with a
peak in the middle and lower values at the sides.
Temperature
readings fluctuated from location to location within a chamber.
By
comparing 20 day old corn plants in five positions within a chamber,
significant differences were found in fresh and dry weight, with the
heaviest plants in the center of the chamber.
They concluded that
light and temperature was a source of "unwanted" variation within a
chamber and suggested some form of calibration.
From their study,
they recommended a randomized complete block design within chambers
with square blocks in preference to complete randomization.
Although no justification was given, working aisles along the walls
were also suggested.
Kalbfleisch (1963) in Canada, in an investigation on
methods of obtaining more uniform artificial light distribution over
plant growth areas, presented several diagrams showing the light
distribution patterns within chambers similar to those of Hammer
and Langhans.
Singh et. al. (1974) reported that although the soybean
is classified as a C3 plant species with low net photosynthetic
capacity, it has the ability to adapt to a high light intensity
under growth chamber studies.
He showed that in terms of
photosynthetic production, the light response curve was linear with
6
light levels from 3,000 to 5,000 foot candles, suggesting that its
behaviour was more like a C4 plant.
Uniformity studies were conducted at the North Carolina
State University phytotron with soybeans (Glycine max) with the
following objectives:
(1)
to study the nature of variation within chambers for the 'A'
and 'B' walk-in chambers and the
'c' reach-in chambers with
growing areas of 9, 3 and 1 square meter respectively;
(2)
to characterize and compare the within chamber patterns of
growth in several chambers repeated over time with chambers
set at one temperature and light intensity;
(3)
to attempt to explain the possible causes of the within
chamber variation;
(4)
to obtain estimates of the size of the between chambers
variation to the within chambers variation;
(5)
to establish whether the variation in plant growth for young
soybean plants persists till maturity and is reflected in
yield;
(6)
and
to recommend suitable between and within chamber designs for
use in the phytotron.
7
3
MATERIALS AND METHODS
Experiments were conducted in the three different sized
chambers, referred to as
'c'
. 'A', 'B' and
Carolina State University Phytotron. l
chambers, in the North
All of the common features
in these experiments will be covered under this section.
Details
specific to each study will be covered under the 'Materials and
Methods' section appropriate to each study.
The variety of soybean, Bragg, was used for all experiments.
The experiments can be divided into two groups.
The first group was
conducted in chambers of all sizes (set uniformly to within specified
limits) in plastic pots of 11.4 cm diameter
(4~
inches), containing
600 m1 of two-third gravel and one-third peat-like substrate.
Pots were arranged on racks of 230m x 23cm, holding four
pots in a two by two arrangement.
In experiments in the 'A' and
'B' chambers, the racks were placed on moveable carts of 46cm x 46cm,
commonly called trucks, with four racks per truck.
In the 'c'
chambers, only racks were used.
Each experiment was terminated after 20 days with
measurements taken on individual p1ant. 2
Five measurements were
1
Floor space in the 'A' and 'B' walk-in chambers were 2.44m x 3.66m
and 2.44m x 1.22m with 2.13m clearance from floor to light barrier.
The 'c' reach-in chambers were 1.22m x .92m with 1.22m clearance.
Other features and specifications in the chambers can be referred to
in Downs, R.J. and Bonaminio, V.P. (1976).
2
The 20-day growth period was decided on as a compromize between
having a short growth cycle to allow for more trials and a long enough
growth period to allow plant reflection of environmental differences.
8
taken on each plant, which was identified within a chamber by its
row and column position. l
The plant measurements referred to as
variables and their measurement units (in metric system) are:
(i)
plant height, a measurement taken from the cotyledonary
node to the tip of the apical bud, in millimeters;
(ii)
leaf area, a measurement of the total surface area of all
fully opened leaves including the primary leaves, in
square centimeters;
(iii)
petiole length, a measurement of the 'stem' portion of the
first trifoliolate leaf from its axis with the main stem
of the plant to the base of the extreme leaflet, in
millimeters;
(iv)
fresh weight, a measurement of the combined weight of stem
and leaves of the plant above the cotyledonary node, in
milligrams;
-(v)
dry weight, the weight of (iv) above, dried for 48 hours in
the oven at 90 degrees Celsius, in milligrams.
All measurements, except for dry weight, were taken on the same day
the experiment was terminated, systematically row by row.
The second group of studies was comprised of two experiments;
one conducted in an 'A' chamber, and another in a 'B' chamber.
The
chambers used had been used previously in the first group of
1
A row is parallel to the door of a chamber and labeled I, 2 ... q,
with row 1 nearest the door.
A column is perpendicular to the door
and labeled 1, 2 •.• r, with column 1 nearest the left side wall.
9
experiments.
Pots of 25.4cm (10 inches) diameter were used with
3,600 ml of the same substrate, and were put directly on the trucks,
with four pots per truck.
The trucks were similarly arranged
inside the chamber as for the first group.
In this study, two
plants in each pot were grown till the 20th day, when a random one
of the two plants was harvested.
The same measurements as recorded
for the first group of experiments were taken with these 20 day old
plants.
The remaining plant in each pot was allowed to develop
until pod and seed formation when the experiments were terminated,
with the following measurements taken on an individual plant basis:
(i)
number of pods per plant;
(ii)
dry weight of pods in grams;
(iii)
dry weight of seeds in grams;
(iv)
(v)
dry weight of aerial parts of plants, in grams;
dry weight of roots, in grams.
This group of experiments will be referred to as studies with mature
plants.
Chamber specifications called for air temperature to
be maintained at + .25 Celsius of the set point as measured with
a No. 24 type "T", welded-bead thermocouple in a shielded, aspirated
housing.
The chambers were set for temperatures of 22/18 degrees
Celsius, in experiments with 'A' and 'B' chambers and either
26/22 or 22/18 degrees Celsius in experiments with the
'c'
10
chambers. 1
Light in the chambers was provided by a combination of
cool white flourescent and incandescent lamps which provided energy
equivalent to an illuminance of 4,000 to 4,600 foot candles for the
'A' chambers, 3,000 to 3,600 foot candles for the 'B' chambers and
2,200 to 2,800 foot candles for the
'c' chambers.
A photoperiod
of 12 hours was simulated using 10 hours of high intensity light
followed by 2 hours with incandescent lamps.
Soybean seeds were soaked in moist vermiculite and kept
at 27.5 degrees Celsius in a seed germinator for 48 hours.
Sprouted bean seedlings were then transplanted to the pots.
Visual
selection for uniform seedlings was made by choosing seedlings with
radical lengths between 2 to 4cm.
Three seedlings were planted in
the 11.4cm pot and six seedlings in the 25.4cm pot.
Further
selection was made at 7 days of growth in a pot by eliminating the
extreme plants.
From 7 until the 20 day growth stage, one plant per
pot for the 11.4cm pots and two plants per pot for the 25.4cm pots
were allowed to grow.
In the maturity studies, the plants selected
for measurements at 20 days were chosen at random, one from each pot.
Deionized water was used up to the 7 day old stage and
the N.C.S.U. Phytotron nutrient solution was used thereafter. 2
1
The 22/18 temperature was adopted as the principle temperature for
experimentation since previous experience in the phytotron suggested
that at this temperature, soybeans showed greater reproducible within
chamber growth pattern.
The higher temperature of 26/22 was used
twice in the 'B' chambers and one-half of the studies with the 'c'
chambers as a check on temperature effect on the within chamber growth
pattern.
2
Analytical constituents found in Downs, R.J. and Bonaminio, V.P. (1976).
11
4
STUDIES IN THE WALK-IN 'A' CHAMBERS
4.1
4.1.1
Materials and Methods
Studies on Juvenile plants.
Four chambers, A4, AB, A9 and AIO out of a possible
twenty-one were available for the studies with juvenile plants in
the 'A' chambers.
Chambers AB, A9 and AIO were adjacent while
chamber A4 was in another part of the same building.
The four
chambers were studied, three at a time in anyone 20-day period,
referred to as a trial.
Four trials were planned.
The design
was a balanced incomplete block, with three replications of each
chamber, and each pair of chambers occurring twice in the same
trial.
This balance was lost in the final trial because AIO,
scheduled for that trial, was unavailable and was replaced by A9.
Table 4.1.1 shows the chambers used and the dates for each trial.
It should be noted that for chamber AB in trial 3, a malfunction
of the light control left the incandescent bulbs on for three
consecutive nights, from September 1 to September 3.
Table 4.1.1 Chamber and trial combination with respective dates for
studies with juvenile plants in the 'A' chambers.
Dates
July 8 to July 30, 1975
July 29 to August 20, 1975
August 19 to September 10, 1975
September 9 to October 1, 1975
Trial
1
2
3
4
Chamber
A8,
A4,
A4,
A4,
A9,
A9,
AB,
A8,
AIO
AIO
A9
A9
12
Because of the size of each experiment, 384 pots each,
the planting and data collection for each trial was done, by
chambers, on three consecutive days.
The order in which the
chambers were planted was also followed in data collection so that
all plants had a 20 day growing period.
A schematic diagram showing the arrangement of trucks
and row, column and truck identity within an 'A' chamber is shown
in Figure 4.1.1.
This arrangement allowed an alley of approximately
23cm in the center which was necessary in order to reach the watering
taps.
It enabled observations to be taken adjacent to both side
walls and the front wall.
The door is located between columns 7
and 12.
Preliminary data on air temperature and light intensity
at several positions within chambers were taken on July 15, 1975
for each chamber occurring in trial 1.
Temperature readings were
taken at the top of the plant canopy in the center of each of eight
trucks labeled 1, 4, 9, 12, 14, 15, 21 and 24.
An unshielded
thermocouple hooked on to a LEEDS and NORTHRUP SPEEDOMAX potentiometer
measured the temperature readings in Celsius, rounded to 0.25 degree.
Light readings were taken at the centers of twelve trucks labeled 1,
4, 7, 8, 9, 10, 14, 15, 17, 20, 22 and 23.
lux using the LAMBDA photometer.
These were measured in
As a follow-up on light and
temperature consistencies within chambers, readings were taken again
for chambers A8 and A9 on June 1, 1977, at the center of all 24
trucks.
No uniformity trials were being run at this time.
2.44m
13
REAR AISLE
24
TRUCK
21
TRUCK
22
TRUCK
23
TRUCK
24
Row Identity
23
22
21
20
TRUCK
17
TRUCK
19
TRUCK
18
TRUCK
20
19
18
17
16
TRUCK
13
TRUCK
14
TRUCK
15
TRUCK
16
15
14
13
~
~
CI.l
H
<
TRUCK
TRUCK
9
10
.
C"1
~
zE-t
~
u
12
TRUCK
TRUCK
11
12
11
10
9
8
TRUCK
6
TRUCK
5
TRUCK
7
TRUCK
8
7
6
5
4
TRUCK
1
I~
TRUCK
4
TRUCK
3
TRUCK
2
DOOR
)\
3
2
1
1
2
3
4
5
6
7 8
9 10 11 12 13 14 15 16
Column Identity
Figure 4.1.1 Schematic diagram within an 'A' chamber showing the
truck arrangement and the column, row and truck identities.
14
However, pots with soil were arranged on trucks and positioned
exactly as was carried out during the uniformity studies.
4.1.2
Studies on mature plants.
The experiment on mature plants was conducted in
chamber A4 from March 17, 1976 to May 28, 1976, a period of 72 days.
Some problems in getting good plant growth at 22/18 temperature led
to the use of 26/22 temperature regime for this study.
Stakes were
used to hold the plants at 14 days, as growth was more luxuriant
compared to the juvenile studies.
27 days after sowing.
First flowering occurred about
Light intensity readings were taken on all
trucks on March 27, 1976.
15
4.2
4.2.1
Results
Preliminary data analyses.
4.2.1.1
Mean, variance (within experiment), range and
coefficient of variation.
experime~t),
The statistics mean, variance (within
range and coefficient of variation (c.v.) were computed
for all variables in each chamber-trial combination, referred to as
an experiment, and are summarized in Appendix 1.
The ranges within
experiment for all variables were consistent,with the tallest and
heaviest plants about three times that of the smallest or lightest
plants in the same experiment.
This gives an indication of the
variation in size of 20 day old soybean plants within a chamber with
all plants treated uniformly.
and
c~v.
Average values of the mean, variance
(average of ratios) are presented in Table 4.2.1 for all
variables.
The values in Table 4.2.1 were for eleven experiments
excluding the experiment in chamber A8 in trial 3.
Table 4.2.1 Means, variances (within experiment) and coefficients
of variation (c.v.) averaged over 11 experiments by variable in
'A' chambers with 20 day old soybeans at 22/18 degree Celsius.
Statistic
plant
height
(mm)
Mean
Variance
c. v. *
77
156
16
Variables
fresh
leaf petiole
area length
weight
(cm2 ) (mm)
(mgm)
133
558
18
27
30
20
318
2,842
17
dry
weight
(mgm)
49
73
17
Note:* These c.v.'s computed from a single plant basis cannot be
compared directly to c.v. 's for field experiments which are computed
(usually) on a plot basis (of several plants).
--
16
A faulty mechanism in chamber A8 in trial 3 caused the
lights to be on for three consecutive nights and resulted in mean
values of approximately one and one half to two times the average
values of the other trials.
Also, the variances were increased
by a factor of two.
4.2.1.2
Correlations among variables.
Simple correlations
among variables were computed for each experiment with 382 degrees
of freedom.
Table 4.2.2 summarizes the average correlations
among variables.
Table 4.2.2 Within experiment correlation coefficients among
variables in 'A' chambers, averaged over all experiments.
e·
Variables
leaf
area
plant height
.61
leaf area
petiole length
fresh weight
Variables
petiole fresh
length
weight
.72
.82
.61
.94
.80
dry
weight
.60
.91
.78
.93
The results showed that variables leaf area, fresh weight
and dry weight were highly correlated, and may be grouped as a set
with close association.
Plant height appeared to belong to another
set having the lowest correlations with the first set.
Petiole
length seemed to be intermediate in correlations and would be placed
in a different set from the above.
This information will be used
later under section 4.2.2.2 regarding the selection of variables for
illustration of growth patterns within experiments.
11
4.2.2
Within experiment variation with 20 day old plants.
4.2.2.1
Position effects.
The variation within experiments
was further investigated to determine whether there were any
systematic positional differences within experiments.
Three
within chamber models were studied; models allowing for the column
effects, for the row effects or for the truck effects.
Each model
is like the randomized complete block model with columns, rows or
trucks as blocks.
Separate analyses of variance were computed,
fitting each model in each experiment.
Coefficients of
determination, R2 , measures of the amount of the total within
experiment variation attributable to the blocking effects in the
model, were obtained for each model in all variables. l
The
averages of R2 (average of ratios) are shown in Table 4.2.3.
The truck effects accounted for 32 to 47 percent of the
total within experiment variation, the column effects for 17 to 33
percent and the row effects for 7 to 10 percent.
It should be noted
that the column and row effects are orthogonal so that the average R2
for a model containing both effects would be the sum of the individual
column and row R2 (s).
A comparison of the truck, column and row
models will be deferred until section 4.2.2.6 pertaining to within
experiment designs.
1
The R2 is a ratio of the sum of squares of each of the effects (row,
column or truck) to the total sum of squares.
Thus, R2 will be a
ratio of the degrees of freedom associated with the two sums of squares
when the variance components for row, column or truck is zero.
The
R2(s) when these components are zero are 6.0, 6.0 and 3.9 for the
row, truck and column effect models respectively.
18
Table 4.2.3 Coefficients of determination in percent averaged over
all experiments in the 'A' chambers for the column, row and truck
within experiment models with ranges given in parentheses.
Variable
Column (15 d. f.)
Model
Row(23 d.f.)
Truck(23 d.f.)
plant height
9.9
(4.3 - 18.9)
6.8
(3.4 - 9.8 )
7.6
(2.9 ~ 17.4)
6.6
(3.3 - 9.0 )
6.7
(3.9 - 9.6 )
4.2.2.2
Characterization and description of growth pattern.
19.2
(10.4 - 34.6)
leaf area
17.3
( 9.3 - 26.0)
petiole length
21.5
(11. 3 - 36. 7)
fresh weight
16.5
( 7.5 - 22.5)
dry weight
23.4
( 7.8 - 39.7)
42.7
(27.9 - 55.0)
37.0
(21. 4 - 46.1)
47.4
(36 . 8 - 58. 2)
31.9
(25.2 - 40.6)
37.2
(21.7 - 48.2)
Estimates of the total within experiment variance were tested for
homogeneity with· Bartlett's test (1937) and were found to be nonhomogeneous for all variables even after the extreme values in
chamber A8 in trial 3 were omitted.
Correlations of experiment
means and variances and of means and standard deviations with 10
degrees of freedom showed that both the variance and standard
deviation were positively correlated with the mean for all
variables and of similar size in both cases.
One feature of non-normal distribution is that the
variance is often related to the mean (Snedecor and Cochran, 1967).
Several transformations in the power family were explored so as to
make the variance or standard deviation more nearly independent of
the mean, at the same time attempting to achieve homogeneity among
the total within experiment variances and making the distribution of
errors within experiments more nearly normal.
The mean-variance
19
correlations and the heterogeneity of the within experiment variances
were determined for the logarithmic, square root and reciprocal
transformations.
For all variables, the square root transformation
seemed to be the best transformation when both reductions in meanvariance correlations and heterogeneity of ~ariances were considered. l
For the purpose of characterizing "the pattern of growth
within chambers and for comparing patterns over chambers and trials,
second degree and higher degree response surfaces models using row
and column position identification as independent variables, were
fitted in each experiment for all variables.
The general form for
the nth degree response surface equation is
y
cr
=
+
+ ~Qklckrl + Q
rn +
~On
€
(4.2.1)
cr
where Ycr is the observation in the rth row and the cth column,
where c
and
€
= 1,
2, ••• 8, 11, 12
18,
r
= 1,
2, ••• 24 and k + 1 < n,
2
cr is the residual error distributed N(o,oF ).2
1
The decision to use this transformation was also made by considering
the reduction in heterogeneity of variances in the 'B' and 'e'
chambers using the different transformations.
A common
transformation was preferred in anyone variable for the 'A', 'B' and
'e' chambers in order that estimates of the between and within chamber
variances, scale of measurements in illustrations and the like may be
compared across chamber types.
See Appendices 2 and 3.
2
The fitting of response surface models is used primarily for the
purpose of smoothing the data.
A gap in c between 8 and 11 is
necessary to account for the empty space between columns 8 and 9.
20
Only the full models which included all possible
regression terms for a given degree response surface were fitted.
Coefficients of determinations, R2 , from the second to the seventh
degree response surface for the square root transformed data,
averaged over all experiments in each variable for the second to
the seventh degree response surface are given in Table 4.2.4.
Table 4.2.4 Coefficients of determination, R2 , averaged over all
experiments in 'A' chambers for the second to the seventh degree
response surface, by variable.
Degree of
response surface
plant
height
leaf
area
2nd
3rd
4th
5th
6th
7th
17.2
23.7
36.8
41.2
50.6
54.9
15.2
19.4
33.7
38.2
46.3
49.2
Note:
* Square
Variables *
petiole fresh
length weight
19.8
26.1
41. 7
54.1
57.2
60.9
13.0
14.9
30.6
34.5
42.4
45.3
dry
weight
. 19.2
24.3
35.5
39.4
46.4
49.0
root transformation applied to all variables.
Examining the average R2 values indicate that there was
little gain beyond fitting the sixth degree response surface models
in all variab1es. 1
Sixth degree response surfaces were thus fitted
in all variables (square root transformed), for each experiment to
characterize the within chamber pattern of variation.
1
Counts were also made on the additional regression terms from a
lower to next higher degree response surface models over all
variables and experiments which were significant at probability less
than 0.001.
The percentage of terms over all experiments and
variables which were significant from the second to the third, third
to fourth, fourth to fifth, fifth to sixth, and sixth to seventh
were 18, 42, 16, 24 and 7 respectively.
21
Dry weight, representing the set of variables (dry weight,
fresh weight and leaf area) and plant height are chosen to show the
fitted response surfaces in each experiment.
These response
surfaces are presented in Figures 4.2.1 and 4.2.2 for plant height
and dry weight respectively.
The growth patterns characterized by these two variables
were very similar in each experiment.
The pattern in each variable
in anyone chamber was also consistent over trials, with two peaks
separated by a valley located approximately in the same position
within each chamber over the trials.
There appeared to be some
similarity in pattern in chambers A9 and AID.
The pattern in AB was
like a mirror image of the pattern in A9 and AID (with the side wall
as the reflecting surface).
The pattern in chamber A4 appeared
different from A9 and AID with some similarity to AB.
Simple correlations were computed to determine the degree
of similarity of the witnineXperiment pattern and to test the mirror
image pattern in A4 and AB.
For each variable, observed values
within any two chambers were paired according to their positions
within chambers.
Correlations were computed for all possible
combinations of experiments, with trials occurring in the same or
different chambers using the same variable in any two experiments. l
I
As an example, six correlations are made between chambers AB (with
three trials) and AID (with two trials) for anyone variable.
With
plant height data, correlations by position within chambers were
made, ·of plant height data in AB with plant height data in AID.
This was carried out with data for all variables.
22
Figure 4.2.1 Contours of individual experiment, arranged by trial,
of a sixth degree response surface for square root of plant
height.
Explanatory note: Chamber identification is shown on the top left
corner in each diagram.
Chamber A8 in trial 3 with a defective
light control is shown with an asterisk (*). Alternate contour
intervals of 0.5 unit on the square root scale are designated by
letters arranged in ascending order as follows:
3.5
4.5
<
<
A <
B <
18.5
19.5
<
<
p
<
Q
<
4.0
5.0
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20.0
The bold-face letters P and V on each diagram show locations
of peaks and valleys in the contour surface.
23
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Trial 1
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Figure 4.2.2 Contours of individual experiment, arranged by trial,
of a sixth degree response surface for square root of dry weight.
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26
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1111111
" ...11.,1.1...... .1..1 II ..........."" 11111111""
""II ~.I~ III "'""""" III
lIt .... G
" .. 11
111 "'.............
tI
111'rl'tGG
~
15
"~II'!GG
H"'"
M....
""GHtt J~
II""
fCGI'i 11.,1.1..1.1 II PI"
""'"
"GHHI~ • .I.... ..1 11H""".....
~PG"IJ.I..I
..I.,1~11 ~ ~
...... ~"
"GSHI4.i
~.. 1 ~
" ~~.....
1;...... ' ..... Ie
.,1..1 II
,............"
"H"
10
" ' " . . . . . . . GGII
"'1'
,. 111111......
"PGHHIJI tIl """
.........
:
AS
AS-
G~:I~
Trial 4
A4
III
a:
C:G
GGG
GG
""" 111
tlh..GG
II .....GG
II~ G
Iltt.GII
.." III
"" II
"" 11
I"!" 11 "
=~:~ ~
hH 11
"" U
~ ..
II •
to""
II
U
lQ •
Il ... GGP
I,j
fl""Ci~
~:..:
u ..... G~
III'tG.pe
I uu. pie
lIt
1 ... ~pe
III
11 .. toteP' .
111.1.1.1.. .1.11111
111
Ilh.. GP I
11 ".I~~ ..I.,I..,l1111
1111
11 .. GP
... 1.1 .. ICKIUC..I.I.J 11111111111111 ...."'G ..
"",.....I••••••• .I.1.J 11111111111 ""GU ..
•
1a
..
'---! ·_10-
-----.+ao
.----.---------+-----.
JC
c
IS
!
SOl:
20
c
A9
2D
j
.i
10
iI
I
,
[
o----···----·-----+-----+
j ..
5
10
lS
2'0
c
.0
+
I
•
"
j!'c----'!""----·,-----!'---~·
10
15
•
Figure 4.2.2 (continued).
C
28
Another set of correlations with the ordering of the
columns in A4 and A8 reversed were also made for all possible
combinations of trials involving A4 or A8 and other chambers.
The results of the computations are summarized in a
two-way table in terms of the minimum and maximum correlations over
all variables for pairwise combinations of chambers, and all
pairwise combinations of trials in the same chamber in Table 4.2.5.
Table 4.2.5 Minimum and maximum correlations by position, over all
variables and all pairwise combinations of all trials for each
chamber and each pairwise combination of chambers, where correlations
with the columns in chambers A4 and A8 reversed as shown in
parentheses.
A4
Chamber
A8
Chamber
A9
A10
A4
min.
max.
.34 -.09
.71 .22
-.16 (-.11)
.19 ( .22)
-.00 (-.13)
.21 ( .16)
A8
min.
max.
.22
.58
-.29 ( .27)
-.00 ( .55)
-.36 ( .20)
-.02 ( .63)
A9
min.
max.
AlO
min.
max.
.36
.64
.24
.52
.41
.56
The correlations suggest that the patterns over trials
were somewhat less consistent for chamber A8 than for the other three
chambers (see the diagonal elements in Table 4.2.5).
While there
were reasonably high and highly significant correlations between
trials for a given chamber, there was likewise a clear indication of
lack of repeatability of the patterns.
29
The interchamber correlations clearly showed that the
growth pattern in A4 was quite unlike the patterns in the other
three chambers, whether or not the column labelling in A4
reversed.
was
Likewise, the pattern in A8 was unlike those of A9 or AlO
unless the column labelling in A8 was reversed.
With the column
labelling in A8 reversed, the interchamber correlations of A8, A9
and A10 were nearly of the same degree as the intrachamber correlations
of the same set.
The differences in the within chamber growth patterns
between chamber A4 and the other chambers (A8, A9 and A10) can be
observed more clearly by comparing the common fitted response surface
for chamber A4, computed from the pooled data over the three trials
for chamber A4 with the common fitted surface for chambers A8, A9 and
A10 (with the columns in A8 reversed) computed from the pooled' data
over the three chambers in trial 1.
In both cases, the transformed
data were used and sixth degree response surfaces were fitted.
Only
data from trial 1 were used for the common surface for chambers A8,
A9 and A10, so that the same total number of plants would be involved
in both data sets.
Predicted values from these sixth degree response surfaces
will be referred to in later sections, one dealing with the use of
prior information (from trial 1) as a covariate and the other dealing
with correlations between measurements taken from the maturity studies
and measurements from earlier trials with A4.
30
Two sets of contours for data pooled by position were
made, one for A4 and the other for three chambers combined (A8, A9
and AlO) for variables dry weight and plant height.
shown in Figure 4.2.3.
These are
The pattern was very similar for the two
variables in either group.
Peaks in A4 were located approximately
in trucks 7, 11 and 17,22 with a valley stretching from truck 14 to
19.
In the other chambers A8, A9 and AlO, peaks were located
approximately in trucks 6, 9 and 19, with a valley stretching
from truck 7 to 18.
4.2.2.3
Consistency of pattern over experiments.
Visual
examination of the predicted contours within experiments (i.e.chambertrial combinations) led to two main conclusions, (i) the within
chamber growth patterns were different among the four chambers and
(ii) the patterns in anyone chamber were fairly consistent when
repeated over time (or trial).
In this section this consistency or
lack of consistency is further examined by the analysis of variance.
Eleven experiments were combined and analysed as a split-plot design. l
1
Chamber A8 in trial 3 was excluded for reasons given earlier.
31
Figure 4.2.3 Contours of a sixth degree response surface for chamber
A4, averaged over three experiments and other chambers (AB, A9 and
AlO, with columns in AB relabeled), one experiment from each
chamber for (a) square root of plant height and (b) square root of
dry weight.
Explanatory note: Alternate contour intervals of 0.5 unit for plant
height and 0.2 unit for dry weight on the square root scale are
designated by letters arranged in ascending order for each
variable as follows:
(a)
(b)
plant height
dry weight
6.5
<
D <
7.0
5.2
<
B <
5.4
9.5
<
G < 10.0
7.6
<
I
7.B
<
The bold-face letters P and V on each diagram show locations of
peaks and valleys in the contour surface.
32
Other Chambers
A 4
Plant
25 +
height
2!! +
PPFF
PPFF
FFFF
eee
~~~Pp ~=~~P
PFFP
eSEEEE
PFF
CEEeEEEE
,.FP """FFF
PFFPFFFFF
FFF
eSEEeEEEEE
FFP
EEEEEEEEEE
FF"FFFFI'
PPP
"'fIFflFFPF
EEUHeUEE
EEEEE EeEe
PP"FF'fIF
"FF
PPFFI'F
FI'F
eEEee EeEEE
FFFF
FFPFI'P
eeEeEeeEEEE
FF
FP,.FFPFFP
EEEEEEEeEe
PFPPPFPPFPF
eEEEEl!Eee
FPP,.PPFPFFP,.
eEEeeeeE
FFPFPFPFFPPPI'
UEEEE
PF,.,.FFI'FPF,.,.I'''
EEEEI!
FFFI'FfI
FFFflF
,.FFfIfI
,.PFI'
,.FflFF
FP,.
PFPflF
GGG
FFP
F,."""
GGGGGG
FFP
PP"FF
GGGGG~G
FPP
PF"FF
GGGGGGG
FFF
FI'PFFP
GGGGGGGG "PI'P
FflFFF
GGGGCGCGG FFFF
FFFPFF
CGGGGGGGG FFP
FFFFFP
CGGQGGGG FFF
FFFPPF
G'GGGGGGG FFF
PFFFFFP
GGGGGGGGG PF
FPFFFF
GGGGGGGG PPF
FPPPPPF
GGGGGGG
FPP
F
FFFPPPFF
GGGGGGG
FFP
FPFPPPFFPFFFPF
CGGGG
PFFP
FPFFPI'FFFF"PP
GGGGG
FFPF F
FI'FFFflFFFFFF
GGGG
FFFFFFF
FFFFFFF
GGG
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GG
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FPP G
ee P,.PF"PFPI'
ee ,.PFFPFP"FP
ee
i
20 +
15 +
10 +
5
+
20 ..
1:> ..
10 +
F"FPPFFI'F,.,.FFPPFPPPPP
F,.fI
PFFPPFFPFFFFFFFFFFP
GG "I'F
P"FFFPPPPPFF"""FF"
CCGGGG FFF
FPFFI'PFPPPFP"FF"FF GGGGGGGG FF
FI'FFFP"FFF PflP I' "" G~GGGGGG ""I'
"PFFFFfI"F
FflFFF GGGGGGGGG PF
FflFPPFPPI'
PFPP GGGCGGGGG PP
FFPflFFflPfI
FflflF GGGGGnGGG PPP
FFFFFFFI'
F~FP
GGGGrGGGG FF
FPFFFFFF
FFFF GGGGGGCGG FF
,.FPI'FFPI'
FFP GGGGGGGG FF
FPFFFFFF
FF,.F CGGGGGGG FP
FPFI'FI'FF
FFF GGGGGG FF
FI'FFFFFFF
'''F GGGGG FFF
PFFFFPFFF
EE
FFF
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FFF
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cEe
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Eee!
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e!!lE
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E E
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FFFFFFFPFFF
ee e
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FFI'FF FPPFP
.FFFFFFPFFFF
eeE
FFFFFFFflFPP
FFFFI'FPFFFF
FFFFFFFFFFF
FFFFF FFPFFF
FFFPFFFFFFF
FFPP FFPPF
FFPFFF,.FFF
FFFF
PFFFF
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FFFFF PFFPFF
FPFFFFFF
G
EE FFFFF FPI'FFFFPFPF"I'FFFFF
G
EE
FFPFFFFFFFFFPPFFFFFFFF
G
EEE FFPPFFFFFFFFPPFFFPFFPP
G
EEEE FFFFFFFFFFFFFI'FFFFFF
G
EEeE FPFPFPPFFFPFFFFFF
G
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G
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G
FFEEEE FI'I'FF
F EE
FFFF
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G
P
5 ..
o ..
R
1..o'----------------+-------+
o
R
I +0.-----+.-----_----_----+
c
c
5
10
20
15
+
~
10
1~
20
Dry weight
2~
20 ..
IS
..
I
IJ
!
I
..
I
5
I
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I
c
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25 +
+
+
I.
o
COCEEFFFFFPF
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OEEFF8FFFFF
FFFFFF
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OE FFFFFFFF EEEEEEEEEEEE
FF
E. FFFFFFF EEEEfeEEEEEEEE
FF
E FFFFFFF eEEEE
EEEEEEE
FF
E FI'FFFF eEEE
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FF
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FF
E FFFFFP EEEE
E FFFFF EEE
,... EEEEE
FFF
EE FP
eEEE
Y EEEE
FFFFFFFFF
EE
EEee
EEEE FFPFPFFPFF
OEEE
EEEE
eEee FFFFFFFFFFF
~ EE
EeeEE
EEEEE FFFF
FF
C eeE eEeeEeeeEeeEE FFF
o EEEEEEEEeEEEeeee FFF
GGG
C EEEeeeEEEEEEEEEE FFF GGGGGGG
o eEEEeEfEeeeEeeE FFF GGGGGG~GG
o eEEEEEeeEeEEEe FFF GGG~ GGGG
C eEEeEeeEcEcEE FFF GG~
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o fEEEeeEEcEE
FFF GGG
GC FF
J e~~EEEEEEE
FFF GGG ~HHH
GG FF
eEeeEEee
FfFP GGG r~HHr
GG PF
FFF G~ rr~H GGFFF
EEeEE
eeeE
FFF GGG HH ." GGGFF
eEEE
FFPF GGG H~~ ~r GG FF
EEEE
FFF GG ~HHHH" GGFF E
eEEE
FFP GG HHHH~ CGGFF e
eEEE
FPF
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HHH GG FFEE
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eEEEe
FFFF GGG
GG FF EE
eeeEE
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o EEeEe
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(GGGGGGGGG PF EEE
CC EEEE
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cc EEEE
FFFFF
G(GGGGGG FFF
CD fEE
FFFFFF
CGGGGGGG FFFPFF
CD EEE FFFFFF
GGGGGGGGGG
FFFFG
+------+-----------.
10
20
I~
20 +
GGGG
GGGG
GGG
GGG
GGG
GGG
GGG
GGG
GGGG
GGGGG
GGGGG
GGGG
I
15 +
FF
~~
10
~
FF
FF
FF
EFF
E F
e F
CiGG ~~~ V
GGGG FFP
GGGGG FFF
GGGGGG FFF
GGGGGGG FFF
GG GGG FFP
GG
GGG FFFF
~~~Fgg
5
..
GGGG ~~~
H~ GGF eeE
GGG ~HH
~~GGFFeEE
FPF GGG ~HH
II
~HG FEEE
FFFFF 4G ~HH IIIIl~~GGFPEE
FFFFFF GGG HH IIIIII~~GFFEE
FFFFFF GG HH 111111 ~GCF e
PFPFPPP GG HH III~II~HGF'E
FFFFFFF GGHH Il1rII:~~G Fe
FFFFFFFPF GG HH III 11 HGGFe
FFFFFFFFFF GG~H 111111 H (iF
FFPP
FFPPGG H~ 11111 ~~GFF
FFFF
FFF GGHHH 1111 HHG F
FPFF
FFF GGH~H III ~~G F
FFF
FFFGG HH
HHcGF
FFF
FF GG HH
HHGGF
FFP
FFFGG HH~~ HHHGGF
P
~;F G~GH~~~~~ ~~F
FF 'GG HH~H GG
FFF GG
GGG
FFF GGG
GGG
FF GG
GG
FFF GGG
GG
FFFF GGG
GG
~~~ F~~~F=~~~~ g~~
G
EFPGG
GGG FFFFFFFF GG~G
EEF G~
GGG FPFFFFF GGGG
OEEFFGGG
GGG FFFFFFF GGGG
COE FFG~GG~GGG
FFFFFF GG~G
OOEEFF GGGGGGG
FFFFF
GGGG
co E FF CGGGGG
FFFFF
GGGGG
CDCEE FP
FFPFF
GGceGG GGG
~OC EE FFF
FF
GGGGGGGGGGG
cc eEE FFFF
GGGGCGeGGGGG
eE
EE FFFF
GGGGGGG
GGGG
FFEEEEE FFFF GGGGGGGGG HrrH ~GG
c ..
" 1+·----------------+-----·
o
5
10
20
c
1~
c
Figure 4.2.3 (continued).
33
The model adopted was
+
IJ
(CR)ik
+ (TR)jk + (CTR)ijk + 0ijkn
where
Yijkn
(4.2.2)
is the observed value on the nth plant within the
kth truck in the (ij)th experiment,
i
= 1,
...
Ci
is the ith chamber effect,
Tj
is the jth trial effect,
j
= 1,
q;
Rk
is the kth truck effect,
k
= 1,
r',
n =- 1, ... 16;
p;
(CR)ik is the effect of the kth truck in .the ith chamber;
(TR) jk is the effect of the kth truck in the jth trial;
(CTR)ijk is the effect of the kth truck in the (ij)th
experiment;
2
is the random "main plot" error, distributed N(O,(J B);
and
oijkn
2
is the random "sub plot" error, distributed N(O,(J w).
The results in the combined analyses of variance using
all trucks in each experiment (r = 24) and numbered as shown in
Figure 4.1.1, page 13, are summarized as mean squares in Table 4.2.6.
The test for consistency of the within experiment pattern
over chambers is made by comparing the chamber x truck mean squares
with the chamber x trial x truck mean squares in F-tests.
As the
data are unbalanced, the ratios of the two mean squares serve as
approximate tests.
The ratios ranged from 9.49 to 18.27, showing
significant differences in the within experiment pattern over
chambers.
-
e
e
Table 4.2.6 Mean squares in the combined analyses of variance as a split-plot design, with 11
experiments as the main plots and 24 trucks of 16 plants each within each experiment. *
Source of variation
Main plots (Experiments)
Chamber
Trial
Chamber-Trial Residual
Sub plots
Truck
Chamber x Truck
Trial x Truck
Chamber x Trial x Truck
Within Chamber-Trial-Truck
Residual
Note:
Df
Variables **
petiole fresh
length weight
plant
height
leaf
area
dry
weight
3
3
4
50.05
48.54
125.72
197.72
310.83
127.67
39.30
36.01
50.07
320.63
464.54
211.63
48.10
48.10
31.43
23
69
69
92
3,960
9.72
7.93
0.48
0.55
0.29
16.90
14.59
1.09
1.13
0.71
5.04
4.75
0.27
0.26
0.15
30.22
24.10
2.58
2.38
1.62
9.14
4.08
0.47
0.43
0.24
* Mean
squares for main effects were adjusted for each other, two-factor interactions
adjusted for each other plus the main effects and three-factor interactions adjusted
for all effects.
**
Square root transformation applied to all variables.
W
.I:'-
35
The test for consistency of the within experiment pattern
over trials in the same chamber is made by comparing the pooled
trial by truck and chamber by trial by truck mean squares, that is,
the truck by trial interaction nested in chambers, with the within
chamber-trial-truck residual mean square in·an approximate F-test.
The ratios ranged from 1.52 to 1.86, which were judged significant.
Thus, the growth pattern within experiments was more consistent
over trials than across chambers.
A comparison for the trial by
truck mean square to the chamber by trial by truck mean square
gives no indication of any part of the pattern of interaction being
associated with trials.
The ratios were 0.9 to 1.1.
This showed
that the pattern of inconsistency as shown in the truck by trial
interaction within chambers mean square being larger' than the residual
was not associated with any average trial differences.
It was noted previously that the within chamber pattern
in A4 was different from the other chambers and the pattern within
chamber A8 was a mirror image of the patterns in A9 and Ala.
The
above analysis of variance was rerun excluding data from chamber A4
and relabeling the data from chamber A8 such that the within chamber
pattern would resemble chambers A9 and Ala.
The results of this
analysis showed a reduction in the ratios of the chamber by truck
mean squares to the chamber by trial by truck mean squares to less
than one-third the previous size.
This showed that a major
portion of the differences in the within experiment patterns over
chambers was due to the within chamber
pattern in A4 and the mirror
36
image pattern in AB.
The ratios of the truck by trials interaction
within chambers to the within chamber-trial-truck residual mean
squares were very similar to the previous size.
4.2.2.4
Temperature and light distribution effects.
Temperature and light intensity readings were taken at several
positions within a chamber, at the start of the uniformity studies
on juvenile plants and after the uniformity studies were completed.
In the first set, readings were taken in chambers AB, A9 and AlO.
This set had eight temperature readings and twelve light intensity
readings with five of the temperature and light readings taken at
the same five trucks.
temperature and light
In the second set, twenty-four readings of
int~nsity,
one for each truck, were taken for
chamber AB and A9 without uniformity studies and two years after
completion of the uniformity studies.
The second set of temperature
and light readings were not taken in chambers A4 and AIO as they were
in use at that time.
Twenty-four light measurements, one per truck, were taken
for chamber A4 during the maturity study seven months after
completion of the uniformity studies with juvenile plants.
This
set of light readings in A4 will also be distinguished from set one
and considered as part of set two.
Table 4.2.7 gives the range
in temperature and light readings, by chamber for the two sets of
readings.
37
Table 4.2.7 Range in two sets of temperature and light readings
by chamber, one taken at the time of uniformity studies for
juvenile plants and the other taken without uniformity studies.*
Chamber
A4
A8
A9
A10
Temperature
degrees Celsius
Set 1
Set 2
23.0 - 26.0
23.3 - 26.8
23.3 - 26.5
24.0 - 26.5
23.5 - 27.0
Light ( x 10 3 lux )
As measured
As rated
2
Set 1
Set
Set 1 Set 2
45 - 49
39 - 45
42 - 47
32 - 46
37 - 46
44 - 52
48
43
48
51
44
51
Note: * Light readings were taken in A4 at the time of the uniform
studies with mature plants.
Correlations of light readings, each with 10 degrees of
freedom, from set one with those on common trucks in set two were
high for both chambers A8 and A9 ( .83 and .87 ).
Correlations
of temperature-readings, each with 6 degrees of freedom, from set
one with those on common trucks in set two were somewhat low in
A9 ( .63 ) compared to A8 ( .87 ).
This showed that the light
and temperature pattern within chambers for A8 and A9 did not
change appreciably over the two year period.
Correlations of both sets of temperature readings were
made by chamber over all trials, with each of the five variables,
plant height, leaf area, petiole length, fresh weight and dry weight.
Similar computations were made for both sets of light readings.
In
all cases, the correlations of light and temperature readings were
made with the mean plant performance of all plants within the same
truck for which the light and temperature readings were taken.
Average correlation over chambers and trials, by variable, are shown
in Table 4.2.8.
38
Table 4.2.8 Correlations, r, and range in parentheses, of
temperature and light readings with each variable averaged over
chambers and trials, the correlations computed on the mean plant
performance on the same truck for which light and temperature
readings were taken.
Temperature Readings
Set 2***
Set 1*
Variables
Set
ht
Lii
1
Readings
Set 2***
plant height
.54
(.15 to .86)
.63
(.54 to .69)
.13
(-.24 to .56)
-.10
(-.39 to .24)
leaf area
.71
(.41 to .98)
.70
(.57 to .84)
.18
(-.24 to .45)
.05
(-.23 to .39)
petiole length
.67
(.17 to .98)
.69
(.51 to .80)
.16
(-.11 to .47)
.01
(-.30 to .33)
fresh weight
.71
(.43 to .96)
.71
(.55 to .87)
.21
(-.21 to .45)
.10
(-.18 to .32)
dry weight
.81
(.61 to .89)
.70
(.61 to .81)
.26
(-.24 to .53)
( .05 to .43)
.20
Note:
* For set 1 of temperature readings, each r value was
estimated with 6 degrees of freedom.
H
p = 0 is rejected if
Irl > .707 at 0.05 probability.
** For set 1 of light readings, each r value was estimated with
10 degrees of freedom. H
p = 0 is rejected if Irl > .576
at 0.05 probability.
*** For set 2 of both temperature and light readings, each r
value was estimated with 22 degrees of freedom.
H
p = 0 is
rejected if Irl > .404 at 0.05 probability.
The results show that the temperature variation within
chambers was more closely associated with the plant growth patterns
than the light variation.
Estimates of the correlations of
temperature with the variables were quite similar in both sets of
readings.
These results should not be taken to imply a 'cause and
effect' relationship between temperature and plant growth.
39
4.2.2.5
Use of prior information as a covariate.
The
purpose in this section is to test the usefulness, as a covariate, of
prior uniformity data.
For this purpose, the experiments in trial 1
will be regarded as prior within experiment information which will be
used to adjust position effects within experiments in trials 2, 3
and 4.
In section 4.2.2.2, the within experiment pattern of
variation in trial 1 was summarized in terms of a common degree
response surface fitted to pooled data, of chambers A8, A9 and AlO.
A
The predicted values, from these surfaces,
YH,
A
YA,
A
Yp,
A
YFW and
Ynw for variables plant height, leaf area, petiole length, fresh
weight and dry weight respectively will be the covariates for
experiments in trials 2, 3 and 4.
Correlations among the covariates
for different variables were high, ranging in values from .89 to .99.
The covariate,
Ynw ,
with the highest correlation with all other
predicted variables on the average was selected for this study to
determine whether additional control of the within experiment
variation may be obtained with a covariate in the truck and columnrow models in each experiment.
Five models for control of the
within experiment variation were compared:
the covariate,
Ynw
(a) a model using only
(b) a model using truck position effects,
(c) a model using truck position plus the covariate
model using both the column and row effects and
A
the covariate,
Ynw '
Ynw,
(d) a
(e) the latter plus
Six of nine experiments belonging in trials 2,
3 and 4 will be used excluding the three experiments in A4 since the
within chamber pattern was different for A4.
In addition to the
analyses for the full chamber, analyses are made with four trucks
40
in each experiment since it is likely that an experiment conducted
in an 'A' chamber will not occupy the full chamber space of twentyfour trucks.
Three groups of four trucks within each experiment
located at different positions were compared:
(b)
trucks located in positions 10, 11, 14 and 15 in a 2 x 2
arrangement and referred to as the center 4 trucks;
(c)
trucks located in positions 5, 9, 13 and 17 in a 1 x 4
arrangement along the side wall and referred to as the
left-side 4 trucks; and
(d)
trucks located in positions 8, 12, 16 and 20 in a 1 x 4
arrangement along the side wall and referred to as the
right-side 4 trucks. l
e.
Analyses of variance/covariance were performed in each
experiment for the five models for all variables.
that is, the full chamber and four trucks,
effective in all variables.
In all cases,
A
YDW was equally
The coefficients of determination for
each model averaged over all five measured variables and the six
experiments are given in Table 4.2.9.
The results showed that without blocking within
experiments, prior information accounted for approximately 33 percent
of the within experiment variation in plant size in later trials,
with the exception of the left-side 4 trucks where the R2 was
21 percent.
1
Refer to Figure 4.1.1, page 13, for location of the trucks within
'A' chambers.
41
Table 4.2.9 Coefficients of determination, averaged over all
variables and experiments for five models, of *nw' truck, truck
and
nw ' column and row and column, row and Ynw in analyses of
variance/covariance by experiment for 24 trucks and three groups of
4 trucks located in the center, left and right-sides of the chamber.*
Y
A
Yw
Truck
Column
~d
~d
Ynw
Location of trucks
Full chamber (24 trucks)
Center 4 trucks
Left-side 4 trucks
Right-side 4 trucks
Model
Truck
33.0
32.6
21.2
33.4
36.2
28.7
16.7
9.1
46.3
43.0
30.0
40.9
Column,
Rowand
Ynw
~w
26.8
31.7
44.0
39.2
45.9
55.2
52.2
57.2
Note: * Each value was an average over five variables and six
experiments.
With blocking by trucks within experiments,.prior
information accounted for an additional 10 to 14 percent of the within
experiment variation in plant size, except for the right-side where
the gain was an additional 32 percent.
This was due primarily to
the reduced effectiveness of the blocking by trucks on the right-side.
With two-way blocking by rows and columns within
experiments, prior information accounted for an additional 18 to 24
percent of the within experiment variation except for the left-side
where the addition was lower, at 8 percent.
A comparison between
blocking by trucks and blocking by columns plus rows within
experiments will be deferred until the next section.
4.2.2.6
Within chamber designs.
Any of the conventional
experimental designs can be used in the within chamber experiments.
The objective of this section is to compare the relative efficiencies
42
of the completely random design (CRD) , the randomized complete
block design (RCB) and the two-way blocking design, by rows and by
columns (L5).
The within experiment error mean square for each design
was computed for the full chamber and for the three groups of four
trucks within experiment as shown in Table 4.2.10. 1
Table 4.2.10 Computational procedures for the within experiment
error mean square in the completely random (CRD), randomized
complete block (RCB) and the two-way blocking (L5) within chamber
designs.
Within chamber design
CRD
RCB
1) with trucks
as blocks
2) with columns
as blocks
3) with rows as
blocks
L5
Within experiment error mean square
among plants within experiments
among plants within trucks within
experiments
among plants within columns within
experiments
among plants within rows within
experiments
columns x rows interaction within
experiments
Estimates of the within chamber error mean square for
the full chamber and the three groups of four trucks are given in
Table 4.2.11.
The results showed that blocking on trucks would reduce
by 33 to 44 percent of the total within experiment error mean square
for the full chamber and by 32 to 48 percent of the within truck
variation in the center four trucks.
Blocking by trucks was also
equivalent to or better than blocking in both rows and columns.
1
The arrangement of the three groups of four trucks within chambers
are given in the preceeding section, page 40.
43
Table 4.2.11 Estimates of the within experiment error mean square
from 11 experiments, for the completely random, randomized block
and two-way blocking designs, from (a) the full chamber (b) 4
trucks in the center of the chamber '(c) 4 trucks at the left-side
of the chamber and (d) 4 trucks at the right side of the chamber.
Variables *
petiole fresh
dry
length weight weight
Within Chamber Design
plant
height
leaf
area
(a) Full chamber (24 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) rows as blocks
LS
0.49
0.29
0.42
0.47
0.39
1.08
0.71
0.93
1.07
0.91
0.27
0.15
0.22
0.26
0.21
2.24
1.62
1.97
2.23
1.94
0.36
0.24
0.29
0.36
0.28
(b) Center (4 trucks)
CRD
RCB 1) trucks as' blocks
2) columns as blocks
3) rows as blocks
LS
0.58
0.37
0.40
0.56
0.36
1.16
0.72
0.84
1.21
0.85
0.31
0.16
0.20
0.31
0.19
2.20
1.49
1. 71
2.30
1. 75
0.35
0.22
0.25
0.37
0.26
(c) Left-side (4 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) rows as blocks
LS
0.27
0.23
0.26
0.24
0.22
0.96
0.83
0.86
0.88
0.74
0.17
0.14
0.15
0.15
0.13
2.14
1.90
1.88
1.98
1.62
0.28
0.26
0.25
0.27
0.23
(d) Right-side (4 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) rows as blocks
LS
0.32
0.30
0.31
0.30
0.28
0.87
0.83
0.79
0.89
0.79
0.20
0.18
0.17
0.20
0.17
2.16
2.09
1.97
2.14
1.90
0.32
0.31
0.29
0.33
0.29
Note:
*
Square root transformation applied to all variables.
Estimates obtained from both the left and right-sides
were lower than those from the center, with or without blocking.
Blocking on trucks appeared to be less effective at the sides
compared to the center.
44
For experiments using the full chamber, blocking within
experiments is clearly advantageous and all indications point to
greatest efficiency using square blocks (in this case, trucks).
Columns as blocks are consistently more efficient than rows as
blocks but both are less efficient than blocking by trucks.
The
two-way blocking design provides only a slight advantage over the
randomized block with columns as blocks.
For experiments involving only four trucks, the results
showed that a 1 x 4 arrangement of trucks on either side is better
(i.e. more uniform) than the center 2 x 2 set of trucks.
For the center 2 x 2 set of trucks, blocking by trucks
is again most efficient, giving gains in efficiency over the CRD
of about 30 to 50 percent.
Blocking by columns is better than no
blocking, giving gains in efficiencies of about 30 percent.
The
two-way blocking design was about the same in efficiency as the
randomized block with columns as blocks.
For the four trucks from either side, the effect of
blocking of any kind is much less than the center four trucks.
This
is due primarily to the more uniform growth at the sides compared to
the center.
The two-way blocking design is best, although only
slightly better than blocking by trucks, giving up to about 10
percent gain in efficiency.
4.2.3
Between experiment variation with 20 day old plants.
4.2.3.1
Between chamber designs.
In experiments where the
treatment factors, such as temperature, to be studied require the
use of a chamber as an experimental unit, it is important to have an
45
appropriate estimate of the experimental error to test treatment
effects.
Because of the expense involved in such experiments, it
is tempting to use a "within experiment" error variance as the
"between experiment" error variance.
This section presents
estimates of the appropriate between experiment error variance (the
whole plot experimental error variance) for three designs;
the
completely random design (CRD), the randomized block design with
trials as blocks
or chambers as blocks (RCB) and the two-way blocking
design with trials and chambers as blocks (LS).
The relative
magnitudes of the between chamber and the within chamber variances
will be given in the next section.
Computations for the between experiment error mean square
for these three designs are shown in Table 4.2.12.
Table 4.2.12 Computational procedures for obtaining the between
experiment variance in the completely random (CRD), randomized
complete block (RCB) and the two-way blocking (LS) designs.
Design
CRD
RCB 1) trials as blocks
2) chambers as blocks
LS
Between experiment variance
among experiments
among chambers within trials
among trials within chambers
chamber x trial residual
(adjusted for average chamber
and trial effects)
Note: An experiment refers to a chamber-trial combination.
In Table 4.2.12, a split-plot design structure is
assumed with the appropriate designs for the whole plot.
There may
or may not be another set of treatments imposed on the sub plot units
46
of pots within experiments.
The expected value of the between
experiment. error mean square is
2
(0 W
2
+ to B)
where
is the
variance among t sub plot units within chambers for a given design
and
2
0
B is the whole plot error component of variance among
chamber-trial experimental units.
The results for the full chambers (t
= 384)
and four
trucks within each chamber taken from the center t left-side and
right-side (t
= 64)
are shown in Table 4.2.13.
The positions of
the trucks within chambers in the center t left-side and right-side
are similar to those used in the computations of the within
experiment error mean squares.
The results showed that the effectiveness of blocking
for whole plots varied with each of the variables and the positioning
of groups of four trucks within chambers.
With plant height and
petiole length data, blocking by trial or by chamber for the full
chambers was no better than a completely random design.
With the
same traits, blocking by trial in the center four trucks and the
left-side four trucks was better than a completely random design,
with gains of 11 to 28 percent.
than the completely random design.
Blocking by chamber was no better
Blocking by chamber for the
same traits at the right-side four trucks was better than blocking
by trial or the completely random design (CRD)t with gains of 32
to 39 percent.
With leaf area, fresh weight and dry weight data,
blocking by trial showed some value in the full chamber and the
different positions of groups of four trucks within chambers.
47
Table 4.2.13 Estimates of the experimental error for whole plots
from 11 experiments in the completely random (CRD), randomized
complete block (RCB) and two-way blQcking designs for the full
chamber and for 4 trucks each from the center, left and rightsides of the chamber.
Design
plant
height
Variables **
leaf petiole fresh
area length weight
dry
weight
(a) Full chamber
CRD
RCB 1) trials as blocks
2) chambers as blocks
L5
93.66
93.29
92.64
125.72*
236.04
157.69
206.67
127.67
48.90
45.46
44.05
50.07*
384.22
258.35
320.02
211.63
43.75
38.57
38.57
31.43
(b) Center 4 trucks
CRD
RCB 1) trials as blocks
2) chambers as blocks
L5
16.46
14.69
14.44
18.11*
29.75
14.99
25.25
13.05
6.35
4.56
6.08
5.98
50.02
32.69
40.72
24.57
4.81
4.03
4.08
3.49
(c) Left-side 4 trucks
CRD
RCB 1) trials as blocks
2) chambers as blocks
L5
10.13
8.65
12.61
13.86*
38.48
25.04
40.26
26.65
7.38
6.53
8.12
8.88*
53.55
34.44
57.40
36.95
5.72
4.52
5.17
4.27
(d) Right-side 4 trucks
CRD
RCB 1) trials as blocks
2) chambers as blocks
L5
21. 79
23.76
14.77
21. 98*
54.89
41.63
40.96
25.68
79.86
60.13
69.85
37.78
12.75
11.25
10.21
6.87
12.85
12.95
7.80
8.78
Note: * Estimates were higher in the design with two-way blocking (L5)
compared to the completely random (CRD) because negative estimates of
the components of variance for trials and/or chambers were obtained.
The best estimates in this case would be the lower values.
** All variables transformed by taking the square root prior to
analysis.
Gains of 12 to 50 percent for whole plots were obtained compared to
the CRD.
The two-way blocking design was best for the full chambers,
the center four trucks and the left-side four trucks, with gains of 11
to 34 percent over the RCB design with trials as blocks.
48
4.2.3.2
Relative sizes of the between to the within experiment
error mean squares.
The relative size of the between to the within
experiment error mean squares is an estimate of the ratio
2
2
(a W
+
2
2
a B is the whole plot component of error variance
among chambers,
a2w is the variance among experimental units within
ta B)/a W' where
chambers and t the number of experimental units within experiments.
The ratio is thus dependent on the number of experimental units (or
pots) within experiments.
With four trucks (t
= 64),
the relative
size of the two error variances would be approximately one-sixth
that for the full chamber (t
= 384)
if
2
a B dominates.!
The ratio
gives an indication of the magnitude of bias in the use of a within
experiment estimate of variance for testing factors requiring
chambers as experimental
unit~.
Relative sizes were computed assuming a completely random
whole plot design and the completely random, and randomized complete
block designs (with trucks as blocks) within chambers. 2
The results
for the full chamber and groups of four trucks in the center and the
two sides are shown in Table 4.2.14.
1
Estimates of the variance component a2B may be computed by taking
the difference of the two error mean squares in Table 4.2.11, page 43
and Table 4.2.13, page 47, and dividing the difference by the
appropriate t.
Similarly, estimates of the between expe~~ment er!~r
mean square may be computed for any t from estimates of a B and a W
from the full chamber.
2
It is assumed that treatments are present within chambers as subplot treatments and that these treatments may be randomly arranged
within chambers or assigned at random to trucks within chambers.
49
Table 4.2.14 Relative sizes of the between to the within experiment
error mean squares. assuming a completely random between
experiment design and two within experiment designs. the completely
random and randomized complete block designs (with trucks "as
blocks) for the full chambers and groups of four trucks located
at the center. left and the right-sides of each chamber.
Within chamber design
(a) Full chamber (24 trucks)
CRn
RCB
plant leaf
height area
Variab1es*
petiole fresh dry
length weight weight
191.1
323.0
218.6
332.5
181.1
326.0
155.5
215.0
121.5
182.3
(b) Center (4 trucks)
CRn
RCB
37.5
44.5
25.6
41.3
20.5
39.7
22.7
33.6
13.7
21.9
(c) Left-side (4 trucks)
CRn
RCB
37.5
44.0
40.1
46.4
43.4
52.7
25.0
28.2
20.4
22.0
(d) Right-side (4 trucks)
CRn
RCB
68.1
72.6
63.1
66.1
64.3
71.4
37.0
38.2
39.8
41.1
Note:
*
Square root transformation applied to all variables.
A within experiment estimate of error variance used in
place of a between experiment estimate of error variance in the full
chambers will be an underestimate by a factor of 120 or more. with a
completely random arrangement of pots within experiments.
The
completely random arrangement may be with or without treatments
within chambers at the sub plot level.
With treatments within
experiments arranged on trucks, with trucks as blocks, the relative
sizes of the between to the within experiment estimates of variance
are higher.
50
In experiments with groups of four trucks positioned in
the center, left and right-sides of each chamber,
t~e
relative sizes
of the two error terms differed by a factor between 13 to 72.
4.2.4
Experiment
4.2.4.1
with mature plants.
Within chamber variation.
The experiment with
mature plants was conducted in 25.4cm pots in chamber A4, selected
at random from the four chambers used previously in the juvenile
studies.
This experiment was run at a higher Day/Night temperature
of 26/22 because an earlier experiment at 22/18 failed, with plants
showing atypical growth thought to be related to the temperature
regime.
Two sets of data were taken at two growth stages, at
20 and 72 days.
In the first set, the same five variables as the
juvenile studies were taken with one of the two plants in each pot
selected at random.
The second set of measurements was taken with
the remaining plant.
The means, variances (among plants), ranges
and coefficients of variation for the variables taken at these two
growth stages are given in Table 4.2.15.
The means (Table 4.2.15) of 20-day old plants were two
to four times larger than the means for the same variables in the
juvenile studies.
Variances were three to eight times as large.
The coefficients of variation in variables at 72 days were 29 to
41 percent.
51
Table 4.2.15 Means, variances (among plants), ranges and coefficients
of variation (c.v.) for variables taken with 20 and 72 day old
plants in the same experiment in chamber A4 at 26/22 temperature.
Variables
20 day old plants
plant height (mm)
leaf area (cm2)
petiole length (mm)
fresh weight (mgm)
dry weight (mgm)
Mean
Variance
304
295
69
742
111
1,348
4,613
92
34,733
622
204
153
45
410
69
-
405
446
91
1,175
172
12
23
14
25
22
46
14
15
2.4
37
182
21
27
1.0
154
2
0.2
4
0.6
6
-
82
23.9
32
5.8
73
29
33
35
41
33
Range
c.v.
72 day old plants
pod number·
seed weight (gm)
shoot weight (gm)
root weight (gm)
whole plant weight (gm)
4.2.4.2
stages.
Correlations between variables at the two growth
Correlations, by position.(or pot), between variables taken
at the two growth stages would show to what degree the pattern of
variation seen with 20 day old plants at the juvenile stage persisted
over time.
These correlations, estimated with 94 aegrees of
freedom, are shown in Table 4.2.16.
These correlations are between different plants from
nearly the same position measured in different ways at different
stages of plant development.
Thus, while these correlations were
not large, there is clear indication of position effects which
persist from the juvenile to the mature stage.
Fifth degree response surfaces were fitted to the square
root of plant height and dry weight (variables taken at 20 days) and
52
Table 4.2.16 Correlations, by position, with 94 degrees of freedom,
between variables taken with 20 day old plants and variables taken
with 72 day old plants in the same experiment in chamber A4 at
26/22 temperature.
Variables
(72 day old plants)
pod number
seed weight
shoot weight
root weight
whole plant weight
Variables (20 day old plants)
plant
leaf petiole fresh dry
height area length
weight weight
.16
.13
.18
.16
.17
.36
.35
.41
.30
.38
.29
.27
.38
.27
.33
.40
.40
.46
.35
.43
.40
.39
.48
.40
.44
the square root of seed weight (variable taken at maturity).l
The
predicted contours for these variables are shown in Figure 4.2.4.
Similarities in the pattern characterized by dry weight and seed
weight can be seen especially. in the center and right side of the
chamber.
There was no obvious similarity in pattern for plant height
and either of the two variables, consistent with the lower correlations
with plant height.
4.2.4.3
Correlations between variables in maturity study with
variables in juvenile studies in A4.
Experiments with juvenile plants
in A4, conducted approximately seven months earlier, were compared to
the experiment
on mature plants to determine whether the within
chamber pattern of variation remained consistent over the given period.
There were two differences between the two experiments, with the
maturity study conducted at a higher temperature (26/22) and in larger
pots (25.4cm).
One large pot occupied the space of four small pots
used in the juvenile studies.
I
Choice of the fifth degree response surface was based on the
coefficient of determination as in the juvenile studies.
53
Figure 4.2.4 Contours of a fifth degree response surface in chamber
A4 for the maturity study, for square root of (a) plant height at
20 days, (b) dry weight at 20 days and (c) dry weight of seed.
Explanatory note: Alternate contour intervals of 0.5 unit for plant
height data, 0.2 unit for dry weight data and 0.2 unit for dry
weight of seed are designated by letters arranged in ascending
order for each variable as follows:
(a) plant height
15.5
<
M < 16.0
18.5
<
P
<
19.0
(c) dry weight of seed
(b) dry weight
8.4 < J <
10.0
<
N
<
8.6
10.2
3.4
<
F
<
3.6
4.6 < I < 4.8
The bold-face letters P and V on each diagram show locations of
peaks and valleys in the contour surface.
R
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FFFFFF G
FFFFFF GG
GG
FFFFFF G
FFFFF GGG
hH
GG
fFFfFF G
FFFFF GG
hhH~
GGG FFFFFF
GFFFFF GGG
H~HH
GGG FfFFFFF
G FFf GGG
h~h~H
GGG fFFFFFF
G FfF
GGG ~hF"',H
G<.G ffFFFIW
FFf GGG hhH~H
GG fFFff~
FfFF GGG hhHAHH
GG FFFFFFfF
FFfF GGG H~HHtIH
GG fFFFFfFF
fFFf GGG h~hhHH
GGG fffFFFF
FffF GGG
HHHHH
GGG FFFFFFF
FFFFf GGG
H~HH
GGG fFfffff
fFFfFF GGG
HHHH
GGG fFFfFFf
fFFfFF GG
~
GeG FFFfFfF
fF ffF GGG
GGG
FFfFF
fF FFF GGG
GGG
fFFfF
f
fFF GGG
G'G
FfFFF
F
FFF 'GG
GGG
FFFF
~ V n~ ~~~
G~~~~G
f
FFF GGGGG
GGGGGG
fF FFFF GGGGG
GGGGGGG
FF FFfFF GGGGG'GC'GGGGGGG
FFFFFfPF
GGGGGGGGGGGGGGG
fFFfffFF
GGGGGGGGGGGG
FfFFFFFFF
~~~~
FFFF
FFF
Ff
FF
FFF
FFF
o •
1·---------------·-------·-------·-------·
o
2
'<
6
8
Ie
c
VI
Figure 4.2.4 (continued).
~
55
Predicted values from sixth degree response surface
equa~ions
fitted to pooled data from the three juvenile experiments
in A4 were used for the correlations with the corresponding
Adjacent sets of four predicted
variables in the maturity study.
values are averaged so that there is a 1 : 1 correspondence, by
The predicted values
position, with data from the maturity study.
for plant height, leaf area, petiole length, fresh weight and dry
A
weight, referred to as
YH'
A
A
YA,
Yp '
A
YFW
A
and
Ynw
and their
correlations, by position, with some variables in the maturity study,
are shown in Table 4.2.17.
Table 4.2.17 Correlations, by position, estimated from 94 degrees
of freedom, between predicted values Y
and Y
p, Y
nw
H, YA' Y
from earlier experiments in A4 with juvenile plants at~2/18
temperature with two sets of variables from the maturity study at
26/22 temperature.
.
Predicted values
from juvenile studies
Variables from maturity study
72 day old plants
20 day old plants
shoot
dry
seed
plant
leaf
weight
weight weight
height area
1H
.50
.04
-.02
-.03
-.07
YA
.48
.08
.06
-.00
-.01
.50
.07
.05
-.01
-.02
YFW
.45
.10
.08
-.00
-.02
YW
.48
.11
.13
.06
.06
~
Yp
~
The results showed that the within chamber variation in
plant height at 20 days was the only variable from the maturity
study showing a significant association with the predicted values
~
~
56
from the earlier trials.
All other correlations, including those
with variables in the maturity study not shown in Table 4.2.17,
were not significantly different from zero.
The similar patterns
of the two studies can be seen by comparing Figure 4.2.3, page 32
with Figure 4.2.4, page 54, for plant height data in chamber A4.
57
4.3
Discussion and Conclusions
All 'A' chambers have top to bottom air flow, specular
reflector aluminum walls and other common features.
Variation in
plant growth within experiments is expected to be at least similar
in pattern over all chambers under a wide range of environmental
conditions.
If all chambers had similar within experiment patterns,
standardized within experiment designs with maximum efficiency could
be recommended for experiments of any size.
Uniformity studies with soybeans have shown that the
within experiment pattern of variation was different over the
chambers sampled.
However, about 40 percent of the total within
experiment variation could be controled by blocking on trucks.
Therefore, in experiments of within chamber treatments, a randomized
block design (complete or otherwise) with trucks as blocks and pots
as the within experimental units, is suitable.
The within experiment variation in four trucks of three
separate groups taken from the center, the left or right-sides of
each chamber showed higher variability among plants at the center.
This has shown that contrary to the usual recommendation to avoid
the sides of the chambers, Collip and Acock (1967), Hammer and
Langhans (1972), it is advantageous to have experiments along the
side.
Maximum use of within chamber space is thus possible.
There was some indication that plant growth within
chambers was associated with temperature, not light.
It would be
useful to take temperature readings within chambers for use as a
58
covariate, particularly where no blocking within chambers is done.
The use of prior plant growth data as a covariate for
later experiments would reduce the total within experiment variation
by one-third.
Its use with blocking within experiments would add
another 10 to 18
~ercent.
In addition to treatments applied within chambers (the
sub plot treatment) between chamber treatment factors (the main plot
treatment) are often studied simultaneously.
In experiments
occupying a whole chamber, a between experiment design with blocking
by time or chambers is no advantage over the completely random
design.
Even in experiments occupying four trucks within experiments
in a compact arrangement and similarly positioned over experiments
a completely random between experiment design is preferred.
Since
the number of 'whole plots' available is small, the additional
advantages of the completely random design will be the flexibility
and the larger number of degrees of freedom for experimental error.
A between chamber treatment factor of Day/Night
temperatures is frequently studied with several within chamber
treatments.
In the N.G.S.U. Phytotron, a standard set of Day/Night
temperatures are provided in four chambers programed at 30/26, 26/22,
22/18 and 18/14, with a nine-hour daylength.
Experimenters with
a few trucks of within chamber treatments will use these chambers.
As several types of experiments will occur within each chamber, and
at different times, trucks will be randomly positioned within a
chamber and moved frequently.
The trucks of one experiment may be
59
placed in a position corresponding to a 'peak' in one chamber and a
'valley' in another, causing part of the within chamber variability
to become between chamber variability.
An experiment where replication of the temperature
treatments is made using the same chamber will have an experimental
error variance that is biased as temperature effects are confounded
with the chamber effects.
Only when the temperature settings are
randomly reallocated to the four chambers will there be an unbiased
estimate of the between experiment error variance.
Without replication of between chamber treatment factors
in time or in chambers, there is no valid estimate of the experimental
error to test chamber treatment factors.
estimate of error
~s
Any within chamber
clearly biased downward.
Relative sizes of the
between experiment to the within experiment error mean squares from
uniformity studies give an indication of the magnitude of this bias.
With a within chamber experiment of four trucks (t '~64), the
magnitude of bias is from 13 to 68 units of error variance.
Higher
estimates are obtained with an efficient within chamber design.
The relative size is thus dependent on the number of within chamber
experimental units (t) and would be higher as t increases.
The
A2
estimate of the between chamber-trial variance component (0 B) was
A2
smaller than the within experiment component (0 W) in all variables.
As an example, the ratio for variable dry weight of
.11/.36.
A2 /A2
0
BOW was
It would be erroneous to believe that the smaller of the
two estimates, that is
A2
0
B' can be ignored and a within experiment
error mean square used in place of a between experiment error mean
60
square, Cooke (1968).
"2
It is only when a
B
is consistently zero
that it can be ignored, but this would be rare.
In the maturity study in chamber A4, the within chamber
pattern of seed yield at maturity was consistent with the growth
pattern seen at 20 days in the same experiment, except for plant
height.
Thus, there is a good indication that the within chamber
growth pattern at the juvenile stage persisted until maturity.
However, the within chamber growth pattern in chamber A4 in the
maturity study was different from the growth pattern seen in earlier
experiments with juvenile plants in the same chamber, except for
plant height.
The similarity in pattern with plant height only
cannot be explained logically.
has
sp~ay
i f needed.
It should be noted that chamber A4
misters to increase the relative humidity within chamber,
It was not known whether the misters were opened or
shut in either of the studies.
Also, a higher temperature setting
in the mature study may have caused a difference in the within
chamber growth pattern.
61
5
STUDIES IN THE WALK-IN 'B' CHAMBERS
5.1
5.1.1
Materials and Methods
Studies on Juvenile plants.
Four chambers, Bl, B2, B3 and B8 out of a possible ten
were available for the studies on juvenile plants in the 'B'
chambers.
The four chambers were studied, three at a time in any
one 20-day period (trial).
planned with four trials.
A balanced imcomplete block design was
However, in trial 2, atypical plant
growth was seen in all three chambers and the experiments were
discarded.
In trial 3, chambers B2 and B8 were run at a different
temperature, the 26/22 and Bl at 22/18 temperature.
This trial
was successfully carried out and the 22/18 temperature was retained
for the rest of the trials.
However, the balanced design was lost.
In trial 6, temperature control within the lamp housing in chamber
B3 failed and all plants were moved into chamber A15, set at 22/18
temperature for three consecutive days until the fault was remedied.
The plants were returned to chamber B3 three days before the
experiment was terminated.
Plants in this experiment were
considerably taller than plants in the other experiments with
abnormal plant growth resembling symptoms seen earlier in trial 2.
Data from this experiment in B3, trial 6 were taken and will be
reported but the results will be given separately when different.
Table 5.1.1 shows the chambers that occurred together
in each trial with the respective dates.
62
Table 5.1.1 Chamber and trial combination with respective dates for
studies with juvenile plants in the 'B' chambers.
Dates
Trial
March 14 ~ April 3, 1975
April 26 - May 16, 1975
May 17 - June 6, 1975
June 7 - June 27, 1975
June 28 - July 18, 1975
1
3
4
5
6
Chambers
B2,
Bl,
B2,
Bl,
Bl,
B3~ B8
B2 , B8*
B3, B8
B3, B8
B2, B3**
Note: * Experiments conducted at 26/22 temperature.
** Experiment where a faulty temperature control mechanism was
recorded.
All experiments in each trial were planted on the same
day and terminated at 20 days from sowing.
Each experiment had
eight trucks with sixteen plants to a truck.
A schematic diagram
showing the arrangement of the trucks with the row, column and truck
identity is shown in Figure 5.1.1.
The door is located between
columns 6 and 12, and was hinged on the left for chambers B2 and B8
and hinged on the right for chambers Bl and B3.
Preliminary light and temperature readings were taken
within chambers Bl, B2 and B3 in trial 6 at the center of all trucks
seven days after sowing.
These readings were taken above the leaf
canopy at a constant distance from the light barrier.
As a check
on temperature and light consistencies, readings were taken on
June 1, 1977 for chamber B2, without any uniformity studies.
5.1.2
Studies on mature plants.
The experiment on mature plants was conducted in chamber
B8 from February 21 to May 21, 1976, at 22/18 temperature with
thirty-two 25.4cm diameter pots.
One of two plants was chosen at
63
Row
Identity
2.44m
8
TRUCK
TRUCK
TRUCK
TRUCK
5
6
7
8
7
6
13
5
.....
4
N
N
TRUCK
TRUCK·
TRUCK
TRUCK
1
2
3
4
3
2
I(
1
2
3
4
5
6
)I
DOOR
I
7
8
9
1
I
10
11
12
13
14
15
16
Column Identity
Figure 5.1.1 Schematic diagram within a lB' chamber showing the
truck arrangement and the column, row and truck identities.
64
random from each pot at 20 days and measurements taken for five
variables similar to the juvenile studies.
The remaining plant was
allowed to grow until seed and pod formation and the experiment
terminated at 90 days.
The same measurements as in the 'A' chamber
maturity study were taken.
Light readings within chamber B8 were
taken for all trucks seven days after sowing as described above.
65
5.2
5.2.1
Results
Preliminary data analyses.
5.2.1.1
Mean, variance (within experiment), range and
coefficient of variation.
The statistics mean, variance (within
experiment), range and coefficient of variation (c.v.) were computed
for each experiment in all variables and are summarized in Appendix
4.
The results show that chamber B2 in trial 3 at the higher
temperature gave means (in original units) two to three times as
large as the averages of the experiments conducted at the lower
temperature.
The within experiment estimates of variance were not
significantly different for the two temperatures in two variables,
leaf area and petiole length and were approximately two times as
large for the higher temperature in the other variables.
In
chamber B8 at the higher temperature, the means were about 25
percent higher than the average of the experiments at the lower
temperature but the within experiment estimates of variance were
similar.
The means of the two experiments at the higher temperature
were different, with means in chamber B2 about two-thirds to two
times as large as the means in chamber B8.
The ranges within
experiments show the largest plants about two to three times the
size of the smallest plants.
Table 5.2.1 gives the means, variances
(within experiment) and coefficients of variation averaged over
twelve experiments in the 'B' chambers at the 22/18 temperature, by
variable.
66
Table 5.2.1 Means, variances (within experiment) and coefficients
of variation (c.v.) averaged over 12 experiments in the 'B'
chambers at the 22/18 temperature, by variab1e.*"
Variable
Mean
plant height, rom
leaf area, cm2
petiole length, rom
fresh weight, mgm
dry weight, mgm
Statistic
Variance c.v.
119
155
39
386
54
717
1,120
107
7,410
119
22
22
26
22
21
Note: * Chamber B3 in trial 6 was excluded.
5.2.1.2
Correlations among variables.
Simple correlations
among variables were computed for each experiment, with 126 degrees
of freedom.
All correlations were positive.
The correlation
coefficients were averaged for each pairwise combination of
variables over twelve experiments and the results shown in
Table 5.2.2. 1
The results showed a similar pattern of correlations
among variables as in the 'A' chambers.
The variables leaf area,
fresh weight and dry weight have very high correlations among each
other.
Plant height and petiole length have lower correlations
with the other variables.
Correlations in the experiment for
chamber B3 in trial 6 were much higher than correlations within
other experiments (correlations> 0.86) due in part to the higher
range of observed values in that experiment.
Correlations within
1
Correlations for experiments conducted at the 26/22 temperature and
correlations in experiment B3 in trial 6 with a faulty temperature
control were excluded.
67
Table 5.2.2 Within experiment correlations among variables in the
'B' chambers averaged over 12 experiments at 22/18 temperature.*
Variables
plant height
leaf area
petiole length
fresh weight
leaf
area
Variables
petiole fresh
length
weight
.59
.80
.83
.63
.95
.83
dry
weight
.64
.94
.82
.95
Note: * The two experiments at 26/22 temperature and experiment in
chamber B3 trial 6 with the faulty temperature control were excluded.
experiments at the higher temperature showed a wider range than
correlations within experiments at the lower temperature
(correlations of .34 to .96).
However, the pattern of correlations
within experiment among variables was similar to that at the lower
temperature.
5.2.2
Within experiment variation with 20 day old plants.
5.2.2.1
Position effects.
The variation within experiments
was further investigated to determine whether there were any
systematic positional differences within experiments.
Three within
chamber models were studied; models allowing for the column effects,
for the row effects or for the truck effects.
The proportions of
the total within experiment sum of squares attributable to the
indicated effects averaged over all experiments are given in
Table 5.2.3.
The results (Table 5.2.3) showed that truck effects
accounted for 37 to 55 percent of the total within experiment
68
Table 5.2.3 Coefficients of determination in percent, averaged
over all experiments in the 'B' chambers for column, row and
truck within experiment models with ranges shown in parentheses.
Model
Column(15 d. f.) Row{7 d.f.)
Truck{7 d.f.)
plant height
49.5
(33.9 - 64.1)
12.8
(5.3 - 22.0)
55.2
(44 •0 - 72.9)
leaf area
41.8
(19.1 - 59.0)
8.2
(4.0 - 13.7)
37.9
(13.6 - 56.4)
petiole length
50.2
(21.1 - 67.4)
12.3
(2.5 - 21.5)
50.8
(20.8 - 63.9)
fresh weight
43.9
(25.8 - 59.4)
8.5
(4.9 - 14.5)
39.8
(17.2 - 59.9)
dry weight
42.2
(24.0 - 55.6)
8.6
(4.3 - 12.4)
36.5
(18.2 - 55.6)
Variable
variation, the column effects for 42 to 50 percent and the row
effects for 8 to 13 percent.
The coefficients of determination
for the experiment in chamber B3 in trial 6 took the lower values
of the ranges in Table 5.2.3 except for variable plant height.
A
comparison of the truck, column and row models will be deferred
until section 5.2.2.6 pertaining to within experiment designs.
5.2.2.2
Characterization and description of pattern of growth.
Second to fifth degree response surface models were fitted to the
square root of all variables for each experiment.
The coefficients
of determination, R2 , averaged over all experiments, by variables,
is given in Table 5.2.4.
Examining the average R2 values indicate
that there was little gain beyond fitting the fourth degree response
69
Table 5.2.4 Coefficients of determination, R2 , averaged over all
experiments in the IB I chambers for the second to the fifth
degree response surface, by variable.
Degree of
response surface
2nd
3rd
4th
5th
No~e:
*
plant
height
leaf
area
54.8
60.0
70.3
73.2
42.1
46.0
51.9
54.4
Variables *
petiole fresh
weight
length
56.6
61.5
68.5
70.9
44.1
47.6
53.4
56.0
dry
weight
42.2
45.4
51.2
53.9
Square root transformation applied to all variables.
surface models in all variables. l
The within chamber growth patterns for each experiment
characterized by the variables plant height and dry weight are shown
in Figures 5.2.1 and 5.2.2, using the fourth degree response
surfaces.
The within experiment patterns for each chamber were very
similar over time (trials).
A closer examination of the contours
for dry weight (Figure 5.2.2), page 73, will indicate similarities
in patterns for chambers Bl and B3 and for chambers B2 and B8, with
the within chamber patterns in one group like mirror images of the
patterns in the other group.
It was noted in page 62 that. the doors
were hinged on different positions for the two groups of chambers
1
Counts made on the additional regression terms from a lower to the
next higher degree response surface model which are significant at
probability less than .01 show percentages of terms over all
experiments and variables which were significant from the second to
the third, third to fourth and fourth to fifth were 9, 17 and 2
respectively.
70
Figure 5.2.1 Contours of individual experiment, arranged by chambers,
of a fourth degree response surface 'for square root of plant height.
Explanatory note: Trial identification is shown on the top left
corner in each diagram.
Chambers B2 and B8 in trial 3 conducted
at 26/22 temperature are shown with an asterisk (*).
Chamber B3
in trial 6 with a defective temperature control is shown with two
asterisk (**).
Alternate contour intervals of 0.5 unit on the
square root scale are designated by letters arranged in ascending
order as follows:
7.5
<
E
<
8.0
18.5
<
P
<
19.0
The bold-face letters P and V on each diagram show locations
of peaks and valleys in the contour surface.
71
8 1
83
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6
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Figure 5.2.1 (continued).
72
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Figure 5.2.2 Contours of individual experiment, arranged by
chambers, of a fourth degree response surface for square root
of dry weight.
Explanatory note: Trial identification is shown on the lop left
corner in each diagram.
Chambers B2 and B8 in trial 3 conducted
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•
c
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76
and this may be one of several factors causing the mirror images.
Although there were differences in pattern between chambers, an
overall pattern for the 'B' chambers can be described.
This
pattern is most easily seen with a fourth degree response surface
model fitted to the square root of plant height and of dry weight
data, averaged over twelve experiments at the 22/18 temperature,
shown in Figure 5.2.3.
The general appearance is that of a hill,
with the peak located near the center of the chamber but displaced
slightly toward the door.
The plants near the side walls were
shorter and weighed less than those in the center.
In the scale
of the original units (taking predicted values from Figure 5.2.3)
plants were 67 percent taller and 75 percent heavier at the center
of the chamber than plants at the sides.
The within chamber pattern in chamber B8 at the higher
temperature was similar to the lower temperature.
In chamber B2,
the pattern was slightly different for dry weight, with a trough
located at the front left corner of the chamber.
For chamber B3
in trial 6 where a defect in temperature control occurred, the
general pattern was similar but with the peak located near the
door.
5.2.2.3
Consistency of pattern over experiments.
Visual
examination of the fitted contours within experiments showed a
consistency of the within experiment patterns for each chamber over
time (trial) and to a smaller degree, over chambers.
The consistency
of pattern over experiments is further tested by the analysis of
variance combining twelve experiments at the 22/18 temperature in a
77
Figure 5.2.3 Contours for 'B' chambers, averaged over twelve
experiments at the 22/18 temperature, of a fourth degree response
surface with the square root of (a) plant height and the square
root of (b) dry weight.
Explanatory note: Alternate contour intervals of 0.5 unit for
plant height data and 0.2 unit for dry weight data on the
square root scale are designated by letters arranged in ascending
order as follows:
(a)
plant height
(b)
7.5
<
E
<
8.0
11.5
<
I
<
12.0
dry weight
5.6 < C
<
5.8
7.6
<
7.8
<
H
The bold-face letters of P and V on each diagram show
locations of peaks and valleys in the contour surface.
78
Plant height
8.
I
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Figure 5.2.3 (continued).
14
16
79
split-plot design similar to model (4.2.1), page 33, for the 'A'
chambers. I
The analysis of variance is given in Table 5.2.5.
The
average position effects over all experiments, that is trucks, were
very large and highly significant, indicating a clearly identifiable
position effect common to all chambers and trials.
Of the
interactions with trucks, the chamber by truck interaction was
largest, and significant
(p
<
0.05) in all cases, indicating a lack
of consistency of the position effects from chamber to chamber.
In no case was the trial by truck interaction significant.
Consequently, the test for consistency of the within experiment
pattern over trials in the same chamber is made by comparing the
trial x truck and chamber x trial x truck error mean squares, with
the within chamber-trial-truck residual mean square in approximate
F-test.
The F-ratios, although small were significant
(p
<
0.05)
indicating a lack of consistency of the position effects from trial
to trial within a given chamber.
Thus, the growth pattern within
experiments was more consistent over trials than across chambers.
5.2.2.4
Light and temperature effects.
A set of eight
temperature and light readings, one per truck, for chambers BI, B2
and B3 and a set of eight light readings taken for chamber B8 is
used to determine whether the light and temperature within chambers
was associated with the pattern of plant growth within experiments.
Another set of temperature and light readings in chamber, B2 was
taken at a later date to check the consistency of temperature and
I
The twelve experiments combined excluded the experiment from chamber
B3 in trial 6 with faulty temperature control.
•
e
e
Table 5.2.5 Mean squares in the combined analyses of variance as a split-plit design with
12 experiments as the main plots and 8 trucks of 16 plants each within each experiment.*
Source of variation
Main plots (Experiment)
Chamber
Trial
Chamber-Trial Residual
Df
3
4
4
Sub plots
Truck
7
Chamber x Truck
21
Trial x Truck
28
Chamber x Trial x Truck
28
Within Chamber-TrialTruck Residual
1,440
Variables **
petiole fresh
length
weight
plant
height
leaf
area
106.42
295.21
20.24
33.14
478.03
34.81
14.16
95.45
16.94
68.67
1,273.85
68.90
15.80
101.41
7.33
137.99
7.99
1.26
1.28
137.40
6.07
3.27
2.11
64.02
3.05
0.82
0.55
333.50
12.69
8.72
6.45
37.74
1.63
0.51
0.52
0.65
1.24
0.32
2.83
0.36
dry
weight
Note: * Mean squares for main effects were adjusted for each other, two-factor interactions
adjusted for each other plus the main effects and three-factor interactions adjusted for all
effects.
** Square root transformation applied to all variables.
(Xl
o
81
light over a two year period.
These readings are summarized in
Table 5.2.6.
Table 5.2.6 Range of temperature and light readings within the 'B'
chambers, set at 22/18 temperature with the range for chamber B2
without uniformity studies in parentheses.
Chamber
Bl
B2
B3
B8
Temperature
degrees Celsius
Light ( x 10 3 lux)
As measured
As rated
23.8
24.0
(25.5
23.5
36
32
(38
37
48
-
26.8
26.0
30.5)
26.8
-
41
35
42)
40
53
39
43
43
52
Correlations, by position, with 6 degrees of freedom,
between light intensity readings across chambers are significant,
showing a consistency in light intensity pattern over chambers, with
correlations from .77 to .96.
A correlation of .87 with 6 degrees of freedom between
the light readings taken on two occasions in chamber B2 is significant,
indicating a consistency of light patterns over time.
A correlation
of .43 between temperature readings taken on two occasions within the
same chamber is not significant.
Correlations, by position, of
temperature readings across chambers are not significantly different
from zero, with values from -.30 to .51.
Thus, from these correlations,
there was no evidence of a consistency of temperature over chambers.
Correlations of light and temperature readings within chambers were not
significantly different from zero, with values from .22 to .42.
Correlations by truck position were made between the
measured variables from each experiment and the corresponding
82
temperature and light readings within chambers to determine the
proportio~
of the variation in plant growth attributable to temperature
and light intensity within chambers.
Only the truck to truck
variation could be studied as only one temperature and light reading
was taken per truck.
These correlations, averaged over all experiments,
are given in Table 5.2.7.
Table 5.2.7 Correlations, r, and range in parentheses, of light and
temperature readings with each variable, averaged over chambers and
trials.*
Variables
Temperature
readings
plant height
.72
(.52 - .86)
leaf area
.62
(.39 - .77)
petiole length
.67
(.38 - .81)
fresh weight
.61
(.42 - .77)
dry weight
.64
(.45 - .78)
Light
readings
.76
(.63 - .90)
.78
(.63 - .95)
.79
(.66 - .92)
.79
(.53 - .92)
.81
(.65 - .93)
Note: * Each correlation is estimated with 6 degrees of freedom.
H : p = 0 is rejected if Irl > .707 at 0.05 probability.
5.2.2.5
Use of prior information as a covariate.
Data from the
first trial in each of the four chambers, trial 1 for chambers Bl, B2
and B8 and trial 3 for chamber B3, provided the prior information
that is used as a covariate to adjust for positional differences
within chambers at subsequent trials.
The data were averaged by
position over the four experiments by variable, with the square root
transformation applied.
Fourth degree response surfaces were fitted
83
to these averages, the predicted values
YR, YA, Yp , Yrw
were obtained for each position in the five variables.
among the predicted values are very high (r
>
YDW
and
Correlations
.94), suggesting that
any of the predicted values would serve nearly equally well as
covariates in controlling the within experiment variation in later
trials.
The effectiveness of the predicted values as covariates
is measured in each of the eleven other experiments in terms of the
proportion of the total within experiment variation attributable to
the covariate.
On the average, anyone of the predicted values is
equally effective, reducing the within experiment variation in plant
height and petiole length by about 53 percent and by about 34 percent
of the variation in the other variables.
e.
A
The covariate,
YR ,
obtained from plant height data, is not more effective as any other
covariate in controlling the within experiment variation in plant
height.
The covariate,
YDW '
is chosen to determine its
effectiveness in controlling the within experiment variation in the
presence of blocking within chambers, using trucks or columns and
rows as blocks.
Five models for control of the within experiment
variation were compared:
(a) a model using only the covariate,
(b) a model using truck position effects,
A
YDW '
(c) a model using truck
A
position plus the covariate,
YDW '
(d) a model using both the column
A
and row effects and
(e) a model using column, row plus
YDW .
The
models were fitted to each of the eleven experiments (experiments in
84
which data were not used in computing
Ynw )'
and the proportion of the
total sum of squares accounted for by each model
computed.
The
average coefficients of determination for eleven experiments are given
in Table 5.2.8.
Table 5.2.8 Coefficients of determination in percent, for five models
of Y
nw ' truck, truck and t nw ' column and row, column, row and t nw
in the analyses of variance/covariance by experiment, averaged over
11 experiments.
plant
height
Model
A
Y
nw
Truck
Truck and Ynw
Column and Row
Column, Rowand Ynw
A
A
55.5
56.0
69.1
63.9
69.3
.
Variables
leaf petiole fresh
weight
area length
dry
weight
32.7
34.6
42.6
46.5
49.0
35.3
35.2
43.8
49.1
51.2
50.8
49.5
62.3
60.2
65.3
35.1
38.0
46.2
49.7
51.8
The results showed that the covariate was about as
effective on the average as blocking by trucks.
With blocking by
trucks the covariate accounted for an additional 13 percent of the
within experiment variation in plant height and petiole length and
8 percent in the other variables.
With a two-way blocking within
chambers by columns and rows, the covariate accounted for an
additional 5 percent of the within experiment variation in plant
height and petiole length and 2 percent in the other variables.
The comparison between the truck effects and the column and row
effects models will be deferred until the next section pertaining to
within chamber designs.
85
5.2.2.6
Within chamber designs.
Treatments applied within
chambers may be arranged in several ways.
The relative
effic~encies
of three within chamber designs, the completely random, the randomized
block and the two-way blocking are compared using the uniformity trial
data from twelve experiments at 22/18 temperature.
Three blocking
systems for the randomized block design are compared, with trucks as
blocks, rows as blocks and columns as blocks.
The computations for
the-within chamber error mean square for each design followed that
given in Table 4.2.10, page 42, for the 'A' chambers.
In addition
to the full chamber in each experiment, estimates of the variance
within four trucks per experiment were also computed.
Three groups
of four trucks were compared:
(b)
4 trucks in the center of each chamber, corresponding to
trucks 2, 3, 6 and 7 in a 2 x 2 arrangement,
(c)
4 trucks located on the left of each chamber, corresponding to
trucks ·1, 2, 5 and 6 in a 2 x 2 arrangement, and
(d)
4 trucks located in the rear of each chamber, corresponding to
trucks 5, 6, 7 and 8 in a 1 x 4 arrangement. l
The results are given in Table 5.2.9.
The results showed that blocking by trucks or by columns
was about equally effective, with reductions of 36 to 54 percent of
the total within experiment variation in the full chambers.
Blocking by rows in the full chambers did not give a significant
1
For truck identity, refer to Figure 5.1.1, page 63.
86
Table 5.2.9 Estimates of the within experiment variance from 12
experiments for the completely random (CRD) , randomized block (RCB)
and two-way blocking (LS) designs with (a) the full chamber, (b) 4
trucks in center of chambers, (c) 4 trucks at the left of chambers
and (d) 4 trucks at the rear of chambers, at 22/18 temperature.
Variables *
petiole fresh
length
weight
plant
height
leaf
area
(a) Full chamber (8 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) row as blocks
LS
1.41
0.65
0.77
1.31
0.61
1.99
1.24
1.24
1.94
1.13
0.67
0.32
0.35
0.63
0.29
4.69
2.83
2.77
4.54
2.47
0.56
0.36
0.35
0.54
0.31
(b) Center (4 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) rows as blocks
LS
1.0.7
0.83
1.07
0.59
0.52
1.44
1.28
1.39
1.17
1.08
0.48
0.39
0.47
0.30
0.26
3.46
3.06
3.31
2.81
2.54
0.43
0.40
0.42
0.35
0.33
(c) Left-side (4 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) rows as blocks
LS
1.40
0.62
0.75
1. 29
0.53
1. 78
1.15
1.17
1. 74
1.03
0.62
0.30
0.34
0.57
0.24
4.37
2.73
2.76
4.22
2.36
0.52
0.35
0.35
0.50
0.30
(d) Rear (4 trucks)
CRD
RCB 1) trucks as blocks
2) columns as blocks
3) rows as blocks
LS
1.28
0.67
0.65
1.22
0.53
1. 76
1.19
1.14
1.68
0.98
0.59
0.35
0.33
0.55
0.26
4.29
2.77
2.65
4.06
2.22
0.50
0.34
0.34
0.47
0.29
Within chamber design
Note:
*
Square root transformation applied to all variables.
dry
weight
87
reduction in the total within experiment variation.
Similar
reductions in the within experiment error variance are obtained for
the four trucks at the rear and four trucks at the side with trucks
or columns as
blocks~
Blocking in rows in the center four trucks
accounted for 42 percent of the variation in plant height and
petiole length and 19 percent of the variation in the other
variables.
The two-way blocking by columns and by rows is best
compared to either blocking on trucks or columns, with an additional
gains of 6 to 26 percent over the randomized block design.
The
plants within four trucks at the left-side have higher total
variability than plants at the rear and center.
However, with a
two-way blocking, the residual error variances are similar in all
cases.
5.2.3
Between experiment variation with 20 day old plants.
5.2.3.1
Between chamber designs.
Three between chamber
experimental designs (whole plot designs) are compared using uniformity
data from twelve experiments, at the 22/18 temperature. l
The three
designs are the completely random design (CRD) , the randomized block
design (RCB) with trials or chambers as blocks and the two-way
blocking designs by trials and chambers (LS).
The error mean squares
for each design were computed following the procedures outlined in
Table 4.2.12, page 45, for the 'A' chambers.
In addition to the full
chamber of eight trucks within each experiment, estimates were also
1
The two experiments at 26/22 temperature and the experiment with a
faulty temperature control were excluded.
88
obtained for three groups of four trucks similarly positioned for
each experiment.
These groups, referred to as the center four
trucks, left-side four trucks and rear four trucks, would be the
likely arrangements within chambers in experiments not occupying the
full chamber.
The positions occupied by each group of four trucks
were given in the previous section, page 85.
The results are given
in Table 5.2.10.
The results showed that blocking by trials gave reductions
of the experimental error variance for the plant growth
measurements of 63 to 87 percent in the full chambers.
chambers was not effective.
•
variation across chambers was
Blocking by
This indicates that the plant to plant
less homogeneous than the variation
within a chamber at different times.
With the three groups of four
trucks taken from the same position within chambers the effectiveness
of blocking by trials was similar with gains of 56 to 94 percent over
the CRD.
The two-way blocking design for whole plots was most
efficient only for plant height, with gains of about 27 percent over
the RCB.
5.2.3.2
Relative sizes of the between to the within experiment
error variances.
Estimates of the between and the within experiment
error variances were given in the preceeding two sections, and these
estimates differred depending on the within and between experiment
designs.
Obviously,
ranges of relative sizes exist with combinations
of any between and any within experiment designs.
For the purpose
89
Table 5.2.10 Estimates of the experimental error variance
whole plots from 12 experiments in the completely random
randomized block (RCB) and two-way blocking (LS) designs
full chamber and for 4 trucks from the center, left-side
of the chamber, at 22/18 temperature.
for
(CRD) ,
for the
and rear
Variables*
petiole fresh
length weight
plant
height
leaf
area
(a) Full chamber (8 trucks)
CRD
RCB 1) trials as blocks
2) chambers as blocks
LS
153.68
57.18
157.73
20.20
204.31
34.09
256.42
34.81
45.62
15.75
56.20
16.94
525.80
68.80
671.37
68.90
43.33
10.96
54.37
7.33
(b) Center (4 trucks)
CRD
RCB 1) trials as blocks
2) chambers as blocks
LS
95.38
39.61
89.68
11.27
100.51
18.86
121. 27
18.27
27.63
9.22
32.19
6.90
295.61
49.61
366.91
47.26
23.23
7.53
28.49
6.20
(c) Left-side (4 trucks)
CRD
RCB 1) trials as blocks
2) chambers as blocks
LS
80.33
32.03
83.76
12.75
102.57
16.83
129.70
15.18
26.21
10.12
31.19
9.47
259.83
16.22
343.69
16.77
19.50
3.90
26.38
3.02
(d) Rear (4 trucks)
CRD
RCB 1) trials as blocks
2) chambers as blocks
LS
76.52
33.99
71.62
11.94
105.95
23.72
130.49
21.60
23.59
11.08
27.76
9.99
266.76
47.17
334.34
40.05
21.68
6.01
26.78
3.54
Between chamber design
dry
weight
Note: * Square root transformation applied to all variables.
of this section, the between experiment design with blocking by trials
is chosen as it is more convenient to carry out than blocking by
chambers.
Two within experiment designs are selected, the randomized
block design with trucks as blocks (RCB) and the completely random
e'
90
design (CRD), to give a comparison.
The relative sizes of the
between to the within experiment error mean squares show by what
factors the two error variances differ.
The results (Table 5.2.11) showed that a within
experiment error mean square erroneously used for testing the between
chamber treatment factors would be an underestimate by factors of 15
to 88 in the full chambers.
Table 5.2.11 Relative sizes of the between to the within experiment
error mean squares assuming the randomized block between experiment
design with trials as blocks and two within experiment designs, the
completely random (CRD) and randomized block (RCB) with trucks as
blocks for the full chambers and groups of 4 trucks located at the
center, left-side and rear of each chamber.
Variables *
petiole fresh
weight
length
plant
height
leaf
area
(a) Full chamber (8 trucks)
CRD
RCB
40.6
88.0
17.1
27.5
23.5
49.2
14.7
24.3
19.6
30.4
(b) Center (4 trucks)
CRD
RCB
37.0
47.7
13.1
14.7
19.2
23.6
14.3
16.2
17.5
18.8
(c) Left-side (4 trucks)
CRD
RCB
22.9
51.7
9.5
14.6
16.3
33.7
3.7
5.9
7.5
11.1
(d) Rear (4 trucks)
CRD
RCB
26.6
50.7
13.5
19.9
18.8
21. 7
11.0
17.0
12.0
17.7
Within chamber design
Note:
*
dry
weight
.
Square root transformation app11ed
to all'
var1abl es.
As the relative size of the two error terms is dependent
on the number of pots within experiments, the relative size for four
91
trucks will be approximately one-half that for the full chambers. The
relative sizes for groups of four trucks were from 4 to 52.
5.2.4
Experiment with mature plants.
5.2.4.1
Within chamber variation.
The experiment in chamber B8
with mature plants had thirty-two pots of 25.4cm with the chamber
selected at random from chambers B1, B2, B3 and B8.
was conducted at the 22/18 temperature.
This experiment
Two sets of measurements
were taken, one at 20 days and the other at 90 days with different
plants and the mean, variance (among plants), range and coefficient
of variation for these measurements are shown in Table 5.2.12.
Table 5.2.12 Means, variances (among plants), ranges and coefficients
of variation (c.v.) for measurements taken with 20 and 90 day old
plants in the same experiment in chamber B8 at 22/18 temperature.
Variables
20 day old plants
plant height ~mm)
leaf area (cm )
petiole length (mm)
fresh weight (mgm)
dry weight (mgm)
90 day old plants
pod number
seed weight (gm)
shoot weight (gm)
root weight (gm)
whole plant weight (gm)
Mean
Variance
Range
c.v.
111
148
34
353
54
462
1,066
112
8,911
166
75'--' 17296 - 224
18 - 59
218 - 577
35 - 82
19
22
31
27
24
35
12
23
4.5
46
126
29
53
3.5
120
12 - 53
3.5 - 26.7
8.8 - 38.9
1.8- 9.7
24 - 69
32
44
31
41
24
The means, variances, range and coefficient of variation
with '20 day' variables are similar to those for the juvenile studies
(Table 5.2.1, page 66).
Coefficients of variation with '90 day'
variables ranged from 24 to 44 percent.
92
5.2.4.2
Correlations between measurements at two growth stages.
Correlations, by position (or pot), between variables taken at the two
growth stages are shown in Table 5.2.13.
All '20 day' variables
showed significant positive correlations with the grain yield
variables, pod number and seed weight, from .37 to .69.
Of the
remaining correlations with vegetative yield, only one reached
significance level, petiole length with whole plant weight.
Table 5.2.13 Correlations, by position, with 30 degrees of freedom,
between variables taken with 20 day old plants and variables taken
with 90 day old plants in the same experiment in chamber B8 at
22/18 temperature.
Variables
(90 day old plants)
pod number
seed weight
shoot weight
root weight
whole plant weight
Note:
Irl
>
.34
Variables (20 day old plants)
plant
leaf petiole fresh
dry
height
area length
weight weight
.573
.669
-.256
-.156
.228
.370
.452
-.134
-.142
.174
.595
.688
-.113
-.085
.355
.397
.475
-.143
-.089
.192
.398
.482
-.198
-.152
.146
is significant at the 0.05 probability level.
Fourth degree response surfaces models with 12 degrees of
freedom were fitted to square root of plant height, of dry weight and
of seed weight data. l
Contours of the response surfaces show the
pattern of variation characterized by these variables at the 20 and
90 day stage (Figure 5.2.4).
The peaks for all variables were
located near the center of the chamber with a slight displacement
1
The quartic term in rows (R4 ) and interaction CR3 were excluded from
the 'full' model as there are only four rows.
93
Figure 5.2.4 Contours for chamber B8 in the maturity study of a
partial fourth degree response surface, with square root of
(a) plant height at 20 days (b) dry weight at 20 days and
(c) dry weight of seed.
Explanatory note: Alternate contour intervals of 0.5, 0.2 and
0.2 for plant height, dry weight and dry weight of seed data,
respectively, in the square root scale are designated by
letters arranged in ascending order as follows:
(a) plant height
8.5
<
F
<
9.0
11.5
<
I
<
12.0
(b) dry weight
(c) dry weight of seed
6.0 <
D < 6.2
1.8
< B
<
2.0
8.4
J
8.6
4.6
<
I
<
4.8
<
<
The bold-face letters P and V on each diagram show locations of
peaks and valleys in each contour surface.
Plant Height at 20 days
94
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+
S
95
toward the right side of the chamber for dry weight and a slight
displacement toward the rear of the chamber for seed weight.
A
valley for dry weight and plant height was located in the rear left
corner of the chamber.
Otherwise the general appearance in all
variables was similar.
5.2.4.3
Correlations between variables in maturity study with
variables in juvenile studies in chamber B8.
The experiment in
chamber B8 with plants taken to maturity was conducted seven months
after the experiments with juvenile plants were completed.
During
that time chamber B8 was used by other research workers. imposing
different temperatures. day length and other conditions.
Correlations
between the variables for the two uniformity studies would indicate
whether the pattern of variation within chamber B8 had changed.
The data from twelve experiments from the juvenile
studies in the 'B' chambers were used. l
A fourth degree response
surface model using row and column positions within experiment as the
independent variables. was fitted to data in all five variables
averaged over twelve experiments.
Adjacent sets of four predicted
values are averaged so that there is a 1 : 1 correspondence. by
position. with data from the maturity study.
The predicted values
1
The experiments were conducted at the 22/18 temperature and
It
excluded was the experiment with the faulty temperature control.
was decided that data from all twelve experiments be used rather than
data from only chamber B8 so that the results would have a wider
inference.
96
for plant height, leaf area, petiole length, fresh weight and dry
weight, referred to as
YH, YA, Yp , YFW
Ynw
and
by position, with variables in the maturity study.
A
with 30 degrees of freedom, of
YH,
A
YA,
A
are correlated
Correlations,
A
Yp '
YFW and
Ynw from
the juvenile studies and all '20 day' variables were significant,
with values from .51 to .79.
Correlations of the same predicted
values with variables seed weight and pod weight were also
significant, with values from .56 to .70.
Correlations with the
remaining variables shoot weight, root weight and whole plant
weight were not significant, with values from -.33 to .18.
These results indicate that the within chamber growth
pattern was consistent in the two studies for the variables
·e
mentioned above.
97
5.3
Discussion and Conclusions
Plant to plant variation within the 'B' chambers with
20 day old soybeans at 22/18 temperature was sizeable with
coefficients of variation on an individual plant basis of 21 to 26
percent.
The variation was clearly patterned, showing a high
degree of consistency over all chambers and trials.
Plants were
taller at the center of each chamber with the plants in the center
generally about 67 percent taller and 75 percent heavier than those
at the sides.
The general pattern of growth within a chamber could
be described as a hill, with a peak located near the center of the
chamber but displaced slightly towards the door.
General predicted
fourth degree response surfaces with data averaged over twelve
experiments for the five measured variables were very similar, with
correlations among predicted values greater than .95.
In the two
experiments at the higher temperature (26/22) the within chamber
growth patterns were similar to those at the lower temperature.
There appeared to be some indication from the
experiments at the higher temperature that the means (of experiments)
for all variables in the same trial did not increase by a similar
proportion as the temperature was raised.
These increases were
about 30 percent for experiment in chamber B8 and 150 percent for
experiment in chamber B2.
This seem to indicate that the chamber
to chamber variation might be higher at the 26/22 temperature than
generally observed at the 22/18 temperature.
98
The gains in efficiency by blocking within chambers was
very
Blocking within chambers by columns, with eight plants
cl~ar.
per block and by trucks, with sixteen plants per block was about
equal in efficiency, accounting for about 45 percent of the within
experiment variation.
Two-way blocking by rows plus columns
compared to blocking by columns alone gave an additional improvement
of about 18 percent for plant height and petiole length and about
10 percent for the other variables.
In experiments occupying one-half of the chamber, a twoway blocking design is appropriate
for trucks positioned at the
sides, center or rear and would account for about 25 to 49 percent
of the variation among plants within four trucks.
Both temperature and light readings within chambers were
associated with all plant growth measurements accounting for 43 and
62 percent of the truck to truck variation in growth measurements,
respectively.
The pattern of light readings within chambers was
consistent over chambers.
Temperature patterns within chambers
were not consistent over chambers and over time.
It is useful to
monitor the light and temperature readings within all 'B' chambers
over a period of time and under a range of operating conditions.
Patterns for light readings over time and across chambers can be
tested for consistency.
If consistent, a common within chamber
light pattern for the 'B' chambers may be useful as a covariate.
The use of temperature readings within chambers can similarly be
explored.
99
An experimenter should be made aware the the temperature
monitored for a chamber is made under a shielded aspirated housing
at one corner of the chamber.
Temperature readings taken at
several positions within a chamber are higher than the nominal
temperatures set for that chamber.
The light readings as rated for
a chamber is taken at the center of each chamber.
Subsequent light
readings measured at different positions within a chamber are lower
than what is rated.
The use of prior information of plant growth data as a
covariate would reduce the total within experiment variation by
one-half for variables plant height and petiole length and by
one-third for the other variables.
An additional 8 to 13 percent
would be gained in the randomized block design model with a
covariate.
Prior information should be regularly updated to
account for the periodic changes of temperature, light and other
factors within chambers.
It would also be desirable to use other
plant species for confirmation of the within chamber growth pattern.
With respect to designing experiments where treatments
are such that chambers must be used as experimental units, several
options are available to the experimenter.
One is that he will be
given one chamber for a sufficiently long period to compare several
treatments.
A chamber will be his experimental unit and the period
of time long enough to run all treatments, is the block.
Alternatively, several chambers may be allocated, each for a short
period, where he assigns his treatments to each chamber.
Here,
blocking will be done across chambers at a given period.
In both
100
situations, it is assumed that the experimenter will replicate his
treatments in chambers and over time.
The results of the uniformity
studies with soybeans showed that the between trials (or runs)
component of variance was more important than the between chambers
component so that blocking on trials would be more effective in
reducing the between chamber variance.
Thus, a between chamber
experimental design with trials (time) as blocks is more efficient
than one with chambers as blocks.
In experiments occupying one-half
of a chamber with trucks similarly positioned in all chambers, the
relative efficiencies of the two blocking systems are about the same
as for the full chambers.
A two-way blocking design using trials and chambers as
blocks was found to be most efficient for variables plant height and
dry weight.
However, this would mean that the experimenter uses
the same chambers in all trials, with re-randomization of his
chamber factors with similarities to a latin square design.
This
may not be flexible and will provide fewer degrees of freedom for
the error than would the randomized block design.
Any within experiment estimate of error used in place of
a proper between chamber estimate to test between chamber factors is
much too small.
The relative sizes of the two error variances show
that for experiments occupying the full chamber, this factor varies
from 15 to 88.
In the experiment in one 'B' chamber where the plants
were allowed to develop to maturity, correlations between measurements
at 20 days with seed yield and pod number were significantly different
101
from zero, with values from .37 to .48 for variables leaf area, fresh
weight and dry weight with seed yield and pod number and from .57 to
.69 for plant height and petiole length with seed yield and pod
number.
It is emphasized that these correlations were between
different plants from nearly the same position in a pot, measured at
different times and on different variables.
Consequently, the only
contributing factor making these correlations different from zero is
the 'common environment the two plants shared.
There is no plant
developmental correlation included as there would be if the
variables, say, pod number and plant height were measured on the same
plant.
Thus, while the correlations were not unduly large, in the
absolute sense there is a clear indication of a persistent position
effect from juvenile to mature stage.
Correlations between '20 day' variables and shoot weight
and root weight were not significantly different from zero.
The
explanation may be that at 90 days almost all of the synthesized food
is directed to grain and fruit development and thus the vegetative
production, i.e. shoot and root production would not be reflective
of earlier vegetative growth at 20 days.
The within chamber patterns
for seed weight and pod number were consistent with the pattern seen
in the juvenile studies.
The within experiment variation in plant height and dry
weight for the maturity study was about similar to the variation in
these two variables for the juvenile study.
Thus the difference in
pot size used for the two studies does not affect the within
experiment variation in these variables.
102
There is some evidence that the within chamber patterns in
the 'B' chambers (as well as the 'A' chambers) can be divided into
two groups, one group being a mirror image of the other.
This
difference in pattern may be due in part to the way the chamber is
constructed, like for example, the position of the hinges on the
door.
It is useful to accumulate data of the 'conformity' of the
chambers with regard to plant response so that appropriate
experimental designs can be planned to take into account this
difference.
A random choice of chambers of one group of similar
conformity as a 'block' in a randomized block design will be better
than a random choice of chambers from both groups in a block.
103
6. STUDIES IN THE REACH-IN 'c'
6.1
CH~~ERS
Materials and Methods
Five chambers, C3, C8, C9, C12 and CIS out of a possible
twenty were available for this study.
All five chambers were
studied in each of four trials for a total of twenty experiments.
Two day/night temperature levels were used, a high temperature of
26/22 degrees Celsius and low temperature of 22/18 degrees Celsius.
For the whole study, each chamber was used twice at each temperature
level.
Table 6.1.1 shows the randomization of temperature levels
for each chamber in the four trials, with dates of trials.
Table 6.1.1 Chamber and trial combination at two temperature levels,
the 26/22 and 22/18 for study in the 'c' chambers.
Dates
Trial
May 31 to June 20, 1975
June 21 to July 11, 1975
July 12 to August 1, 1975
August 12 to August 22, 1975
1
2
3
4
Temperature. degrees Celsius
26/22
22/18
C3,
C9,
C3,
C8,
C8
C12, CIS
C12
C9, CIS
C9,
C3,
C8,
C3,
C12, CIS
C8
C9, CIS
C12
All experiments were terminated after 20 days, with measurements taken
on the same five variables as in the 'A' and 'B' juvenile studies.
The arrangement of plants within each chamber is shown in
Figure 6.1.1, with the rack, column and row identities.
The maximum
number of plants within each chamber is 60 if the four-plant racks
are used.
Although it is possible to have more plants without the
use of racks, they greatly facilitate handling of the potted plants.
104
Row
Identity
1. 22m
6
RACK
11
RACK
13
RACK
12
RACK
14
RACK
15
5
EMPTY SPACE
4
s
RACK
A
0 '\
6
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RACK
7
8
RACK
RACK
10
9
3
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--
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2
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1
3
1
2
3
4
5
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~I
D~OR
't
I
RACK
RACK
4
RACK
6
1
7
8
9
10
Column Identity
A
=
Aspirator
Figure 6.1.1 Schematic diagram within a 'c' chamber showing the
rack arrangement and the column, row and rack identities.
105
As a result, there was a gap of about llcm between rows 2 and 3 and
between rows 4 and 5 and a gap of 17cm between the plants and the
left side wall where the aspirator was located.
The door is located
between columns 3 and 7.
In trial 3, defective temperature controls were reported
on July 16, in chambers C8, C12 and C15.
The defect was due to a
delay of up to fifteen minutes before the temperature changed from
the high temperature to the low temperature.
The plants remained
in the respective chambers until the fault was checked and corrected
in three days.l
In trial 4, chamber C15 had a similar defect on August 20,
and the plants were moved to chamber C14, with the same temperature
setting.
The fault was corrected after one day and the plants
returned to C15.
Light readings were taken above the leaf canopy seven days
after sowing for all chambers in trial 4.
One reading was taken in
the center of each of nine racks, labeled 1, 3, 5, 6, 8, 10, 11, 13
and 15.
1
The reason for not moving the plants out of the defective chambers
was that no other chamber was available.
106
6.2
6.2.1
Results
Preliminary data analyses
6.2.1.1
Mean, variance (within experiment), range and coefficient
of variation.
The means, variances (within experiment), ranges and
coefficients of variation for each chamber-trial combination are
The ranges, on the average,
summarized by temperature in Appendix 5.
showed that the differences in plant size of the largest plant within
experiments were as large as three times the size of the smallest
plant at the high temperature level and about twice as large as the
smallest plant at the low temperature level.
The means, variances
(within experiment) and coefficients of variation averaged over ten
experiments for each temperature level are shown in Table 6.2.1.
Table 6.2.1 Means, variances (within experiment) and coefficients of
variation (c.v.) averaged over 10 experiments in the 'e' chambers
for each temperature level, by variable.
Temperature
High
Low
mean
variance
c. v.
plant height (mm)
leaf area (cm2 )
petiole length (mm)
fresh weight (mgm)
dry weight (mgm)
295
257
76
565
75
1,636
992
69
5,467
96
14
12
11
plant height ~mm)
leaf area (em )
petiole length (mm)
fresh weight (mgm)
dry weight (mgm)
99
110
24
254
37
283
515
21
2,155
38
17
20
19
18
17
Variable
13
13
The results show that the means at the high temperature
level for variables plant height and petiole length are about three
times those at the low temperature level.
For the remaining
107
variables, the multiples are about two.
There is about a six-fold
difference . in the within experiment error variance between the two
temperature levels for plant height and a two to three-fold difference
in variance for the remaining variables.
The coefficients of variation were lower for the high
temperature level than the low temperature in all variables.
This
showed that a three-fold increase in the mean plant height between the
two temperature levels resulted in less than a three-fold increase in
the within experiment error variance.
6.2.1.2
Correlations among variables.
Correlations among
variables within each experiment averaged over ten experiments for
each temperature level are shown in Table 6.2.2.
These correlations
are smaller at the higher temperature level than the lower temperature
level but are similar regarding the pattern of relationship among
variables.
Grouping of variables from their correlations would place
the variables leaf area, fresh weight and dry weight
±rt~ne
set with
plant height and petiole length in different sets.
The correlation
pattern is similar to that for the 'A' and 'B' chambers.
Table 6.2.2 Within experiment correlations among variables in the
'c' chambers, averaged over 10 experiments for the two temperature
levels. *
Variables
plant
height
plant height
leaf area
petiole length
fresh weight
dry weight
leaf
area
.46
.51
.69
.55
.50
.75
.93
.94
Variables
petiole fresh
length
weight
.42
.47
.74
.71
.36
.81
.55
dry
weight
.32
.74
.44
.86
.94
Note: * The upper and lower triangular matrices refer to the
correlations at the high and the low temperatures respectively.
108
6.2.2
Within experiment variation with 20 day old plants
6.2.2.1
Position effects.
The variation within experiment was
further investigated to determine whether there were any systematic
positional differences within experiments.
Three within chamber
models were studied; models allowing for the column effects t for the
row effects and for the truck effects.
The proportion of the total
within experiment variation attributable to each of the effects
averaged over all experiments is shown in Table 6.2.3. 1
Table 6.2.3 Coefficients of determination averaged over 20
experiments t by variable t for the column, row and rack within
experiment models, with the ranges in parentheses.
Variables
Column (9 d. f.)
Model
Row(5 d. f.)
Rack(4 d.f.)
plant height
36.8
(15.6 - 63.6)
16.8
(5.7 - 41.3)
52.8
(33.2 - 74.2)
leaf area
34.7
(14.1 - 57.9)
11.9
(4.1 - 24.4)
48.1
(27.6 - 69.3)
34.7
( 9.4 - 54.1)
12.9
(3.7 - 39.0)
51.8
(31.0 - 72.8)
28.0
(11. 6 - 47. 7)
15.2
(3.3 - 32.0)
46.6
(23.5 - 68.4)
27.2
13.1
(3.4 - 27.2)
41.6
(27.8 - 61.9)
petiole length
fresh weight
dry weight
( 9.5 - 50.0)
The results showed that the within experiment variation
attributable to each of the effects is 27 to 37 percent for the
column effects t 12 to 17 percent for the row effects and 42 to 53
1
The coefficients of determination t R2 t are not different for the
two temperature levels.
ConsequentlYt the R2(s) are shown here
averaged over all experiments.
109
percent for the rack effects.
A comparison of these three models
will be differred until section 6.2.2.6 pertaining to within chamber
designs.
6.2.2.2
Characterization and description of pattern of growth.
Response surface models from the second to the fifth degree with
columns and rows as independent variables are fitted to the square
root of data for all variables, by experiment. l
The proportion of
the total within experiment variation attributable 'to each model,
the coefficient of determination, averaged over all experiments, by
variable, are shown in Table 6.2.4.
Table 6.2.4 Coefficients of determination averaged over all experiments
in the 'c' chambers for the second to the fifth degree response
surface, by variable.
Degree of
response surface
2nd
3rd
4th
5th
Note:
*
plant
height
leaf
area
43.8
55.7
64.3
68.7
32.4
37.6
48.3
57.0
Variables *
petiole fresh
length
weight
36.4
43.7
52.7
60.5
29.2
34.7
45.7
54.8
dry
weight
25.6
30.7
41.8
51.1
Square root transformation applied to all variables.
The results show differences in the additional contribution
to the sum of squares for the various models for variable plant height
I
Rows were assigned values from I to 8, to take into account the two
gaps between rows 2 and 3 and rows 4 and 5, which could be occupied
with a row of plants at each gap.
110
and the remaining variables.
11.9
pe~cent
With plant height, there was a gain of
from the second to the third degree, a gain of 8.6
percent from the third to fourth degree and a gain of 4.4 percent from
the fourth to the fifth degree, showing a continued decline at every
stage.
In the other variables, the gain in information from the
second to the third degree was about 5.8 percent, the gain from the
third to the fourth degree was about 10.5 percent and the gain from the
fourth to the fifth degree was about 8.7 percent.
This would indicate
that a fourth degree response surface model may be sufficient in the
case of plant height data while a fifth degree response surface model
may be better than a fourth for the remaining variables.
As there
were only six rows in each experiment, a fifth degree model would
give complete accountability of the row
effects~
Thus, fourth degree
response surface models are used to characterize the within experiment
patterns in the
'c'
chambers to avoid fitting models giving complete
accountability of the row effects. l
Variables plant height and dry
weight are chosen to illustrate the within experiment growth patterns
in the
'c' chambers (Figures 6.2.1 and 6.2.2 respectively).
The fitted contours within each of the figures are
arranged by temperature and chambers for ease of comparison across
chambers and temperatures.
Although there seems to be a lack of
any consistent pattern in all experiments, a closer examination of
1
Counts were also made on the additional terms in the regression model
from a lower to the next higher degree response surface model over all
variables and experiments which were significantly different from zero
at probability less than .05.
The percentage of terms which were
significant from the second to the third, the third to the fourth and
the fourth to the fifth were 13, 14 and 7 respectively.
III
Figure 6.2.1 Contours of individual experiment, arranged by
temperature and chamber, of a fourth degree response surface with
the square root of plant height.
Explanatory note: Trial identification is shown on the top left
corner in each diagram.
Alternate contour intervals of 0.5
unit on the square root scale, are designated by letters arranged
in ascending order as follows:
3.5
<
A
<
4.0
19.5
<
Q
<
20.0
The bold-face letters P and V on each diagram show locations
of peaks and valleys in the contour surface.
112
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temperature and chamber, of a fourth degree response surface with
the square root of dry weight.
Explanatory note: Trial identification is shown on the. top left
corner in each diagram.
Alternate contour intervals of 0.2 unit
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ascending order as follows:
4.8
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116
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Figure 6.2.2 (continued) •
10
119
the diagrams in Figure 6.2.1 shows a clear and fairly strong tendency
for the spread of the taller plants towards and across the rear of the
chamber in most experiments with the low spot usually near the door or
across the right hand corner near the door.
For dry weight (Figure
6.2.2), the pattern of consistency was not as clear as for plant height.
However, the low spots tend to be near the right side of the door.
The location of the peaks within chambers varied but seems to be at the
left side of the chamber and towards the rear.
With some exceptions
in C9 in trial 2, C8 in trial 3 and C3 in trials I and 4, there was
some recognizable consistency between the two variables.
There was some concern that the response surface models may
be overly refined, with only sixty observations from each experiment,
with the peaks (or valleys) in the fitted surfaces being
at least in some cases, by a few plants.
dete~mined,
Plots of the observations
for the two variables show recognizable consistencies between the
observed and fitted data.
Thus, the response surface models are not
overly refined.
6.2.2.3
Consistency of pattern of growth.
section the growth patterns for experiments in the
In the preceeding
'c'
chambers with
plant height data from illustrations, show a fair degree of consistency
over all experiments at the two temperature levels.
In this section
the consistency of pattern is further examined, firstly, by computing
correlations by position over all possible pairwise combinations of
experiments by variable and, secondly, by the analyses of variance to
test the average position effects, as determined by racks, and the
interaction effects of racks by experiment.
The correlations are
120
divided into groups between experiments within each temperature
levels (two groups) and between experiments in different temperature
levels.
The results show that for all three groups, only the
correlations for plant height are consistently positive and significant
in most cases.
Table 6.2.5 summarizes the range of correlations for
plant height and for the other variables, as a group, over all pairwise
combinations of trials and all pairwise
combinations of chambers.
With the exception of the correlations of experiments in chamber C15
in trials 1 and 3, experiments in chamber C12 in trial land
experiments in chamber C3 in trials 1 and 2, the positive correlations
varied from .11 to .74, most are significantly different from zero,
indicating some similarity in growth pattern for plant height over all
experiments.
Comparisons of the diagonal elements (intrachamber
correlations) with the off-diagonal elements (interchamber correlations)
~n
Table 6.2.5 show that the correlations between trials within
chambers for variable plant height are similar to the correlations
between trials across chambers in the same variable, indicating that
the interchamber correlations are approximately of the same degree as
the intrachamber correlations.
For the remaining variables, the lower
ranges of correlations are almost always negative, although the average
correlations are positive.
These positive average correlations may
suggest some consistency in the within experiment patterns for these
remaining variables over all experiments but of a lower degree of
consistency than for plant height.
Table 6.2.5 Range of correlations, by position over all pairwise combinations of trials for each
chamber and each pairwise combination of chambers for variable plant height with the
correlations for the remaining variables in parentheses.*
Chambers
C3
C8
C9
C3
C8
Chambers
C9
.33 to .57
(-.29 to .29)
.21 to .57
(-.18 to .48)
.22 to .55
(-.24 to .40)
.26 to .60
(-.19 to .46)
-.07 to .59**
(-.21 to .41)
.29 to .57
( .13 to .51)
.29 to .65
(-.36 to .50)
.25 to .67
( .06 to .48)
.20 to .57
(-.18 to .46)
.29 to .65
(-.36 to .48
.11 to .74
(-.27 to .53)
.22 to .57
(-.12 to .58)
.25 to .55
(-.08 to .49)
-.01 to .59 **
(-.12 to .50)
C12
CIS
C12
CIS
.16 to .52
( .11 to .48)
Note:
* Each correlation estimated at 58 degrees of freedom.
** Low correlations of -.07 to .04 were obtained between chamber CIS in trials 1 and 3 with
chamber C3 in trial 1 in the first case and a low correlation of -.01 in the second case between
chamber CIS in trial 3 with chamber C12 in trial 1.
.....
IV
.....
e
e
e
122
The combined analyses of variance assume a split plot
design, with experiments in the main plot and racks (positions)
within experiments in the sub plot.
With the main plot factors of
chambers, trials and temperatures treated as cross classified factors,
the data are unbalanced.
Partial sum of squares are computed such
that the main effects are adjusted for other main
effect~,
the
two-factor interactions are adjusted for all main effects plus other
two-factor interactions and the three-factor interactions are adjusted
for all effects in the model.
The results of the analyses for each
variable are shown in Table 6.2.6.
For the analyses of variance, the significance of
positions within chambers, as measured by the rack effects, and the
consistency of position effects, as measured by the various
interactions with racks, are of interest.
The approximate F-test
of the average rack effects is made by synthesis, with the rack mean
squares plus the chamber x trial x rack mean squares as the numerator
and the chamber x rack mean squares plus trial x rack mean squares as
the denominator in the F-test.
Rack effects were significant in all
variables, with an F-ratio of 10.4 for plant height and F-ratios of
4.3 to 5.6 for the other variables, indicating a significant within
experiment consistency for plant height over all experiments and to
a smaller degree, a significant within experiment consistency for
the other variables over all experiments.
Table 6.2.6 Mean squares in the combined analyses of variance as a split-plot design, by variable,
with 20 experiments in the main plots and 15 racks of 4 plants each within each experiment.*
Source of variation
leaf
area
Variables **
petiole fresh
length
weight
Df
plant
height
dry
weight
4
3
1
4
3
43.35
39.57
14,816
10.26
57.17
33.71
115.49
8,578
14.61
16.80
25.26
30.20
4,017
8.53
7.85
69.79
396.20
16,918
35.47
37.69
13.75
15.12
1,861
11.33
7.89
4
23.06
1.56
1.71
3.74
0.97
14
56
42
14
56
42
56
30.75
1.77
1.25
4.04
0.59
1.14
0.81
19.27
1.88
1. 74
1.97
1. 38
1. 34
0.86
3.57
0.30
0.51
0.93
0.22
0.27
0.17
30.05
3.44
4.18
3.39
2.68
3.18
2.67
3.80
0.36
0.41
0.46
0.27
0.38
0.31
900
0.58
0.74
0.13
1.65
0.22
Main plots (experiment)
Chamber
Trial
Temperature
Chamber x Temperature
Trial x Temperature
Between chamber-trial within temperature
residual
Sub-plots
Rack
Chamber x Rack
Trial x Rack
Temperature x Rack
Chamber x Temperature x Rack
Trial x Temperature x Rack
Chamber x Trial x Rack
Chamber-trial-rack within temp~rature
residual
I
Note: * Mean squares for main effects were adjusted for each other, two-factor interactions
adjusted for each other plus the main effects and three-factor interactions adjusted for all effects.
** Square root transformation applied to all variables.
.....
N
W
e
e
e
124
6.2.2.4
Light effects.
Preliminary data on light taken at
trial 4 for all chambers are used to determine whether there was an
association of light with plant growth within chambers.
Table 6.2.7
gives the minimum and maximum light readings taken within chambers
and the light, as rated, for each chamber.
Table 6.2.1
Ranges of light readings by chamber and readings as rated.
Chamber
C3
C8
C9
C12
C1S
Light( x 10 3 lux)
As measured As rated
24
23
27
27
23
-
29
31
32
33
29
31
30
31
29
31
Correlations, by position (rack), of light readings across
chambers show high correlations among four chambers C3, C8, C12 and
C1S, with r > .92, and correlations of chamber C9 and the other
chambers from .45 to .69.
These results indicate a consistency of
light patterns within chambers over four chambers.
Correlations by position (rack) between light readings and
the five variables over all trials in each chamber show most
correlations not significantly different from zero, with values from
-.78 to .68.
A count over all variables and experiments show 77 out
of the 100 negative, with all correlations (20) of light and plant
height, negative.
These results indicate a negative association of
light with plant height and possibly, negative association of light
125
with the other variables.
However, this is not to be interpreted
as a 'cause-and-effect' relationship between light and the other
variables.
6.2.2.5
Use of prior information as a covariate.
Prior
information on plant performance within experiment can be used as a
covariate for subsequent experiments.
The usefulness of prior
information will depend on whether the position effects persist over
time.
To test this, data from ten experiments in trials 1 and 2 are
used.
The square root of each variable is pooled over the ten
experiments by position and fourth degree response surfaces are fitted.
The predicted values for plant height, leaf area, petiole length,
1H,
fresh weight and dry weight data are referred to as covariates
A
Y , and
FW
and
A
Y~,
A
Y
nw
respectively.
Correlations are high among
with correlations greater than .90.
remaining correlations are from .37 to .84.
This seem to suggest
that two covariates, combining any of these three
Yp
with either
or
YH
The
YA,
may be more effective than a single covariate
for control of the within experiment variation in later trials.
Coefficients of determination computed from data in trials
3 and 4 using the covariates show that
A
YA is best among all covariates
with an average of 22.7 percent accountability in the within experiment
error variance in all measured variables and over all experiments. The
A
covariate,
Y , is the most effective covariate for plant height data,
H
with a 34.2 percent accountability of the within experiment variation
A
in plant height.
However,
Yp
A
was a better predictor than
YH over
126
all measured variables and experiments, with an average coefficient
of determination of 23 percent.
A
The two predictors,
A
Y plus
A
Yp
jointly account for an average of 29.6 percent of the within experiment
variation in all variables over all experiments.
Thus, the use of two covariates in a within experiment
design with blocking is explored to determine the efficacy of these
covariates (singly or jointly) for control of the within experiment
error variance.
Five models were compared: (a) a model using rack
A
position effects,
(b) a model using rack position plus
Yp ' (c) a
Yp
model using both column and row effects, (d) the latter plus
A
(e) a model using column and row effect plus
and
A
YA and Yp '
The
coefficients of determination, averaged over ten experiments are
shown in Table 6.2.8.
Table 6.2.8 Coefficients of determination, averaged over 10
experiments in trials 3 and 4, by variable, for within experiment
designs with blocking and the use of covariates.
Blocking
Df
Model
Covariates *
by racks
by racks
by columns
and rows
by columns
and rows
by columns
and rows
Note: * Covariates
trials 1 and 2.
plant
height
leaf
area
Variables
petiole fresh
weight
length
dry
weight
Yp
14
15
58.9
62.4
53.2
55.7
53.9
56.3
51.4
53.4
44.4
46.7
none
14
55.5
53.3
53.1
50.3
45.9
Yp
15
63.9
56.2
56.3
54.2
49.3
YA + Yp
16
65.7
57.2
57.0
55.0
50.0
YA
Yp
none
and
are predicted values computed from
127
The largest gain in information of 8.4 percent from the use
A
of prior information in conjunction with blocking occurs when
used in the columns and rows model for plant height data.
Yp
is
In all
other cases, marginal gains in information of less than 3 percent are
obtained from the use of prior information with blocking.
A
comparison between blocking by racks and by columns and/or rows is
deferred until the next section.
6.2.2.6
Within chamber designs.
Several within experiment designs
with blocking are compared to the completely random design to determine
the efficacy of blocking within experiments.
Comparisons are also made
between the use of square blocks of four plants using rack effects
model and the use of rectangular blocks of ten plants using the row
effects model or blocks of six plants using the column effects model.
It was shown in section 6.2.1.1, page 106, that the within experiment
error variances in original units of measurements were different for
the two temperature levels.
However, the within experiment error
variances for the two temperature levels with the square root
transformation are similar in four variables except for plant height,
where the within experiment error variances between the two temperature
levels differred by a factor of two.
Thus, the within experiment
error variances for the different models are made using the square
root transformation on all variables, by pooling over the two
temperature levels.
The computations follow those given in
Table 4.2.10, page 42, for the 'A' chambers with 'racks' in place of
'trucks'.
The results (Table 6.2.9) showed that compact blocks within
~
128
experiments are as effective as blocking by columns plus rows.
Blocking by rows alone is not as effective as blocking by columns.
Blocking by racks show a reduction of 24 to 47 percent of the within
experiment error variances in all variables over all experiments.
Table 6.2.9
Estimates of the within experiment error variance from
20 experiments for the completely random design (CRD) , the
randomized block design (RCB) and the two-way blocking design (LS).
Within chamber design
plant
height
CRD
1.09
RCB 1) racks as blocks
.58
2) columns as blocks .82
3) rows as blocks
.97
LS
.64
.e
Note:
*
6.2.3
leaf
area
1.12
.74
.85
1.08
.76
Variables *
petiole fresh
length
weight
.22
.13
.17
.21
.15
2.35
1.65
2.01
2.13
1.71
dry
weight
.29
.22
.25
.27
.23
Square root tranSformation applied to all variables.
Between experiment variation with 20 day old Elants.
6.2.3.1
Between chamber designs.
Variation among the 20
experiments form the basis of this study comparing three between
chamber designs (the whole plot designs), the completely random (CRD),
the randomized block(RCB) and the design employing a two-way blocking
by trials and chambers (LS).
Table 6.2.10 shows the computational
procedures used in estimating the error mean squares for the three
designs from the experiments in the
levels.
'c' chambers with two temperature
129
Table 6.2.10 Computational procedures for the between experiment
error variance in the completely random design, randomized block
design and the two-way blocking design from experiments in the.
'c' chambers with two temperature levels.
Design
Between experiment error variance
C~
RCB 1) trials as blocks
2) chambers as blocks
L5
experiments within temperatures
chambers in trials in temperatures
trials in chambers in temperatures
chambers x trials interaction
(adjusted for average trial and
chamber effects in temperatures)
Note: An experiment refers to a chamber-trial combination.
The results (Table 6.2.11) show that the between
experiment estimates of error variance with a design using chambers
as blocks are not different from the estimates of error variance in
the
C~.
Hence, the variation from trial ,to trial for a given
chamber is as great as the variation over both chambers and trials.
Blocking by trials, however, show gains in information in all
variables over the
C~
of 60 percent in the error variances for leaf
area and fresh weight and about 22 percent for the other variables.
The estimates of experimental error for the two-way
blocking (L5) show gains in information from 88 to 96 percent over
the
C~
for leaf area, petiole length, fresh weight and dry weight
data and a gain of 30 percent in. information over the
height data.
C~
for plant
Accepting these estimates at face value, they clearly
indicate a strong advantage in the two-way blocking by both trials
and chambers.
However, because of the limited degrees of freedom
130
Table 6.2.11 Estimates of the between experiment error variance for
whole plots from 20 experiments in the completely random design(CRD),
the randomized block design (RCB) and the two-way blocking design(LS).
Design
CRD
RCB 1) trials as blocks
2) chambers as blocks
L8
plant
height
leaf
area
33.16
27.86
37.46
23.06
33.13
14.98
41.46
1.56
Variables *
petiole fresh
length
weight
14.23
10.56
12.76
1.71
96.54
31.90
134.26
3.74
dry
weight
9.62
7.36
9.76
0.97
Note: *All variables transformed by taking the square root prior to
analyses.
for the estimation of the experimental error variances, the results
should be viewed with some reservations.
It is likely that the
two-way blocking is not as efficient as the estimates may suggest.
6.2.3.2
Relative sizes of the between to the within experiment
error mean squares.
The relative size of the between to the within
experiment error mean squares from this uniformity study would provide
some indication on the magnitude of bias in the
'c'
chambers
resulting from the erroneous use of any within experiment error mean
square in place of the proper between experiment error mean square.
As a comparison, two between chamber designs, the completely random
design (CRD) and the two-way blocking design (LS) for the whole plot
are selected.
These are combined with two within chamber designs,
the completely random design and the randomized block design with
racks as blocks.
The relative sizes of the between to the within
experiment error mean squares (Table 6.2.12) are from 30 to 65 times
131
Table 6.2.12 Relative sizes of the between to the within experiment
error mean squares, assuming a completely random (CRD) and a two-way
blocking by trials and by chambers (LS) chamber designs and two
within chamber designs, the completely random design and the
randomized block design (with racks as blocks).
Design
Between
Within
chamber
chamber
CRD
CRD
LS
LS
Note:
* Square
CRD
RCB
CRD
RCB
plant
height
leaf
area
30.5
48.1
21.1
39.8
29.7
44.6
1.4
2.1
Variables *
petiole fresh
length
weight
64.9
81.5
7.8
13.2
41.3
59.4
1.6
2.3
dry
weight
32.7
43.6
3.3
4.4
root transformation applied to all variables.
with a completely random between chamber design.
Estimates of the
relative size of the two error terms are smaller than those aboveusing the between chamber design, LS.
However, this result is to
be taken with caution for the same reason as given in the preceeding
section.
132
6.3
Discussion and Conclusions
The variation within experiments in the
'c'
chambers was
patterned, with some consistency for variable plant height, over all
chambers, trials and temperature levels.
Plants were generally
taller on the left-side of the chamber near the aspirator, the height
decreasing as the plants were further away from the left-side.
Plants nearer the door were shorter.
For the remaining variables,
the pattern was less consistent than for plant height, but there was
generally better growth on the left-side of the chamber with poorer
growth near the door.
A significant proportion of the within experiment error
variances can be controlled by blocking, with compact blocks.
Blocking by racks of four plants reduced the within experiment
variation by about one-half.
The use of prior information as a
covariate in a model containing the rack effects is no better than
a model containing the rack effects only.
Preliminary data on light within chambers indicate a
possible negative association of light with all variables particularly
with plant height.
A between chamber experimental design with blocking by
trial gave reductions of the experimental error variance in all
variables from 20 to 60 percent over the CRD.
A design with
blocking over chambers did not give a significant gain in information
over the CRD.
A two-way blocking design by trials and by chambers
appeared to be particularly efficient in its accountability of the
133
between experiment error variances.
While the estimates of the
reduction of the experimental error variance exceeded 88 percent for
four variables, these estimates should be taken with some reservations
as they are estimated with only four degrees of freedom.
the use of a two-way blocking design in the
However,
'c' chambers is not to be
ruled out.
Relative sizes of two experimental errors for whole plots
and
su~
plots may be as high as 80.
Hence any within experiment error
variance used in place of a between experiment error variance will be
an underestimate.
Correlations among variables within experiments in the
'A', 'B' and
'c' chambers show a common pattern of relationship among
the variables.
Leaf area, fresh weight and dry weight have higher
correlations with each other while plant height and petiole length
have lower correlations with the latter.
The means for four variables leaf area, petiole length,
fresh weight and dry weight (22/18 temperature) of experiments for
each chamber type were lower in the
'c' chambers than the means for
the same variables in the 'A' and the 'B' chambers.
these four variables in the
The means for
'c' chambers were 16 percent lower than
the means in the 'A' chambers and 33 percent lower than the means in
the 'B' chambers.
The mean
for plant height in the
'c' chambers
was 28 percent higher than the mean for plant height in the 'A'
chambers and 20 percent lower than the mean in the 'B' chambers.
134
The pooled within chamber error variances in the
'e'
chambers in variables leaf area, petiole length, fresh weight and dry
weight, in the original units of measurements was 10 to 50 percent
lower than the 'A' studies and was from 50 to 80 percent lower than
the 'B' studies, at 22/18 temperature.
error variance for plant height in the
The pooled within chamber
'e'
chambers was 80 percent
higher than the 'A' chambers and 40 percent lower than the 'B'
chambers.
As the chamber size increases from
e
to A, it is
expected that variability among plants within chambers would also
increase.
Excepting plant height, the pooled within chamber
estimates of error variance in the other variables for the square
root transformation show that the error variances are similar for
the 'A' and
'e'
chambers and both are lower than the 'B' chambers.
Thus, for experiments where it is considered necessary to combine
data from different chamber types for a wider inference, this
non-homogeneity of the within chamber variances across chamber types,
even with the square root transformation, should not be overlooked.
135
7
GENERAL CONSIDERATIONS FOR EXPERIMENTS IN PHYTOTRONS
Uniformity studies at the North Carolina State University
Phytotron in several chamber types and repeated over time and chambers
(experiments) have clearly shown that there was considerable variation
in the within chamber growth of soybeans at 20 days.
Further, this
variation in plant growth within chambers was highly patterned (that
is, it is not random) and there was a reasonably high degree of
consistency of this pattern for a given chamber type over all
experiments.
This variation can be controlled to a significant degree
by the use of appropriate within chamber experimental designs.
The
use of compact blocks in a within chamber design using blocks was
worthwhile in all classes of chambers studied.
A convenient unit of
block for the 'A' and 'B' walk-in chambers is a truck, which will take
sixteen pots (experimental units) of ll.4cm diameter.
In the
'c'
reach-in chambers, a rack is a suitable unit of block which will take
four pots of ll.4cm diameter.
The experimental design within chambers
with blocking was more efficient than a completely random design.
The value of blocking within chambers in the control of the within
experiment error variances with 20 day old soybeans is obvious and does
suggest that blocking within chambers would likely be of value for
other plant species.
The tendency for the pattern of variation within experiments
to show consistency over chambers and trials raises the possibility of
obtaining some control of the within experiment variation using prior
136
information to adjust for the position effects within chambers.
In
these studies, this idea was tested by using information from the early
trials to develop predictors of plant response for the later trials.
The use of these predictors as covariates to adjust for the position
effects was effective in most cases, particularly in the absence of
blocking within chambers and thus warrants further consideration.
Data should be accumulated on the nature of response pattern in each
chamber, the degree to which they continue to persist over time and
over chambers and the extent to which these patterns can be generalized
to different species and stages of plant growth.
In the studies with the walk-in chambers, there was some
indication from preliminary data that temperature and/or light readings
within chambers was associated with the within chamber growth pattern.
It would be advantageous for temperature and light measurements to be
taken in each study for use as covariates within chambers.
Also,
- consistencies in the light and temperature patterns within experiments
can be checked periodically and any deviation from a 'common' pattern
over time can be used to detect faulty controls.
Leaving aisles along the side wall or treating plants at
the edges of the chamber as guard plants has been cited, due to poorer
plant growth and/or higher variability at the edges compared to other
positions within chambers.
The results with the 'A' chambers of
nine square meters growing area showed that variation in growth among
plants within four trucks located at the two sides of the chambers
was small than variation within four trucks located approximately at
137
the center of the chambers.
Thus, it is not necessary to leave the
space along the side walls empty and the whole chamber can be used
for experiments.
Maximum use of space will bring down the cost of
experimentation per unit area.
The idea of rotating trucks systemmatically over several
positions within chambers in the course of an experiment to minimize
the within chamber experimental error has been suggested.
While
such an idea is worth considering, the management of such experiments
is certainly less practical and more complex than would an experiment
without rotation.
Absolute uniformity of experimental material is
unnecessary so long as the experimental design and/or other statistical
controls are effective (which is so from these uniformity studies)
in controlling the experimental error.
In experiments where treatment factors require a chamber
and treatment effects are of interest, replication of treatments over
time and/or space (that is, in other chambers) is necessary to
provide a valid estimate of the experimental error.
The use of any
within chamber estimate of the experimental error is not only
erroneous but will underestimate the valid experimental error by as
much as 300 times.
Among the many proposals for experimental designs on whole
chamber treatments are the use of response surface type designs and
factorial replication to reduce the number of combinations of
treatments combined factorially.
However, the use of such designs
will still require a large proportion of the total chambers in any
138
phytotron.
Lang (1963) proposed that one method of increasing the
'capacity' of a phytotron was by moving plants between a number of
chambers set to provide certain standard conditions.
Although no
data were provided, he reported that experiments in Australia and
California showed no differences in behaviour of plants kept
stationary and those moved daily to different chambers set under
identical conditions.
The uniformity studies at N.C.S.U. in all
chamber types show a very significant chamber by trial effects.
Hence, a few chambers set to provide standard conditions and
repeatedly used without change in all replications (that is, with no
rerandomization) yields an experiment in which the chamber and
treatment effects are completely confounded.
Estimates of the
experimental error wouid be biased.
Experimental designs for whole chamber treatments employing
blocking over time was generally more effective in reducing the
experimental error variance among whole plots than blocking by
chambers.
There is also some indication that a two-way blocking with
trials (time) and chambers would give an additional control of the
experimental error.
Appendix 1
Experiment
Su~mary
of means, variances (within experiment), ranges and coefficients of variation
in the original units of measurements for each experiment in the 'A' chambers.
Plant height (II),
,2
Leaf area(A) , cm 2
UlID
_
A2
Fresh weight(FW), mgw
_
A2
FW
a FW
Dry weight(DW), mgm
-
Range
cv
48
76
28 - 87
18
I 299 2,301 151 - 475 16 44
52
23 - 76
17
48
63
24 - 78
16
53
86
18 - 79
18
30 - 108
19
57
21 - 67
19
56
94
31 - 87
17
16
52
70
27 - 80
16
2,467 150 - 460
16
49
75
24 - 78
18
211327
3,164 176 - 503
17
52
91
27 - 86
18
14 - 45
2J 1317
3,264 165 - 489
18 149
84
28 - 80
19
19 - 43
16
2,155 207 - 485
13 153
52
34 - 78
14
16 1141
628
67 - 246
181 28
41
16 - 57
23 1321
47 - 124
19 1124
566
51 - 203
191 25
30
13 - 45
22
158
32 - 122
181120
481
35 - 210
181 23
21
12 - 44
20 I 302
2,324 147 - 517
16
81
147
51 - 130
15 1141
735
35 - 200
191 29
37
11 - 55
21 I 337
3,795 108 - 490
18
161
442
112 - 243
13 1 171
923
76 - 279
191 43
74
10 - 68
20
4
57
107
35 - 95
18 1 91
454
23 - 198
231 18
14
11 - 32
21 1239
2,213 124 - 408
20
39
A9
1
75
176
32 - 119
181149
640
80 - 215
171 ~8
35
15 - 50
211357
3,350 211 - 532
16
A9
2
76
123
52 - 117
15 1146
565
58 - 212
161 27
25
14 - 48
18 I 347
3,013 182 - 504
A9
3
87
139
60 - 129
J4 1136
480
47 - 199
161 28
29
16 - 49
19 I 308
A9
4
83
239
53 - 140
19 1136
570
61 - 211
18130
40
15 - 53
AlO
1
78
147
52 - 115
15 1132
627
7; - 201
191 '27
32
AlO
2
84
111
54 - 114
13 1 150
391
94 - 210
13
23
H
0
A4
2
82
183
54 - 147
A4
3
74
187
A4
4
71
A8
1
A8
3*
A8
Note:
*
H
A
cvl P a p
29
A2
a DW
Range
Trial
cv
Petiole 1ength(P), mm
a A
Chamber
Range
A2
Range
cv
Range
3,216 187 - 600
cv I DW
18
I 434 5,863 220 - 688 18 63 144
349
Faulty light control mechanism caused continuous incandescent bulbs to be on for 3 successive nights.
t-'
W
\0
e
e
--
-
e
--
Ai>pendix 2 Correlations, by chamber type, between mean (of expe~iment) and the within experiment error va~iance, by expe~iaent,
the original units of measurements, the square root, logarithmic and reciprocal transformations of each variable.
Chamber
type
Number
of
Expts.
Plant height(H)
H
III
Leaf area(A)
log (H)
II"
A
IA
I/A
P
.92
.93
12
.91
.38
-.60
.90
.62
.13
-.61
A*
11
.31
-.09
-.56
.96
.33
-.31
-.76
8
15
.79
.37
-.54
.85
.17
-.28
-.64
.86
8**
14
.83
.20
-.64
.83
.23
-.42
8***
12
.89
.67
-.17
.87
.68
C
20
I .85
.63
-.54
-.56
.37
C(Low)
10
Note;
1-'
29
.58
,!p,
10ll:(A)
A
COUgh) 10
Fresh weight(FW)
Petiole length(P)
loe.(P)
lIP
IFW
weight(DW)
loe.(FW)
l/FW
OW
low log(DW)
Irl at
(n-2) d.f.,
I/DW 0.05 level
.81
.48
-.30
.84
.79
.54
-.11
.82
.52
.03
-.52
.86 1.58
.15
-.47
.87
.55
.14
-.35
.86
.84
.40
-.46
.86
.514
.80 I
.67
.07
-.60
.86 I .91
.40
-.72
.85 I
.532
.17
.76
.86
.68
-.15
.90
.81
.39
-.60
.90 I
.576
.23
-.84
.91
.86
.21
-.69
.67
.74
.'37
-.66
.86
.444
.78
.33
-.40
.82
.79
.57
.27
.03
.36
.39
-.23
.79
.632
.70
.42
-.04
.69
-.01
-.50
-.74
.76
.39
.15
-.11
.48
.632
.78
.07
.90
.55
-.07
.38
.28
.06
93
•
.81
-.79
.87 I .34
-.10
-.43
.11
-.67
.91
.77
.56
.95
.13 -.29
-.77
.7 /•
.94
-.71
.85
.53
.21
-.10
.33
.09
.38
.10
.18
-.38
.44
.95
FW
D~y
fo~
I'
75
1
.576
1
.602
* excluded chamber A8 in trial 3 with faulty light control.
** excluded chamber 83 in trial 6 with faulty temperature control.
*** excluded chamber 83 in trial 6 and chambers 82 and 88 in trial 3 (at 26/22 temperature).
~
o
Appendix 3
Chamber
Number
of
Expts.
type
I
2
values for homogeneity test of the within experiment error variances, by chamber type and with the original
units of measurements and the square root, logarithmic and reciprocal transformations of all variables.
X
Plant height(H)
III
log (H)
367
110
151
II
I
Petiole length(P)
Leaf area(A)
A
IA
1011 (A)
831
190
105
158
1/11
Fresh weight(FW)
2
X•.()S '
Dry we1ght(DW)
low 1011(DW)
P
Ip
101lCP)
883
369
132
57
329
189
93
91
412
188
99
70
270
210 \ ,•• ,
1/A
lIP
FW
IFW
lOl(FW)
l/FW
OW
(n-1)
l/DW
19.7
A
12
A*
11
113
92
121
416
62
70
156
833
179
84
57
379
76
58
90
270
88
57
70
0
15
255
77
85
1,578
165
172
291
1,976
660
465
406
1,281
233
134
161
1,308
184
65
59
1,457
0**
14
177
51
84
1,308
113
121
249
1,865
186
145
206
1,224
161
74
119
1,189
132
31
59
1,2931 22.4
0***
12
132
50
30
751
75
39
68
1,101
179
103
70
563
138
56
33
864
48
17
23
809
19.7
C
20
I 445
119
77
811 1138
84
283
1,351
257
39
186
1,266
192
65
176
1,168
217
47
95
682
30.1
C(lIigh)
10
24
33
52
119
43
40
49
125
19
9
10
41
46
36
37
32
81
9
9
C(Low)
10
15
12
11
16
1 33
36
52
154
44
29
24
43
22
28
·53
58
36
33
35
I
Note:
21.7
32116.9
58
16.9
* excluded chamber A8 in trial 3 with faulty light control.
** excluded chauwer 03 in trial 6 with faulty temperature control.
***
excluded chamber 03 in trial 6 and chambers B2 and B8 in trial 3 (at 26/22 temperature).
I-'
.pI-'
•
e
--
-
e
Appendix 4
Experiment
Summary of means, variances (within experiment), ranges and coefficients of variation
in the original units of measurements for each experiment in the 'B' chambers.
Leaf area(A) , cm 2
Plant height(lI), mm
.2
e
Petiole 1ength(P). mm
.2
Fresh weight(FW).mgll
.2
Dry weight(DW), IIgll
-
cv I DW
a2DW
Range
cv
6,815 212 - 613
221 50
107
30 - 79
21
211 408
5.300 262 - 557
18 I 63
122
41 - 86
18
19 - 79
271 445
10.650 259 - 726
23 I 59
128
37 - 92
19
13 - 48
251281
23 I 44
91
15 - 71
22
Chamber
Trial
iI
B1
3
83
324
44 - 134
22 1145
1,240
78 - 234
241 35
B1
5
113
547
67 - 210
21 I 168
790
93 - 222
171 39
B1
6
125
656
78 - 223
21 I 184
1,689
103 - 285
B2
1
102
549
55 - 170
23 I 115
808
30 - 184
B2
3*
324
1.574 197 - 427
12
330
1,174
248 - 419
10
95
120
60 - 124 12
906
12.211 609 - 1,187 12
B2
4
\57
1.362
93 - 235
23
203
1,622
95 - 291
20
54
168
20 - 81
24
508
13.081 234 - 753
B2
6
\57
1,243
97 - 244
Z2
\67
1,524
70 - 264
23
44
~84
22 - 76
31
403
B3
1
95
411
55 - 148
21
I 106
903
43 - 179
281 28
50
15 - 48
B3
4
132
666
87 - 206
19
1169
1,372
61 - 252
22141
100
83
5
127
571
85 - 220
19
I 161
921
80 - 231
191 40
79
B3
6** 1209
2.114 122 - 319
22
1173
2.620
63 - 270
30164
861
B8
1
88
a II
Range
cv
A
,2
o A
Range
cv
Pap
Range
cv I FW
94
18 - 60
281372
68
22 - 64
221 44' 147
54
251 30
0 FW
4,155
Range
99 - 453
116
327
73 - 159
16
23
65
186
33 - 98
21
9,030 242 - 630
24
53
112
32 - 81
20
261264
4.412 128 - 430
25 I 43
102
23 - 65
24
18 - 71
241428
7.703226 - 654
20 I 58
115
33 - 89
19
19 - 70
22 I 378
5.537 216 - 589
20 I 52
99
28 - 79
19
20 - 119 461494
19,771 200 - 278
28
34 - 112
24
92
486
51 - 156
24
I III
684
60 - 193
241 29
42
18 - 49
22 I 268
2.962 178 - 472
20
3*
147
485
97 - 233
19
I 195
485
132 - 246
III 54
76
35 - 76
16 I 496
5,481 308 - 700
15
88
4
115
924
64 - 119
26
1162
1,765
75 - 268
26\38, 177
12 - 72
351434
B8
5
132
867
76 - 213
22
1174
1,077
84 - 250
19
21 - 69
26
Note:
*
**
43
126
denotes trials conducted at 26/22 temperature.
denotes
trial where temperature controls were defective.
437
I 73 306
I
39
71
21 - 71
21
70
100
45 - 92
14
11.860 242 ";" 727
25 I 57
167
33 - 92
23
7.417 203 - 644
20 I 61
131
32 - 92
19
~
N
Appundix 5 Summary of means, variances (within experiment), ranges and coefficients of variation
in thu origInal units of measurements for each experiment in the 'c' chambers.
Experiment
Chamber
Leaf area(A), cm 2
Plant heIght (II), mm
Trial
"
lIigh Temperature
C3
1
280
C3
326
3
2
2
Petiole length(P), mm
,,;l
Range
cv
A
II A
Range
cv
P
2,293
1,526
165 - 360
241 - 405
17
12
288
256
401
1,373
187 - 292
187 - 336
9
14
67
73
il
II
Fresh weight(FW), wgw
2
Range
cv
FW
llFW
52
61
49 - 84
53 - 90
11
11
506
)63
2,576
7,579
397 - 615 10
390 - 737 15
67
72
P
Range
Dry
-
cv
weight (DW), mgm
2
DWO DW
Range
cv
57
118
50 - 88
49 - 87
11
15
C8
C8
1
4
246
335
1,482
1,425
132 - 344
172 - 407
16
11
218
289
1,026
831
127 - 308
208 - 349
15
10
64
84
68
74
45 - 81
67 - 103
13
10
473
642
4,558
5,859
308 - 667 14
450 - 809 12
63
80
79
99
40 - 87
62 - 104
14
12
C9
C9
2
4
300
351
1,795
1,421
200 - 400
255 - 425
14
11
256
308
1,255
1,533
92 - 333
209 - 394
14
13
79
100
58
105
57 - 92
73 - 118
10
10
540
709
3,763
8,266
358 - 713 11
500 - 910 13
76
90
102
135
53 - 101
62 - 115
13
13
C12
C12
2
3
256
285
2,523
1,032
159 - 371
218 - 353
20
11
248
269
521
954
174 - 296
186 - 354
9
11
72
81
75
93
51 - 98
61 - 100
12
12
532
609
3,715
7,229
376 - 676 11
331 - 803 14
82
79
99
85
53 - 107
57 - 107
12
12
C15
C15
2
4
294
278
1,852
1,008
201 - 415
193 - 376
15
11
225
272
828
1,195
171 - 290
206 - 305
13
13
63
76
45
55
52 - 82
63 - 94
11
10
474
605
3,130
7,992
344 - 632 12
205 - 752 15
61
79
73
111
47 - 84
58 - 108
13
102
100
305
299
65 - 144
54 - 126
17
17
94
120
290
375
45 - 128
68 - 164
18
16
22
25
11
14
16 - 31
15 - 34
15
15
218
293
2,217
1,680
54 - 321 22
173 - 377 14
32
37
29
24
18 - 44
23 - 47
17
13
Low Temperature
C3
2
C3
4
14
C8
C8
2
3
101
109
332
284
65 - 167
80 - 156
18
15
112
123
813
351
42 - 184
91 - 164
25
15
25
28
24
33
15 - 41
19 - 41
20
18
255
265
2,933
1,583
108 - 418 21
192 - 350 15
41
36
71
23
10 - 68
24 - 48
21
13
C9
C9
1
3
108
97
328
273
67 - 144
55 - 142
17
118
111
416
686
57 - 159
24 - 173
17
17
24
28
25
34
28
19 - 49
16 - 38
21
21
257
260
1,490
2,708
152 - 348 15
107 - 409 20
39
37
27
45
25 - 52
15 - 55
14
18
C12
C12
1
4
91
89
151
236
65 - 122
(;0 - 137
13
17
10'1
125
361
684
53 - 152
51 - 187
18
21
22
26
10
28
15 - 31
15 - 39
14
20
243
298
1,326
2,810
138 - 354 15
168 - 453 18
37
38
29
41
20 - 50
22 - 55
17
C15
C15
1
3
92
97
244
375
59 - 119
61 - 144
17
20
97
98
560
617
39 - 155
47 - 154
24
25
21
21
16
21
13-33
12 - 32
19
22
230
225
2,136
2,670
100 - 346 20
122 - 349 23
36
33
50
42
15 - 55
18 - 49
20
20
15
t-'
~
W
-
e
e
144
9
LIST OF REFERENCES
Bartlett, M.S. (1937). Journal of Royal Statistical Society
Supple 4:137.
Collip, H.F. and Acock, B. (1967). University of NottinghamReport of School of Agriculture 1966 - 67, 81-87.
Cooke, D. (1967). Symposium on the Techniques of Experimentation
in greenhouses. Acta Horticulturae, 5-13.
Downs, R.J. and Bonaminio, V.P. (1976). Phytotron Procedural Manual.
Tech. Bul. No. 244. North Carolina Agricultural
Experimental Station.
Downs, R.J., Hellmers,H. and Kramer,P.J. (1972). ASHRAE Journal. 47-55.
Hammer, P.A. and Langhans,R.W. (1972). HORTSCIENCE Vol.7(5) ,481-483.
Hruschka, H.W. and Koch, E.J. (1964). American Society of
Horticultural Science Vol.85, 677-684.
Kalbfleisch, W. (1963). Symposium on Engineering Aspects of
Environment Control for plant growth. Melbourne, Australia.
159-174.
Kramer, P.J. (1963). Symposium on Engineering Aspects of Environment
Control for plant growth. Melbourne, Australia. 39-43.
Lang, A. (1963). Symposium on Engineering Aspects of Environment
Control for plant growth. Melbourne, Australia. 5-39.
Snedecor, G.W. and Cochran, W.G. (1967). Statistical Methods. The Iowa
State University Press, Ames, Iowa. Sixth Edition.
Singh, M., Ogren, W.L. and Widholm, M. (1974). Crop Science: 14, 563-566.
Went, F.W. (1955). The Experimental Control of Plant Growth.
Chronica Botanica No.17.