f
AN APPLICATION OF SEGMENTED LINEAR REGRESSION
MODELS TO THE ANALYSIS OF DATA FROM A
CROSS-SECTIONAL GROWTH EXPERIMENT
by
Katherine L. Monti
Ra1tech Scientific Services
St. Louis, Missouri 63188 U,S.A.
Gary G. Koch and John Sawyer
Department of Biostatistics
University of North Carolina
Institute of Statistics Mimeo Series No. 1200
November 1978
An Application of Segmented Linear
Regression Models to the
Analysis of Data from
A Cross-Sectional Growth Experiment
Katherine L. Monti
Raltech Scientific Services
A Division of Ralston Purina Company
St. Louis, Missouri 63188 U.S.A.
Gary G. Koch and John Sawyer
Department of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, North Carolina 27514 U.S.A.
ABSTRACT
Several different methods of analysis are applied to data consisting of weight measurements, taken at specified post-treatment times,
of harvested thyroids from rats given one of four treatments.
Previous
studies of this type of data indicated that the growth is initially
rapid, and that a second phase of less rapid glUwth is followed by a
final phase in which little additional growth occurs.
The data are fur-
ther characterized by increasing variance through time.
The primary purpose of the analysis is to study the effects of
the treatments at the end of the study period.
One-way analysis of
variance tests among groups are performed on each day, but the results
are not particularly helpful.
of
var~ance
However, results from two-way analyses
(over subsets of days and groups) are consistent with the
three-growth-phases model and accordingly indicate significant group
differences during each.
Moreover, this conclusion is obtained whether
or not a log-transformation is applied to the data.
Essentially the
same results emerge when unweighted least squares, weighted least
squares, and maximum likelihood methods are used to fit descriptive
~
regression models.
As discussed in the text, there are problems in-
volved in applying each of these analys·es to this data set. but the
fact that the conclusions drawn are essentially the same throughout
illustrates the inherent robustness of the procedures applied.
PREFACE
In the spring of 1974 a research assistant of Dr. Colin Thomas
(University of North Carolina Medical School) brought a set of data to
Dr. Gary Koch for statistical analysis.
Katherine Monti, a graduate
student in Biostatistics at the University of North Carolina, Chapel
Hill at the time, worked under his supervision to generate the results
which are essentially those contained here in Sections 3.1 through
3.3.1.
Some time later further analyses were performed, in a some-
what piecemeal fashion, to generate the results in Section 3.3.2 and,
with the help of John Sawyer, Section 3.3.3.
As a result of the
unsteady flow of progress and several shifts in direction, not all
analyses are "symmetric" in the sense that a test performed in one
analysis may not, for historical reasons, have been performed in a.
similar analysis even though the test is equally applicable in both
situations.
The purpose of this report is to document the results of the
many directions which were taken as this particular set of data was
explored.,· The reader is asked to overlook the "asYmmetry" of this
report and should note that the essential message is that when these
data were subjected to a variety of statistical attacks, none of which
was a clearly superior method, the conclusions which addressed the
concerns of the investigator remained essentially the same.
It is
hoped that the reader overlooks the occasional missing statistic
and instead directs his attention to the robustness of the analyses.
ACKNOWLEDGMENTS
Katherine Monti's contribution to this research was supported in
part by National Institutes of Health, Division of General Medical Services
Grant No. 5T01 GM00038-20.
The contributions of Gary Koch and John
Sawyer were supported in part by joint research agreements with Burroughs We11come Company and
Hoechst~Roussel
Pharmaceuticals, Inc.
The authors would like to thank Dr. Colin G. Thomas, Jr., for
providing the data.
Thanks are also due to Sam Caltagirone and Virginia
Kettenbach for their assistance in generating various computer runs.
typing assistance of Jean McKinney and Brenda Lehmuth is gratefully
acknowledged.
The
1.
Introduction
Often an investigator, primarily interested in the long-range
effects of different treatments, collects data at fixed time periods
after treatment and monitors the effects of treatments through time.
Such data are sometimes collected on the same experimental units
through time so that each unit is observed at post-treatment time
points.
The resulting growth curves of each treatment group can then
be studied and comparisons among groups can be undertaken.
example, Allen and Grizzle [1969].)
On
(See, for
the other hand, in some experi-
mental situations it is necessary to sacrifice the experimental unit
in order to obtain the required measurement, and therefore it is impossible to monitor post-treatment growth patterns on the same experimental
unit through time.
from a specific
In this paper we discuss the analysis of data resulting
expe~iment
of the latter type.
These data consist of
weights of rat thyroids which were harvested at fixed intervals after
treatment.
The investigator is interested in studying the time-response
patterns ?f each treatment group and in comparing these groups; more
specifically, he is interested in among-group comparisons for the final
portion of the study.
When the variable under study in such time-response experiments
is a weight, volume, or some other quantity which is inherently multiplicative in nature, there is often a tendency for the variability
of the respective measurements to
increases.
incr~ase
as the measurement itself
Consequently, the usual linear model analysis of variance
2
assumptions of normality and variance homogeneity are not realistic
for such situations.
One way of dealing with this potential source
of difficulty is to analyze the log-transformation of the data in the
hope that sufficient linearization of the multiplicative nature of the
response is achieved that normality and variance homogeneity assumptions are reasonable on the log-scale (i.e., a lognormal probability
model is applicable).
Another aspect of such time-response studies is that growth is
frequently regarded as having several stages.
The stages of the growth
pattern may be readily identifiable or previously known, and the number
of stages may differ with the variable being studied.
Results from
this experiment and a previous one (Philp et al., 1969) indicate that
rat thyroid growth is initially quite rapid and that a second phase
of less rapid growth is followed by a plateau-like third phase during
which little growth occurs.
Thus, the analysis of the data should take
into account the characteristics of each distinct phase, yet all timeresponse models should be continuous in the sense that the end of one
phase is the beginning of the next phase.
Furthenmore, for the example
under consideration here, the analysis should provide a basis for predicted values that are specifically relevant to the last growth phase and
are not overly sensitive to the nature of the pattern of growth in the
earlier phases.
In short, the characteristics of the data set which is under
investigation here are:
(1) the measurements are taken through time
on independent sampling units; (2) the underlying distribution is
basically unknown but may be more nearly lognormal rather than normal;
and (3) the time-response growth pattern is characterized by three
3
successive phases, for which the growth rates decreased.
2.
The Data
The data obtained in this investigation consist of weight measure-
ments of the thyroids of three hundred and forty-one rats.
Ten-week-
old Wistar rats weighing 180 + 30 gm. were housed two per cage and
fed a standard chow diet.
A solutiml of 0.1% propylthiouracil was
prepared every other day in distilled water and used as drinking water.
All rats received an initial intraperitoneal injection of 10 mI. of
0.1% propylthiouracil and were maintained on goitrogen in their drinking
water thereafter.
An injection of
75
through a tail vein at the same time.
Selenomethionine was administered
The control group of rats was
injected with 0.5 ml. of saline solution.
Ten rats were sacrificed on day 0 so that a pretreatment value
of thyroid weight could be estimated.
The remaining rats were separated
into four treatment groups:
Group 1
co~trol
Group 2
3 microcuries 75Selenomethionine/kg body
weight (0.54 microcuries) and goitrogen
group - goitrogen and saline solution
Group 3 -- 100 micro curies 75Selenomethionine/kg body
weight (18 microcuries) and goitrogen
Group
4
1000 microcuries and 75Selenomethionine/kg body
weight (180 microcuries) and goitrogen.
Then, usually five animals from each group were sacrificed and
thyroid weights measured on days 2, 5, 7, 9, 12, 14, 16, 19, 21, 23, 26,
28, 30, 33, 35, 37, and 40 after injection.
(On certain days, only
four such measurements were obtained for one or more of the groups.)
Other technical details concerning the design of this experiment are
given in Heaton et al. (1973).
4
A summary of the structure of the overall experiment is given in
Table 1 with the initial pre-treatment group of animals which were sacrificed on day 0 being arbitrarily assigned to Group 1.
Table 1 also in-
cludes means and standard deviations for the observed weight measure-·
ments for each of the 69 group x day subpopulations under study.
A
corresponding graphical display of these means is given in Figure 1.
Finally 9 Table 2 contains the means and standard deviations for the
(natural) log-transformed data and a corresponding graphical display
of these means is given in Figure 2.
A listing of the data is given
in Appendix I.
3.
Analysis
Several different statistical strategies were employed in the
analysis. of these data with some being applied in parallel fashion to
both the actual and the log-transformed data.
Because of the potential
multiplicative structure of the data, the analyses of the log-transformed data are of definite interest from a technical point of view.
Furthermore, there is the added advantage that the assumption of
variance homogeneity is more realistic if the data are log-transformed
than if X:O transformation 1.s imposed,
norma~ity
A Kolmogorov-'Smirnov test of
of the distribution of the observatious and of the
log~
observations about their respective cell means was used as a heuristic
indicator.
It failed to "detect" non-normality in either case.
Appendix II.)
(See
Upon examining the residuals of the log-transformed data
about their means it can be seen that the
asst~ptions
of variances and normality are not seriously violated.
of homogeneity
There is
siderable heterogeneity if the data remain untransformed.
con~'
(See Tables
5
TABLE 1
OBSERVED MEAN THYROID WEIGHTS AND CORRESPONDING
STANDARD DEVlATIONS§ FOR SPECIFIED POST-TREATMENT GROUPS
Day
Within
group
sample size
Group 1
Group 2
Group 3
Group 4
Pooled
within group
root mean
square
One-way
analysis of
variance F
0
10
8.27
(1.31)
2
5
9.94
(1.14)
10.16
(0.25)
.7.92
(0.40)
8.12
(1.49)
0.97
7.38 **
5
5
22.22
(2.13)
18.76
(4.47)
15.68
(2.15)
14.86
(1. 78)
2.84
6.92 **
7
5
33. 1St
(2.88)
28.90
(4.25)
22.64
(6.10)
22.68
(4.48)
4.66
5.41 **
9
5
48.10t
(9.57)
30.44
(3.82)
34.15t
(5.86)
26.84
(7.77)
6.96
7.68**
12
5
43.50
(9.65)
37.28
(l0.53)
41.92
(9.49)
38.52t
(3.54)
9.00
0.50
14
5
55.18
(9.36)
41.68
(3.26)
32.82
(2.45)
49.72
(6.66)
6.10
12.78 **
16
'5
58.10
(5.90)
50.02
(8.85)
37.56
(7.13)
47.02
(9.23)
8.10
5.48**
19.
5
62.74
(11.08)
48.70
(13.02)
40.30
(11.29)
41.58t
(8.76)
11.28
4.04
21
5
56.86
(14.49)
56.30
(7.87)
50.30
(5.79)
44.12t
(4.46)
9.20
1.85
23
5
63.06
(8.88)
64.06
(16.55)
52.68t
(14.36)
56.42
(8.34)
12.40
0.87
26
5
63.50
(16.11)
58:20
(15.10)
65.64
(11.01)
55.68
(16.03)
14.71
0.49
28
5
.'
66.50
(8.50)
63.16
(11.63)
61.72 t
(15.21)
56.98
(8.41)
10.97
0.65
30
5
65.62
(12.40)
62.24
(15.17)
53.64
(3.48)
58.16
(16.06)
12.79
0.82
33
5
73.18
64.08 .
(18.70)
58.08
(2.80)
60.48
10.62
1.95
(7.78)
'"
(5.75)
35
5
74.00
(10.36)
69.92
(10.12)
59.92
(9.57)
53.36
(8.83)
9.74
4.65 *
37
5
75.88
(21.54)
67.40
(13.26)
48.50
(9.50)
56.00
(5.19)
13.76
3.88 *
40
5
64.42
(19.50)
68.40
(15.83)
61.90
(7.01)
59.3S t
(14.53)
14.95
0.30
§ weights are expressed in mg's; standard deviations are in parentheses; standard errors can be
computed by dividing standard deviations by square roots of sample size.
t
within group sample size equals 4 instead of 5.
*-
means signific3nt at a=.05;
**
mc~ns
~1gniflcant
at
Ct&.Ol.
Figure 1
ObBe~ed Mean Thyroid Weighte*
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e
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7
TABLE 2
OBSERVED MEAN LOGARITHMS OF THYROID WEIGHTS
AND CORRESPONDING STANDARD DEVIATIONS§ FOR
SPECIFIED POST-TREATMENT DAYS AND TREATMENT GROUPS
Day
Within
group
sample size
0
10
2.10
(.16)
2
5
Group 2
Group 3
Group 4
2.29
(.12)
2.32
(.02)
2.07
( .05)
2.08
Group 1
Pooled
within group
root mean
square
One-way
analysis of
variance F
0.11
6.76
1r1r
( .19)
1r1r
5
5
3.10
(.10)
2.91
(.22)
2.75
(.13)
2.69
( .12)
0.15
7.47
7
5
3.50t
(.09)
3.36
( .14)
3.09
(.26)
3.11
(.19)
0.19
5.12
9
S
3.86t
(.21)
3.41
(.13)
3.52t
(.18)
3.25
(.31)
0.22
5.91
12
5
3.75
(.22)
3.58
(.29)
3.72
(.21)
3.65t
(.09)
0.22
0.55
14
5
4.00
(.17)
3.73
(.08) .
3.49
(.07)
3.90
0.12
17.52**
16
5
4.06
(.10)
3.90
(.18)
3.61
(.21)
3.83
(.22)
0.18
5.20 **
19
5
4.13
(.18)
3.86
(.24)
3.67
(.27)
3. n t
( .21)
0.23
3.97 *
21
5
4.01
(.26)
4.02
(.15)
3.91
(.11)
3.78t
(.10)
0.17
1.83
23
S
4.14
3.93t
(.30)
4.02
(.15)
0.21
0.99
(.lM
4.14
(.23)
1r
1r*
(.13)
26
S
4.12
(.31)
4.03
(.28)
4.17
( .16)
3.99
( .28)
0.26
0.49
28
S
4.19
(.12)
4..13
(.19)
4.l0 t
( .24)
4.03
(.15)
0.17
0.71
30
S
4.17
(.20)
4.11
(.25)
3.98
(.07)
4.03
(.28)
0.22
0.73
33
5
4.29
(.11)
4.12
(.33)
4.06
(.05)
4.10
(.09)
0.18
1.51
35
S
4.30
(.14)
4.24
(.15)
4.08
(.17)
3.97
(.16)
0.16
4.62
37
S
4.29
(.31)
4.19
( .20)
3.86
( .22)
4.02
(.09)
0.22
3.78*
40
S
4.12
(.33)
4.20
(.23)
4.12
4.06 t
(.26)
0.24
0.27
§
t
*
(.11)
.*
These results are expressed in natural logarithms of weights in mgj standnrd deviations are in
parentheses; standard errors can be computed by dividing standard deviations by square roots of
sample size.
within group sample size equals 4 instead of 5.
means significant at a-.Ol.
me~~e significant at a-.05; ••
Figure 2
Observed Mean Thyroid Weights. Log Scale
[~.'.i.
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e
e
9
1 and 2.)
For preliminary screening purposes, one-way analyses of variance
were used to compare the four treatment groups at each of the 17 posttreatment days on which data were collected.
These tests, which are
reported in Section 3.1, were not particularly revealing so that more
powerful analytical methods became of interest.
In Section 3.2, the
results of two-way analyses of variance are reported for each of the
three phases mentioned previously.
The two-way analyses were under-
taken in this manner so that group differences at the end of the study
(which are of primary interest for this particular investigation) could
be analyzed separately.
In order to provide a descriptive framework
for predictive purposes as well as a more refined basis for treatment
group comparisons, various models were investigated.
A discussion
of the problems which were considered is included in Section 3.3.
mented linear models are used rather than the more
well~known
Seg-
polynomial
models for reasons which will become clearer after the results of
Section 3.2 are given.
Thus the specific justification for this
type of model is deferred until Section 3.3.
The results from ordinary
least squares regression models of the data are reported in Section 3.3.1;
the results
from certain types of weighted least squares analyses are
v
reported in Section 3.3.2; and the results from certain types of
maximum likelihood analyses are reported in Section 3.3.3.
3.1.
One-Way Analyses of Variance
The results from one-way analysis of variance F-tests for group
differences at each of the 17 post-treatment days are reported in
Table 1 for observed weight measurements and in Table 2 for the (natural)
10
log-transformed data.
For both of these types of analysis, group dif-
ferences at most of the earlier days of the experiment are found to be
significant (a=O.lO).
To test the assumption of variance homogeneity, Cochran's
{s2
II
max '
2
sh } criterion and the {s2
max
Is 2Uli n }
the Biometrika I Tables) were applied.
criterion (as given in
The former indicated significant
(a=0.05) variance heterogeneity at days 21 and 33 for the untransformed
data and days 2 and 33 for the transformed data; and the latter at days
2, 30, and 33 for the untransformed data.
Thus, from a within day
point of view, variance heterogeneity does not appear to be a source
of difficulty for either type of data.
In summary, the one-way analyses provided inconclusive statistical
results regarding the treatment group differences during the later days
of
t~e
study even though it seems apparent from examination of Figure 1
that some long-range differences do exist.
Thus, the remainder of the
paper will be concerned with more powerful procedures which take into
account long-range trends.
3.2.
Two-Way Analyses of
Varianc~
The primary questions of interest in this investigation pertain
to treatment group comparisons, especially during the last phase of growth.
A two-way analysis of variance over the entire range of post-treatment
days could be used to test the significance of group differences in an
overall sense, but the implications of such results specifically to
the patterns of growth at the end of the study would not be obvious,
particularly if there is si.gnificant group x day interaction.
Fur-
thermore, variance heterogeneity represents a more serious problem
11
for the untransformed data if all seventeen post-treatment days are
used in an overall two-way analysis of variance.
To facilitate inter-
pretation of the results and to reduce variance heterogeneitYt three
separate two-way analyses of variance are undertaken t one for the data
covering each of the three growth phases.
The results of this experi-
ment as well as previous work reported by Philp et ale (1969) suggested
that the first rapid-growth phase extends through about day 12, the
second period of less-rapid growth extends through about day 23, and the
final phase, in which little growth occurs, extends through the final
day 40.
Thus these analyses correspond to days 2 through 12 inclusive t
days 12 through 23 inclusive and days 23 through 40 inclusive.
Before proceeding to the two-way analyses of variance for these
three growth periods, it is of interest to investigate the extent to
which the assumption of variance homogeneity is realistic for each of
them.
{s
2
max
For this purpose, Cochran's {s2
/s
max
II
s~} criterion and the
2
i} criterion (as given in Biometrika I Tables) are applied
mn
to the corresponding.sets of one-way analyses of variance error (within
day) mean squares for each of the three growth periods.
These test
statistics indicate that variance heterogeneity does not appear to be
a source of difficulty for the analysis of either the untransformed
.-
data from the second and third growth periods or the log-transformed
data from any of the growth periods.
However, they are clearly significant
(a=O.Ol) for the untransformed data from the first growth period; and
thus the results of analysis here should be interpreted with some caution.
The test statistics obtained from two-way analyses of variance
are summarized in Table 3a for the untransformed data and in Table 3b
for the log-transformed data.
Group effects are significant (a=O.Ol)
12
TABLE 3
13
during all three growth phases and day effects are significant (a =0.01)
during the first two phases, but they are non-significant (a=0.25)
during the third.
In addition, there is
signific~nt
(a=0.05) group
x day interaction with respect to the untransformed data during the
first growth period.
These conclusions and the differences among them
across the three time periods provide further support for the assumption
that the growth pattern under study has three phases.
Since the interest
of the investigator is primarily concerned with the final days of the
study, another two-way analysis of variance is undertaken on days 28
through 40 inclusive.
Although its results are not reported here,
the conclusions are the same for both types of data as those for days
23 through 40; and the potential concern that the "tail" (days 23-40)
is too long to be representative of the "final days" of the study is
alleviated.
In view of the general significance of group differences for
all three growth phases, pairwise group comparisons become of interest
for purpos.es of interpretation.
In this regard, corresponding F-sta-
tistics are given in Table 3a for the untransformed data and Table 3b
for the log-transformed data.
For both types of data, the principal
conclusiqns suggested by these results are:
i.
ii.
Group 1 (the control group) animals have significantly
(a=O.Ol) . larger thyroid weights than their respective
counterparts in the other three groups during the first
two growth periods and this relationship continues to
apply with respect to group 3 and group 4 during the
final growth period;
Group 2 animals
thyroid weights
the last growth
thyroid weights
growth pe ri od •
have significantly (a=0.05) larger
than group 3 and group 4 animals during
period and significantly (a=0.05) larger
than group 3 animals during the middle
14
In addition, if days 28 through 40 are used for the final growth
phase rather than days 23 through 40, then essentially the same conclusions are obtained.
Further inspection of Table 3 indicates that the error mean
square estimates of variance increase considerably across the three
time periods for the untransformed data, but that they are relatively
similar for the log-transformed data.
This result suggests that a
homogeneous variance lognormal probability model appears to be a more
plausible framework for the analysis of these data than a homogeneous
variance normal probability model, particularly for those analyses
which are directed at the overall set of days as opposed to adjacent
subsets.
In summary, two-way analyses of variance permit group differences to be investigated for selected sets of days including those
corresponding to the final phase of growth.
Since different post-
treatment days are analyzed together, since the weights tend to increase with days, and since weight is inherently multiplicative in
nature, the variability of the respective weights also tend to increase with days.
Thus, the assumptions of the homogeneous variance
normal probability model are less plausible for the two-way analyses
of variance across days than for the one-way analyses of variance
within days discussed in Section 3.1.
The implications of this poten-
tial source of difficulty are reduced to some
e~tent
by separating the
days into three growth phases, a strategy which also facilitates the
interpretation of the final-phase results.
Moreover, this final
phase represents the part of the experiment where the assumption of
variance homogeneity is realistic for the untransformed data.
It also
15
appears reasonable for the second phase but is definitely inappropriate
for the first.
Nevertheless, the general similarity of the conclusions
from the two-way analyses of variance for the untransformed data for
each of the three growth phases to those from their corresponding logtransformed data counterparts (for which the assumption of variance
homogeneity is realistic for all days, both within and across phases)
may be regarded as an illustration of the robustness of such analyses
to variance heterogeneity.
3.3.
Segmented Regression Analyses
The two-way analyses of variance discussed in the previous section
represent a reasonable straightforward method for assessing the statistical significance of group differences during the three growth
stages.
However, except for this inferential information, they are
not particularly useful for describing the relationships between
thyroid weight and days within the respective groups.
To this end,
statistical models were fit to the data using multiple regression
(unweighted and weighted) and maximum likelihood methods.
Before
discussing the relative merits of various models which could be employed,
it is important to recall the objectives of the experimenter.
The
primary purpose of the experiment was the comparison of the groups at
the end of the study.
There was no intention on the part of the re-
searchers to provide a growth model which accurately portrayed the
weight increase of the thyroid through time under different treatments.
Rather, a simple, easily-interpreted descriptive model which reflected
group differences in the final phase was sought.
Although the re-
searcher was not particularly concerned with extrapolating predictive
16
values beyond 40··days
post~treatment,
the merits of providing a model
which could be used for limited extrapolation are obvious,
Although several different types of models could have been
used for this purpose, attention was restricted to segmented linear
regression models,
the
three~phase
The structure of these models was motlvated by
growth pattern that the data tend to follow and thereby
involves three corresponding linear segments which have different slopes
but are intersecting at the respective endpoints.
1be principal ad-
vmltages for this type of model with respect to the data under consideration are as follows:
i.
ii.
iii.
It provides an effective characterization of the
variation in the data in terms of a relatively small
number of parameters;
Its parameters have a relatively simplistic interpretation as growth rates during the three periods;
Its linear structure during the final phase is useful
for predictive purposes.
On the other hand, the principal disadvantage of a segmented
model is that it does not correspond to an underlying biological
model for growth and is somewhat misleading in the sense of indicating
that growth rates change at distinct points in timeo
data
are.~ross-sectional rather
Hawever, the
than longitudinal, so the data are
not even "continuous" 1.n a direct sense.
thyroid weight' situation under study
hare~
Furthermore, for the
an appropriate biological
growth model with meaningful and interpretable parameters is not
readily available.
In addition, general polynomial models were con-
sidered unrealistic because of their inability to provide a
satis~
factory framework for the final phase \-]here little grovJth occurred
The concern was that the statistical properties of the rcgress:Lon
0
17
estimates of the parameters of the polynomial models would be dominated
by the data from the first two stages in which there was the most growth,
and the extent to which this occurred would weaken the appropriateness
of such models for predictive purposes during the following final phase.
For comparative purposes polynomial models were actually fit.
(See Appendix III for details.)
They did indeed fit poorly and, as
anticipated, were increasing far too rapidly at the end of the study
to be considered useful for future predictive or even descriptive purposes.
In view of the inadequacy of polynomial models and lacking any
known theoretical model, segmented linear models were fit because of
their simplicity and descriptive usefulness.
From a general point of view, the fitting of segmented linear
regression models to data can require relatively complex analytical
and computational effort since both the number of segments and their
respective boundary (origin and termination) points may be unknown.
In this case there are more parameters to estimate and the model is
no longer linear in its parameters.
Such problems are discussed in
Hudson (1966) and Gallant and Fuller (1973), but are beyond the scope
of this paper where it is reasonable to assume at the beginning that
there
ar~.three
growth phases (or segments) and that the boundary points
are at least approximately at 12 days and 23 days.
The analysis chosen to use in fitting the data had to consider
the heterogeneous variance structure.
Unweighted least squares ana-
lysis of the data ignored the problem and would violate the homogeneity
assumption.
Unweighted least squares analysis of the log-transformed
data would provide estimates within a homogeneous structure, but the
estimated line segments would be transformed to arcs on the regular
18
scale and the arcs are unlikely to be adequately descriptive of the
data.
Of course, if the original scale is not of fundamental interest,
this regression model on the log-data could be of interest.
The results
of unweighted regression analyses on the untransformed data and, briefly,
on the log-transformed data are given in Section 3.3.1.
Another method of accounting for the variance heterogeneity
using the untransformed data is provided by weighted least squares
methods.
There are problems with these analyses, however, when the
true variances are unknown.
The problems and different "solutions"
are explored in detail in Section 3.3.2.
Finally, the results obtained using maximum likelihood methods
are given in Section 3.3.3.
These methods provide predicted values on
the original scale of the data even though the log-structure is utilized
in the computations.
Interestingly, the results obtained by maximum
likelihood methods, which require considerable more complicated
com~
putations, are for the most part relatively similar to the unweighted
and weighted regression analyses.
Thus, in spite of the different
issues which pertain to the appropriateness of the various methods
discussed in this paper, in terms of their corresponding assumptions,
they all yield essentially the same results for this example.
As a
consequence of this general robustness, the choice among them in such
applications appears to be mostly a matter of personal taste and
computational convenience.
Nevertheless, an underlying concern for
the issue which pertains to this choice is still entirely warranted,
and the actual verification of the general similarity of their corresponding results or the resolution of their differences remains a
fundamental component of the application of statistical methodology
19
to biological as well as other types of data.
3.3.1-
Unweighted Regression Model Fitting
Let Yhik denote the observed thyroid weight for the k-th animal
in the h-th group which is sacrificed on the i-th day where h == 1, 2, 3, 4;
i
= 0, 2, S,
~i
= S for i
... , 40;
and k
= 1, 2,
... ,
~i
> 0, as specified in Table 1.
(with n
lO
= 10 and usually
For the animals in each
group, the three-phase segmented regression model with boundary points
at 12 days and 23 days may be formulated as
(1)
where
[i
if i
2
12
L12 if i ~ 12
(0
~
i f i < 12
(i-12) if 12
~i ~
23
(11 if i .2'.. 23
=
so that
So
~a
l
if i
~
23
(i-23) i f i
~
23
corresponds to a common baseline intercept at day
a
for
correspond to the respective
all four groups, and 8 , 8 , and 8
2h
3h
1h
slopes (or growth rates) during the first stage, second phase and
final phase for the h-th group.
This model may then be expressed
in the general matrix notation
(2)
ZO
by having y denote the entire set of Y
arranged in an animal within
hik
day within group manner and using the definitions of the {x
{x
Zhik
}, and {x
of the design
3hik
lhik
},
} in (1) to specify the structure of the columns
matrix~.
In particular,
!.10
1
""t1
!S
=
~l
1
1
""Il
(341x13)
Z
(3)
~Z
~3
1
"t1
3
~4
1
"t1
where 1
~c
4
is a (c x1) vector of l's and ~
=
I
~i' blanks are matrices
i
of z"eros., and the sub-matrix X. for the h-th group is formed by deleting
""!l
unnecessary rows corresponding to the days with
2
0
5
0
7
9
0
12
0
1Z
2
4
~O = 15 ® 12
12
12
12
12
12
12
12
12
12
2
l2:
for which
09
0
7
9
11
11
11
11
11
11
11
11
~i
< 5 from
0
0
0
0
0
0
0
0
(4)
0
0
3
5
7
10
12
14
17
denotes Kronecker product matrix multiplication.
For
21
example, the fifth rows in the sets for day 7 and day 9 would be deleted
from ~ to form ~l'
Unweighted least squares estimates for the parameters of this
model and their standard errors are given in the last column of Table 4.
Test statistics for group comparisons with respect to the three types
of slope parameters, both separately and simultaneously, are given in
the last column of Table 5; and test statistics for group comparisons
at the end of the study with respect to model predicted values at
day 37 and day 40 are given in the last two columns of Table 6.
As
mentioned previously, the estimated parameters and the test statistics
for the various hypotheses should be regarded cautiously here because
the unrealistic assumption of variance homogeneity underlies their
basic formulation.
More specifically, the covariance matrix for the
unweighted regression estimates parameters vector '"B has the form
.....u
where
~
is a diagonal matrix with the elements v
hik
= Var(y
the main diagonal, but the estimated covariance matrix for
hik
8
;;:.u
) on
which
is obtained from such analysis is
(6)
where
s
-4
2
=
n hi
LLl
h=l i k=l
is the overall pooled within group within day mean square with respect
to corresponding means
22
e'
TABLE 4
ESTIMATED PARAMETERS AND STANDARD ERRORS FOR PRELIMINARY
MODEL ! AS BASED ON WEIGHTED LEAST SQrARES WITIl RESPECT TO
FOUR ALTERNATIVE VARIANCE STRUCTURES
Parameter
One-day
pooled
variance
Three-day
pooled
variance
Six-day
pooled
variance
Overall
pooled
variance
(unlo1eighted)
(0.26)
5.00
(0.42)
4.82
(0.99)
4.81
(1. 79)
13 11
3.53
(0.13)
3.57
(0.14)
3.79
(0.17)
3.87
(0.28)
13 12
2.88
(0.13)
2.79
(0.14)
2.86
(0.17)
2.86
(0.28)
813
2.21
(0.13)
2.34
(0.14)
2.79
(0.17)
2.50
(0.28)
1314
2.42
(0.13)
2.34
(0.14)
2.66
(0.18)
2.81
(0.29)
a2l
1.65
(0.30)
1.63
(0.30)
1.23
(0.30)
1.15
(0.36)
622
1.90
(0.30)
1.94
(0.30)
1.89
(0.30)
1.90
(0.35)
B23
2.09
(0.31)
1.85
(0.31)
1.41
(0.31)
1.81
(0.37)
624
2.06
(0.31)
2.05
(0.31)
1.59
(0.31)
1.41
(0.37)
831
0.49
(0.31,>
0.36
(0.31)
0.51
(0.29)
0.52
(0.26)
832
0.55
(0.31)
0.54
(0.31)
0.53
(0.29)
0.53
(0.26)
633
0.31
(0.31)
0.34
(0.31)
0.32
(0.30)
0.23
(0.26)
634
0.04
(0.32)
0.15
(0.33)
0.26
(0.31)
0.27
(0.27)
13 0
4.71
e
e
e
TABLE 5
CHI-SQUARE TEST STATISTICS Q FOR PRELIMINARY MODEL ~ AS BASED ON
LEAST SQUARES WITH RESPECT TO FOUR ALTERNATIVE VARIANCE STRUCTURES
~EIGHTED
One-day
pooled
Hypothesis
D. F.
varl:anc~
Three-day
pooled
Six-day
pooled
v~rianc~ __ ~ __'\T~I"iance_ __
Overall
pooled
variance
J unweighted)
81l=B 12=8 13=8 14
3
78.65 **
71.39 **
43.74 **
22.90 **
821=822=823=B24
3
1.32
LOS
2.72
2.91
831=B32=833=834
3
1.59
0.76
0.63
1.14
SZl:SZZ:S23: SZ4
6
3.13
2.85
6.37
5.94
12
5083.65 **
1979.10 **
1160.88**
56
199.014 **
J
S
B31-832-S33-S34
Sll=S12- S13- 14- SZ1=SZZa :}
. **
3846.85
823=824=B 31 =S32=8 33 =S34= 0
Goodness of fit residual
Pure error
D.F.
118.11**
means significant at aaO.05;
•
NOT DEFINED BUT IMPLICITLY INCORPO~~TED IN
WEIGHTED LEAST SQUARES ANALYSIS THROUGH
ESTI~~TED VARIANCE MATRIX
272
*
77.50 *
55.35
Mean
square
N
99.6
** means signficant at a=O.01.
w
TABLE 6
ChI-SQUARE TEST STATISTICS Q FOR GROL~ COMPARISONS
AT DAY 37 AND DAY ·40 AS BASED ON ~JEIGHTED :'EAST SQUARES FITTING O?
P:RELIMINARY MODEL ;.s 'wITH RESPECT TO FOUR ALTERNATIVE VARIAJ'iCE STRUCTtJ1lliS
Overall
Three-day
pooled
One-day
pooled
Group
t
comparison
1 vs 2
Day 37
Day 40
Day 37
1.11
0.62
O~72
0.32
G.8S
9.78'**
6.10"''*
10 . 90 **
-; .il:;'
*i<
6.90 **
1L20 **
7.79
2 VB 4
7.41
:3 vs 4
0.14
1 vs 2
**
5.66
2 vs 3
VB
3 vs 4
t
D.F ..
1f
mean:.s significant
1l1li
1 fOK' the
40
Day 4C
1.4.1l)
1 vs 4
variance
variance
pooled
variance
(unweighted)
37
D,~y
Jl1.77
1 vs 3
Six-day
poo2ed
variance
20.10
pai!""~ise
a~ID .. :J5J;
~t~
'"
:">:*
**
2.0 .. 26 **
:7081
,;7J:
4.03
5.89
'"
0 .. 20
,<:<
1~:·
<>
33
*~:;.
10.20
:II:
5.57 '"
4.30 '"
5.9D *
tj
C.02
0.04
0.01
0.02
·It'"
15.7~
10.64
1<
Cl
17.25
',[;:r;r
3.97
"
31
Day 40
0.77
1.30
'fi.t'k
3 .. 63
5.59
Day 37
0.,53
5.19 "
cO'wpari:scns in t'be fi:r,st six rOl.. r. S and D.F ..
*-;)
n;;l~
~-~'
"'*
,.
:!l
22.01 **
17.30
*t:r,
16.77 **
12.01 **
11.37
**
9.12 **
6.69 **
8.87 **
6.32
a.ao
0.00
26.70 '* . .
18.60 **
..
3 for the overall compsrison in the last TOW.
m,eane 6:~S'TI.:r.ficdJ.-::'~: ,:::.t a~~O .. Ol ..
N
.f::'-
e
e
e
25
Thus, if the respective variances v
hik
are sufficiently similar that
~~£v~ is approximately equal to (~~~)v where
4
v
={
\'
L
,
L
n
,hi
L
h=l i k=l
(7)
v hik /341}
denotes the average variance for the {Yhik}, then V§
is a reasonable
....u
Va..,
estimate for ~(6
); otherwise, the estimated variances based on
...... u
....u
A
will be too small for some linear functions of 8
.... u
others.
and too large for
Moreover, similar remarks apply to the test statistics for
various full rank linear hypotheses of the type H :
O
~
= 0 which
have the form
(8)
and to model goodness of fit statistics which have the form
(9)
where R is an orthocomplement matrix to
~.
Since the variance heterogeneity in the data seems to indicate
that
~B
is not a reasonable estimate for
~(~),
these considerations
....,u
imply that the results of the unweighted regression analysis should not
be used as a basis for statistical inferences pertaining to group comparisons and the other sources of variation under study in this experiment.
Nevertheless, for purposes of comparison with the more
complex analyses given in Sections 3.3.2 and 3.3.3, it is of interest
to note that the test statistics from such analysis in Tables 5 and 6
suggest that:
1.
The model (1) does provide an adequate characterization
of the data since the goodness of fit statistic is nonsignificant (a=O.25). Also the model accounts for
26
100{1160.88/l2l6.23} = 95% of the variation among
the mean thyroid weights for the 69 group x day
subpopulations.
2.
The differences among the first stage slopes for
the groups are significant (a=O.Ol) , but the differences among the second phase slopes and third
phase slopes are clearly non-significant (a=0.25).
3.
The differences among the predicted values for the
groups at day 37 and day 40 are significant (a=O.Ol).
In this regard, group 1 and group 2 animals have significantly (a=O.OS) larger thyroid weights than their
counterparts in group 3 and group 4. Otherwise, the
pai.rwise difference betvleen group 1 and group 2 is
non-significant and so is the pairwise difference
between group 3 and group 4.
In view of these conclusions, the model (2) can be simplified to
(10)
for which
S2*
represents the common second slope faT all groups and
B3* represents the common third slope.
TIle vector of parameters is
thus reduced to
and the design matrix
X~
"'I'
for this final model can be constructed
from the preliminary design matrix X in (3) by adding columns 3, 6,
9, and 12 together, adding columns 4, 7, 10, and 13 together and
leaving all other colmuns in tact. The resulting unweighted least:
squares estimates for the parameters of this model and their standard
errors are given in the last coluUUl of Table 7 and the predicted values
are plotted in Figure 3.
Test statistics for group comparisons with
respect to. the first stage slope parameters are given in the last
column of Table 8.
Since the slope parameters for the second and
third phases are the same for the four groups, comparisons among first
phase slope parameters are logically identical to comparisons among
e
e
e
TABLE 7
ESTIMATED PARAMETERS AND STANDARD ERRORS FOR FINAL
AS BASED ON WEIGHTED. LEAST SQUARES WITH RESPECT TO
FOUR ALTERNATIVE VARIANCE STRUCTURES
MODEL)~F
Parameter
One-day
pooled
variance
Three-day
pooled
varia.nce
Overall
pooled
variance
(unweighted)
Six-day
pooled
variance
80
4.71
(0.26)
5.00
(0.4.2)
4.82
(0.99)
4.80
(1. 79)
811
3.46
(0.11)
3.48
(0.12)
3.69
(0.15)
3.64
(0.23)
812
2.91
(0.11)
2.85
(0.12)
3.04
(0.15)
3.13
(0.23)
813
2.26
(0.11)
2.34
(0.12)
2.72
(0.15)
2.59
(0.23)
814
2.41
.(0.11)
2.37
(0.12)
2.65
(0.15)
2.66
(0.23)
82*
1.92
(0.16)
1.86
(0.16)
1.53
(0.16)
83*
0.36
(0.16)
0.35
(0.16)
0.41
(0.15)
•
1.56
(0.19)
0.39
(0.13)
N
......,
Figure 3
Predicted Thyirold Weights from Model ~f
Based on Un~{eighted Lesst Sq,uares 1!:.nalysia
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Thy ,owe
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Day
e
29
TABLE 8
CHI-SQUARE TEST STATISTICS Q FOR FINAL MODEL ~ AS BASED ON WEIGHTED LEAST SQUARES
WITH RESPECT TO FOUR ALTERNATIVE VARIANCE STRUCTURES
Hypothesis
D.F.
One-day
pooled
variance
Three-day
pooled
variance
Six-day
pooled
variance
Overall
pooled
variance
(unweighted)
*tr
76.42 **
70.48 **
22.13**
26.58 **
23.84 **
13.17 **
1
103.54**
87.12 **
53.44 **
54.61 **
811-8 14
1
78.63 **
81.51 **
60.50 **
47.39 **
812-8 13
1
30.67 **
17.92 **
'.20 *
14.49 **
812"'814
1
17.92 **
15.64 **
8.95 **
11.01**
813"'814
1
1.65
0.06
0.26
0.21
82*"' 0
1
140.65**
133.98**
97.30 **
70.70 **
8 3*- 0
1
5.10 *
5.01 *
7.41 **
8.97 **
811-812"'813-614-82*-83*- 0
6
5080.51 **
3844.00 **
1972.73**
1154.94 **
62
202.18 **
120.96*
83.87
61.29
811-812-813-814
3
126.95 **
811-8 12
1
811-8 13
Goodness of fit residual
*
means significant at a=0.05;.**
114.29
means significant at a"'O.Ol.
30
model predicted values for any given day at or after day 12, and hence
specifically day 37 and day 40 at the end of the study.
Finally, the
conclusions suggested from the analysis of this final model are essentially the same as those indicated for the preliminary model (1) except
that in this framework the pairwise difference between group 1 and group 2
is also significant
(a~O.Ol).
The original model,
~,
was also fit for the log data.
The
parameter estimates and standard deviations are given in Table 9 and
a plot of the predicted values is given in F1.gure 4.
The pairwise dif-
ferences at the end of the study are given in Table 10.
Although the
significance levels are lower than those observed for the untransformed
data, the general conclusion remains:
groups 3 and 4, which are nearly
identical, are significantly different from group 1, with group 2
being intermediate between groups 3 and 4 and group 1.
Further atten-
tion was not given to the log-model because the primary interest of
the researcher was in an interpretation based on the original scale
of the data.
Since other methods were available for handling the
problem of heterogeneity, there was no purpose in pursuing the reduc·tion of the model for the log-transformed data.
3.3.2.
Weighted Regression Hodel Fitting
One approach for dealing with the implications of variance
heterogeneity to model fitting is weighted least squares.
Here, how-
ever, the weights must be based on estimated variances since the true
variances are unknown.
This aspect of such analysis gives rise to an
awkward dilemma in the sense that the statistical properties of
weighted least squares model fitting with estimated variances are
31
TABLE 9
ESTIMATED PARAMETERS AND STANDARD ERRORS FOR
PRELIMINARY MODEL ~ BASED ON THt LOG-TRANSFORMED DATA
AND UNWEIGHTED LEAST SQUARES ANALYSIS
Parameter
Estimate
Standard Error
80
811
812
813
814
2.0877
0.0386
0.1635
0.00612
0.1396
0.00605
0.1274
0.00612
0.1338
0.00621
821
0.0063
0.00778
822
623
0.0268
0.00765
0.0313
0.0244
0.00792
0.00790
0.0087
0.00555
0.0097
0.00555
0.0064
0.00568
0.0059
0.00584
824
831
632
633
634
..
Figure 4
Predicted Thy~oid Weights, Log Scale,
from Model ~ Based on Unweighted Least Squares Analysis
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33
TABLE 10
F-STATISTICS FOR PAIRWISE GROUP DIFFERENCES
BETWEEN THE VALUES ON DAY 40 AS PREDICTED BY MODEL X
BASED ON THE LOG-TRANSFORMED DATA AND UNWEIGHTED ~
LEAST SQUARES ANALYSIS
Groups
Difference
F-statistic
1-2
1-3
1-4
2-3
2-4
3-4
.045
.197
.203
.135
.158
.006
0.32
6.26*
6.38*
t
3.75tt
3.86tt
0.005
t The F's are distributed with 1 and 272 degrees of freedom.
*. Significant at a=O.OS.
tt B.orderline significant at a=O. 05 since critical value is 3.88 •
....
34
generally unknown (and may well be poor) unless the specific application in question may be regarded in an asymptotic regression framework;
in other words, the sample sizes on which estimated variances are based
are sufficiently large that the estimates and test statistics which are
obtained with them via weighted least squares are approximately the
same as those which would have been obtained with the corresponding
unknown true variances.
For the experiment under study here, this
issue causes some difficulty because the sample sizes for the respective group x day subpopulations are relatively small (i.e., n
i
=
4 or 5).
However, as indicated in Section 3.1, the variances corresponding to
the same day appear relatively homogeneous and can accordingly be
pooled to form "one day estimated variances" with 14, 15, or 16 degrees
of freedom (except day 0, where no pooling can be done and D.F. = 9).
Although this level of sample size is not really considered large enough
for asymptotic regression, the "one-day estimated variances" are nevertheless used in one of the weighted least squares analyses because they
directly reflect
in the data.
ess~ntially
the full range of variance heterogeneity
Thus, the results which are obtained with them are of
interest for comparative purposes.
The other weighted least squares
analyses are based on pooled estimates of variance for consecutive sets
of three days -("three day estimated variances") and six days
estimated varianees")respectively.
("six~day
The principal advantage of these
latter types of analysis is that the sample sizes corresponding to
the estimated variances are essentially large enough for the application of weighted least squares in an asymptotic regression sense.
However, the variance heterogeneity present in the data across days
implies that such pooling is not really appropriate, particularly
35
during the first growth phase.
Hence, the results from these analyses
also have their limitations, but are of interest for comparative purposes.
Thus, the three different types of variance estimates are used
to define the weights.
These are:
one-day pooled estimated variances
1.
4
ii.
~i
= {I I
h=l k=l
'"
v'"hikl = vhi*l
(11)
three-day pooled estimated variances
4
{L
(~i" - I)}
h=l
(12)
where the sets {Si,3} correspond to the following
consecutive three-day subsets:
(0,2,5), (7,9,12),
(14,16,19), (21,23,26), (28,30,33), and (35, 37,40);
iii.
six-day pooled estimated variances
"hik3 =
4
{L )
vhi *3
=
nhi
I
(13)
h=l i £Si 6 k=l
,
where the ~ubsets {Si,6} correspond to the following
consecutive six-day subsets:
(0, 2, 5, 7, 9, 12),
(14, 16, 19, 21, 23, 26), and (28, 30, 33, 35, 37, 40).
For computational convenience, the weighted least squares model fitting
procedures are applied to the group x day means
mated variances of the type
{Yhi }
for which esti-
36
for t = 1, 2, 3 are used to define the weights.
The specific weighted
least squares analyses are undertaken with the computer program GENCAT, as described in Landis et al. (1976).
In this regard, the direct input mode 'is ap-
plied with respect to the vector of observed means
{Y hi } in combination with
each of the three types of estimated variances {vhi*t}'
design matrices are the appropriate analogues of
~
The corresponding
in (1) and
~F
in (10) which
are obtained by replacing the successive sets of 4 or 5 rows for the respective group x day subpopulations (and 10 rows for day 0) by unique single rows
so that their dimensions are (69 x13) and (69 x 7) respectively.
The estimated parameters for the three different types of
weighted least squares analyses and their corresponding standard
errors are shown in the first three columns of Table 4 for the preliminary model
model~.
~
and the first three columns of Table 7 for the final
In addition, test statistics for the various hypotheses
discussed in Section 3.3.1 are shown in Table 5 and Table 6 in reference
to
!'
and in Table 8 in reference to
~F.
Thus, it can be seen that the
results from the three different types of weighted least squares analyses are relatively similar to each other as well as to those from
the unweighted analysis.
More specifically, the differences among
the methods with respect to estimates of parameters are approximately
of the same order as the standard errors of the estimates involved.
In addition, the conclusions from the various tests of hypotheses are
essentially the same in terms of non-significance vs. significance
(with respect to a=0.05 or a=O.Ol).
On
the other hand, there are
substantial differences among the specific values for the test sta-
37
tistics corresponding to many of the hypotheses where significance
is indicated.
In particular, this occurs for virtually all of the
hypotheses which are considered under the model
~
and the hypotheses
for overall model explained variation (i.e., all parameters are equal
to 0 except the constant term
the preliminary model
~
80 ) and model goodness of fit for both
and the final model
~F.
Although these dif-
ferences appear worthy of some technical concern, they do not really
represent a methodological paradox.
The critical issue here is that
the weighted least squares analyses are undertaken in an asymptotic
regression framework.
As such, the means
{Yhi } are regarded as having
approximately normal distributions with variances which are presumed
to be known and approximately equal to the respective types of variance
estimates {Vhi*t}.
However, the means
{Yhi }
are based on relatively
small sample sizes (i.e., the {vhi*l}} at one extreme and unrealistic
pooling with respect to the variance heterogeneity (i.e., the {v
at the other.
hi
*3}}
In spite of these potential sources of difficulty,
weighted least squares model fitting is still considered reasonable
here for purposes of estimation and for testing hypotheses involving
model parameters at significance levels a=0.05 and a=O.OI because the
statistical properties of the estimated parameters are the components
of such analyses, which are usually robust to the limitations of the
underlying asymptotic framework.
However,
tes~s
of hypotheses in-
volving model parameters at significance levels a<O.Ol are considered
unrealistic for this type of application.
In addition, the model good-
ness of fit statistics often tend not to be robust to asymptotic assumptions because they are defined in terms of the residuals between
the observed means Yhi and their corresponding model predicted values.
38
Thus, their interpretation is inherently difficult and unclear.
More
specifically, for the data under consideration here, the goodness of
fit statistics for the preliminary model
~
are significant (a=O.OS) for
each of the three types of weighted least squares analyses with the
greatest amount of deviation being indicated for the one involving
the {vhi*l}' which are based on the fewest degrees of freedom, and
the least amount of deviation being indicated for the one involving
the {v *3}' which are based on the most degrees of freedom.
hi
these differences in magnitude appear disturbing, the model
Although
~
never-
theless accounts for at least 96% of the corresponding "weighted variation" among the mean thyroid weights for the 69 group x day subpopulations; and this result suggests that the fits to the data which are
obtained from the weighted analyses are at least as good as the one
obtained from the unweighted analysis in Section 3.3.1 i for which the
goodness of fit statistic is non-significant.
model
~
For this reason, the
is considered to provide an adequate characterization of the
data for prediction and statistical inference purposes via weighted
least squares methods.
Given this preliminary framework, the simplifi-
cation of the model X to the model
~
is justified for all three types
of weighted least squares in view of the non-significance (a=O.25)
of the corresponding test statistics for the model reduction hypothesis of equal second stage slopes and equal third stage slopes.
In
this sense, model fitting logic represents another type of similarity
among the weighted least squares and unweighted analysis beyond the
general similarity of results mentioned previously.
Thus, although
it is conceivable that a better fitting model can be found for the
data by either using different boundary points. additional linear
39
segments, or even certain types of non-linear segments or curves, such
model building is beyond the scope of this paper as well as the objectives of the experiment.
I
Here, segmented regression is used as
a reasonable and simplistic framework for statistically describing the
relationship between thyroid weight and days within the respective
groups in a manner consistent with the three-stage growth pattern,
which such data are thought to follow.
The previous discussion has emphasized the similarity of the
results for the models X and
analyses.
~
from the unweighted and weighted
However, some statement needs to be given concerning which
one is considered to be the most appropriate for estimation purposes.
In this regard, a compromise must be reached between sample size requirements on the one hand and variance heterogentiy representation
on the other.
As a consequence of these competing issues, the "three-
day pooled estimated variance" weighted analysis is considered to be
the most appropriate method for this application (because the asymptotic basis of the "one-day pooled estimated variance is unrealistic
with respect to the former and the unweighted and "six-day pooled
estimated variance" weighted analyses involve pooling which either
ignores or provides a less sensitive reflection of the latter).
How-
II~ addition, a considerable improvement in goodness of fit of ~F
can be achieved by including an additional parameter in the model for
the difference between the common intercept and the predicted value
for day O. Specifically, the goodness of fit statistics are 81.13,
71.39, 77.32 with D.F.=61 for the I-day, 3-day, and 6-day weighted
methods, respectively. The details of the design matrix ~; are given
in Section 3.3.3. Other details of the analysis using this model with
weighted least squares methods are not available. However, the major
changes in predicated values occurred in the first phase; predicted
values in the third phase were not altered much and the pattern of
pairwise group differences remained unchanged.
40
ever, this method only partially accounts for the variance heterogeneity
in the data; and thus, some caution should be given even to the interpretation of the results it yields for estimated standard errors and
statistical tests because of the
approxin~tions
involved.
In this
regard, the covariance matrix for its estimated parameter vector
§Wz
has the form
(15)
where ~2 is a diagonal matrix with the elements {v
} (which are
hikZ
viewed as known constants) on the main diagonal, but the estimated
"-
covariance matrix for
~Z
from the weighted least squares is
(16)
Thus, if the
{vhik2 }
are sufficiently similar to the true variances
{Vhik } that
"-
then Y~2 is a reasortable estimate of ~{fw2}.
Similar remarks apply
to the test statistics for full rank linear hypotheses of the type
H :
O
~ ~
0 which have the
for~
(18)
and to model goodness of fit statistics which have the form
(19)
where R is an orthocomplement matrix to X.
However, as indicated
previously, the valid use of the latter requires both that
41
R D
(20)
""!2
and that the sample size is large enough for Central Limit Theory to
imply that
Sl
approximately has a multivariate normal distribution
(which may not be realistic for the data under study here).
To conclude this discussion, final model predicted values
(21)
based on the "three-day pooled estimated variance" weighted analysis
are given in Tablell and the corresponding relationships are graphically
displayed in Figure 5.
of the final model
~
There, a further indication of the plausibility
can be obtained by the comparison of the respec-
tive observed and predicted values.
Moreover, the general scatter of
the data about the plotted segments does not appear to follow any
specific pattern and thus suggests that more complex models may not
provide sufficiently better characterizations of the variation in the
data to merit any attention.
3.3.3.
Maximum Likelihood Model Fitting
The weighted least squares methods discussed in Section 3.2.2
only partially take into account the variance heterogeneity in the data
because of the sample size requirements for the asymptotic framework
in terms of which they are applied.
Thus, although their results are
considered useful for previously indicated reasons, they are, at the
same time, approximations to those which might be obtained by more
sophisticated methods.
In this regard, one alternative approach for
dealing with variance heterogeneity is the formulation of models which
42
TABLE 11
PREDICTED MEAN THYROID WEIGHTS AND CORRESPONDING STANDARD
ERRORS BASED ON FINAL MODEL ~; AND THE THREE-DAY POOLED VARIANCE WEIGHTED ANALYSIS
Day
Group 1
0
5.00
(0.42)
2
Group 2
Group 3
Group 4
11.97
(0.33)
10.71
(0.33)
9.68
(0.33)
9.74
(0.33)
5
22.42
(0.48)
19.28
(0.48)
16.70
(0.48)
16.85
(0.49)
7
29.38
(0.67)
24.99
(0.67)
2L37
(0.67)
21.59
(0.68)
9
36.35
(0.89)
30.70
(0.88)
26.05
(0.89)
26.33
(0.90)
12
46.79
(1.23)
39.26
(1.22)
33.06
(1.23)
33.44
(L24)
14
50.52
(1.13)
42.99
(L12)
36.79
(1.13)
37.16
(1.14)
16
34.23
(1.11)
46.72
(1.10)
40.52
(L12)
40.89
(1.13)
19
59.84
(1.26)
52.31
(1.25)
46.11
(1.27)
46.49
(1.28)
21
63.57
(1,44)
56.04
(1.43)
49.84
(1. 45)
50.22
(1.46)
23
67.30
(1.66)
59.77
(1.66)
53.57
(1.68)
53.94
(1.68)
26
68.36
(1.40)
60.82
(1.40)
54.63
(1.42)
55.00
(1.42)
28
69.06
(1.29)
61.53
(1.29)
55.33
(1. 31)
55.70
(1. 31)
30
69.77
(1.26)
62.23
(1.26)
56.04
(1.27)
56.41
(1. 28)
33
70.82
(1. 35)
63.29
(1.35)
57.09
(1.36)
57.47
(1.37)
35
71.52
(l.50)
64.00
(1. 49)
57.80
(1.50)
58.17
(1.51)
37
72.23
(1.69)
64.70
(1.68)
58.50
(1.69)
58.88
(1. 70)
40
73.29
(2.03)
65.76
(2.03)
59.56
(2.03)
59.93
(2.04)
..
e
e
e
Figure 5
Predicted Thyroid Weights from Model !F
Based.. ~.~eighted Least Squares Analysis Using Three-Day Pooled
I'~ T-rl'rll'~-"r'T
80
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44
specify the relationship of both thyroid weight and its corresponding
variance to day and group effects.
By using maximum likelihood tech-
niques it is possible to find estimates based on a lognormal probability
structure that can be used to provide predicted cell means on the
original scale of the data.
The difficulties of heterogeneity of
variances are overcome and yet the problems which arise when predieted values are found using least squares methods on the log-data
are avoided.
The proposed model, based on ghi
= loge (Yhi),
which are
independent, homoscedastic, normally distributed random variables, is
E(g)
= ~(e-v/2
X
~),
(22)
so that
(23)
Here
1
is the identity matrix and
!£2e
and
~
denote operators which
transform a vector into a corresponding vector of natural logs or
exponentials respectively,
As mentioned above, (22) provides estimates
based on a lognormal probability structure, yet, as can be seen from
(23), predicted values can be found on the original scale.
The parameters
methods.
~
and v were estimated using maximum likelihood
The general iterative search program MAJaJIK (Kaplan and
Elston, 1972) ·is used to find estimates which maximize the log-likelihood.
Weighted least squares estimates of
~
and the overall pooled
log-transformed data variance estimate of v were used as starting
values.
The variance-covariance matrix of the estJ.mates of ~ and v
were calculated using the information matrix. (See Bradley and Gart,
1962. )
For the model (1) the parameters were estimnled and
;m
~
($21
(D·
iB:n.
wasteste,dto justify l\ll,seof
chi-square
,-
~;u '- rtJ>23 - ft32~
= fBJ'l = {~33
1£F'
'B34
i
'Sln!ce :Q(Q"tihJe lco;g-likelj!]h'o'om
s:t,atistic..aasa value of .2 • .52 (bas'ed onD d. If .),~ the
model eould be reduced to seven parameters;i. e",
D
where 1J2*
and S3* are the connnon slop"es for growth phases two and
three respectively.
reduced model is fit.
As before,
~
substitutes X in the model and the
The resulting estimates and associated sta-
tistics are included in Tables
l2~
13, and 14.
The goodness of fit statistic for
is 141.80, which is large.
!F~
given in Table 14
A plot of the fitted model to the data
suggests that a substantial portion of that large value could be accounted for if a single extra parameter were added to the model.
Instead of forcing the day 0 observations and the intercept of the
first phase line segments to be incorporated into a single estimate
So'
allow the day 0 (untreated) observations to be estimated by one
parameter, say SI' and the intercept of the four first-phase slopes
to be estimated by another parameter, say SR'
The fit of the new
model
(25)
is substantially improved over the model based on X with
46
TABLE 12
ESTIMATES OF PARAMETERS AND STANDARD ERRORS
FOR MODELS ~ AND ~* BASED ON
MAXIMUM LIKELIHOOD ESTIMATION
Model ....
X
Parameter
Estimate
S.D. t
aO
aI
aa
6.46
.37
all
a 12
3.36
Estimate
S. D. t
8.32
.51
2.55
.52
.19
4.13
.19
2.69
.16
3.28
.16
a13
2.28
.14
2.84
.15
a 14
2.33
.14
2.89
.15
1.49
.38
1.10
.36
a22
1.96
.34
1.55
.32
a2:3
1.82
.31
1.47
.29
B24
1.91
.32
1.55
.30
a31
.37
.35
.45
.33
B32
.51
.34
.58
.31
a 33
.34
.30
.41
.28
a 34
.19
.32
.28
.29
v
.0451
.0034
.0381
.0029
e21
t
Model ....
X*
TW·se estimates are based on the inverse of the information
matrix.
47
TABLE 13
ESTIMATES OF PARAMETERS AND STANDARD ERRORSt
FOR MODELS ~F AND ~;
BASED ON MAXIMUM LIKELIHOOD ESTIMATION
Model
~
Parameter
Estimate
S.D. t
eO
eI
6.46
.37
6R
Estimate
*
~
S.D.
8.32
.52
2.54
.53
t
ell
621
3.40
.14
3.98
.15
2.78
.13
3.37
.14
~31
2.28
.12
2.85
.13
2.33
1.82
.12
2.90
.13
.17
1.45
.17
6.3*
.35
.17
.43
.15
v
.0454
.0035
.0384
.0029
e 41
62*
t
Model
These estimates are based on the inverse of the information
matrix .
..•.
48
TABLE 14
LOG-LIKELIHOOD RATIO TEST STATISTICS
FOR MODELS BASED ON MAXIMUM LIKELIHOOD ESTIMATION
D.F.
Chi-squared
statistic, Q~
55
139.28
6
2.52
~
62
141.80
Goodness of fit to ,...
X*
55
81.90
6
2.89
61
84.79
3
78.58
Source of variation
1.
Goodness of fit to ,...
X
821 ... 822
B31
or:
623 ... 824
832 ." 833
Goodness of fit to
2.
:2
Goodness of fit to
*
~
I<:
83 4
49
1
'""Ill
x*
..,
:Sl
=
(26)
34l X 14
(27)
where the notation
~l'
etc. is as defined previously.
Upon reducing
the model so that there is a common slope 8 * for the second line
2
segment and a common slope 8 * for the third line segment, the design
3
matrix
*
~
reduces to
*
~
in the manner that
:s was
reduced
to~.
The
estimates of the parameters for both the reduced and the unreduced
II
two-intercept" models are also included in Tables 12 and 13.
The
test statistics, shown in part 2 of Table 14 still indicated that the
goodness of fit statistic is large (84.476 is the a=O.25 critical
value.).
It is not excessively large in view of the fact that sample
sizes in the cells are rather small so that their log-means do not
2
necessarily satisfy the asymptotic theory for the X -approximation.
Also, as can be seen from the plot of the means (Figure 1), no model
can fit the data well unless more parameters are used.
The means zig-
zag up and down in places, and any reasonable model will have a large
goodness-of-fit statistic.
Of the models fit by maximum likelihood methods,
fitting design.
*
~F
is the best
The addition of this single parameter can be justified
on both practical and technical grounds.
Practically, only one degree
50
of freedom is lost and the goodness of fit statistic is substantially reduced.
Forcing the single intercept to accommodate both the pre-treatment
and post-treatment observations seriously affects the fit of the data
in the first growth phase.
More technically, one can note that the
pre-treatment rats may have been selected or managed somewhat different1y in the sense of not being given a specific diet over time,
as were the treated animals.
In any case, the predicted values at
the end of the study are nearly the same for both models, and it is
the final phase of the study which is of interest to the researcher.
The predicted values found using
plotted in Figure 6.
*
~
are given in Table 15 and are
The residuals and standardized residuals of the
cell means from their respective predicted values are given in Table 16.
Although there are patterns of pluses and minuses in these residuals,
an
~xamination
of the plot of the cell means indicates that such pat-
terns would seem to emerge from any simple model and that any more
complex model which might minimize this problem would be difficult to
interpret and might represent an overfit model.
4.
Conclusion
Ia this paper some of the problems of analyzing a complex data
set have
be~n
discussed.
One-way analysis of variance methods were
mentioned, but they proved inadequate to detect group differences.
Two-way methods, used on relevant subsets of the dataset, did detect
the group differences which were suspected.
Furthermore, the two-way
analysis indicated that the identified phases of growth did, in fact,
have different characteristics; in particular, the first phase had
substantially more group x day interaction.
51
e-
TABLE 15
PREDICTED MEAN THYROID WEIGHTS AND CORRESPONDING
STANDARD ERRORS BASED ON MODEL ~; AND
MAXIMUM LIKELIHOOD ESTIMATION
Group
Day
1
3
4
0
8.32
(0.52)
2
10.49
(0.42)
9.28
(0.41)
8.24
(0.40)
8.34
(0.40)
5
22.42
(0.61)
19.38
(0.54)
16.78
(0.49)
17.04
(0.50)
7
30.37
(0.86)
26.12
(0.75)
22.48
(0.68)
22.84
(0.70)
9
38.32
(1.13)
32.86
(1.00)
28.18
(0.91)
28.64
(0.93)
12
50.24
(1.56)
42.96
(1. 38)
36.73
(1. 28)
37.34
(1.30)
14
53.14
(1.44)
45.85
(1.26)
39.62
(1.15)
40.23
(1.17)
16
56.03
(1.39)
48.75
(1.22)
42.51
(1.10)
43.12
(1.12)
19
60.37
(1.47)
53.08
(1.31)
46.85
(1. 22)
47.46
0..23)
21
63.26
(1.61)
55.98
(1.47)
49.74
(1. 39)
50.35
(1. 40)
23
66.15
(1. 79)
58.87
(1.68)
52.63
(1.62)
53.24
(1- 62)
67.43
(1.57)
60.14
(1. 44)
53.91
(1.37)
54.52
(1. 38)
28
68.28
(1.49)
61.00
(1.35)
54.76
(1. 26)
55.37
(1. 28)
30
69.13
(1.46)
61.85
(1.32)
55.61
(1.23)
56.22
(1. 25)
33
70.40
(1.54)
63.12
(1.40)
56.88
(1.32)
57.50
(1. 34)
35
71.25
(1. 66)
63.97
(1.53)
57.73
(1. 46)
58.35
(1. 47)
37
72.10
(1. 82)
64.82
(1. 71)
58.59
(1.64)
59.20
(1.65)
40
73.38
(2.13)
66.10
(2.03)
59.86
(1.97)
60.47
(1. 98)
26
.
0-
-
2
Figure 6
rr'r~fcn~:~Yl,~b Fill··-f:f·;+.~' ~.! ~J'~~i'~:. rU'-'!~Tl-"lf-!-=j
Predicted Thyroid Weights from Model
L.1.: ~-TI -r-I "~.;;r.~J~ '-'- ~~T'~TJ~~a~Y~.OFnr~l
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Day
e
53
,~
,
TABLE 16
RESIDUALS AND STANDARDIZED RESIDUALS OF THE CELL*
MEANS FROM THEIR PREDICTED VALUES BASED ON MODEL ~
AND MAXIMUM LIKELIHOOD ESTIMATION*
Group
Day
2
3
4
0
-0.05
-0.10
2
-0.55
-0.59
0.88
1.07
-0.32
-0.44
-0.22
-0.30
5
-0.20
-0.10
2.78
0.93
9.78
2.58
-0.62
-0.36
-1.10
-0.74
-2.18
-1.44
2.78
1.20
0.16
0.08
-0.16
-0.08
-:2.42
-0.83
5.97
2.14
-1.80
-0.71
7
9
-6.74
-1.52
2.04
0.43
-5.68
-1.49
5.19
1.60
1.18
0.32
-4.17
-1.03
9.49
2.66
16
2.07
0.42
1.27
0.30
-6.80
-1.94
-4.95
-1.32
19
2.37
0.44
21
-6.40
-1.14
-3.09
-0.53
-4.30
-0.93
0.32
0.07
-6.55
-1.58
0.56
0.13
-5.88
-1.25
-6.23
-1.25
5.19
1.00
0.05
0.01
11. 73
2.46
6.96
1.28
3.18
0.67
1.16
0.24
1.61
0.33
12
14
23
3.90
1.02
26
-3.93
-0.66
28
-1. 78
'-0.29
-1.94
-0.37
2.16
0.40
30
-3.51
-0.57
0.39
0.07
-1.97
-0.40
1.94
0.39
33
2.78
0.45
0.96
0.17
1.20
0.24
2.98
0.59
35
2.75
0.44
5.95
1.05
2.19
0.43
-4.99
-0.97
8.78
0.59
-8.96
-1.38
2.58
0.45
2.30
0.39
-10.09
-1.94
2.04
0.38
-3.20
-0.61
37 .
40
*
Top number is the residual. 1
residual
e<
1
rn; (Yi -
i
-
1i ;
Yi )/ (Yi~ )
-1.09
-0.18
bottom number is the "standardized"
,
54
Various methods of modeling were explored.
The maximum likelihood
method is perhaps the most suitable method since it simultaneously
(1) handles the problem of heterogeneity, (2) avoids the problems of
using estimated weights in least squares regression, and (3) provides
predicted values which are not on the log-scale.
Because of the apparent
suitability of the model from a theoretical standpoint, more attention
was devoted to the fine points of the model; 1. e., another parameter
was added and residuals were examined.
The computational difficulty
of the estimation might lead one to consider one of the weighted least
square regressions.
Of the various weighting schemes, the three-day
pooled variance model might be the preferable scheme, for reasons
discussed earlier.
It is particularly noteworthy that the results of all regression
analyses. and even the two-way ANOVA are essentially the same.
All
point to the same pairwise group difference at the end of the study:
the control group has the highest thyroid weights, the low dosage
group had weights in a middle range and the two high-dosage groups
have virtually the same low weights.
Furthermore, most of the re-
gression models have plots which are very nearly the same regardless
of the
d~tai1s
of the model.
plots. "look" about the same.
Only a few plots are
g~ven
but other
Notice in particular that the eight-
parameter maximum likelihood model and seven-parameter three-day pooled
variance weighted least squares model are virtually indistinguishable.
1~e
agreement would probably have been closer if eight parameters
were used in both models.
The fact that the various methods pro-
vided such similar results and conclusions regarding the end-of-study
group differences illustrates the robustness of the methods.
As noted
55
earlier, the choice of method can by and large be considered a matter
of personal taste in this case.
56
APPENDIX I
LISTING OF THE DATA
PRE-THEATMENT
OBSERVATIONS:
6.7
6.9
6.9
7.2
7.8
GROUP
DAYS
1
2
8.8
9.2
9.3
9.9
10.0
GKOUP
DAYS
3
3
IJ
34.0
49.6
60.1
6 -/ • 0
46.5
51.1
55.1
62.2
67.2
66.3
74.2
75.5
36.7
50.0
63.3
65.3
75.7
55.5
58.9
5'J.2
73.9
80.7
42.1
1+2.7
47.4
73.0
73.2
1
lJ
2
50.2
8.7
8.8
10.2
10.8
11.2
9.9
9.9
10.2
10.4
10.4
1.5
7.6
7.9
8.1
8.5
6.2
7.0
8.7
8.8
9.9
23
18.8
22.1
22.3
23.5
24.1+
15.5
16.2
16.4
19.4
26.3
13.5
14.7
14.8
16.3
19.1
12.9
13.3
14.9
16.2
17.0
26
7
29.8
32.8
33.2
36.8
25.8
26.0
28.2
28.3
36.2
16.1
18.3
23.1
23.8
31.9
18.5
19.4
22.0
23.7
29.8
28
59.9
60.8
61.0
71.9
78.9
50.8
52.2
66.6
67.7
78.5
47.1
58.1
58.6
83.1
49.1
49.7
54.5
64.1
67.5
9
36.3
46.2
50.6
59.3
21+.0
30.6
31.2
32.5
33.9
26.6
33.336.1
1+0.6
16.5
22.8
26.2
32.8
35.9
30
1+8.2
61.1
63.8
75.13
79.2
1+3.9
51.2
62.8
71.8
81.5
48.8
51.3
55.3
55.5
57.3
38.8
If8.2
54.a
72.0
77.0
12
33.8
34.3
43.2
50.9
55.3
23.7
32.0
36.7
42.4
51.6
31+.2
35.0
38.7
44. 1f
57.3
33.8
38.3
39.8
42.2
33
60.8
71.1
75.7
77.3
81.0
35.9
59.0
69.2
69.3
87.0
55.8
55.9
60.5
61.7
56.5
55.4
55.4
58.7
64.7
68.2
llJ
44.8
48.1
54.8
60.0
68,2
37.4
39.7
41.7
44.2
45.4
29.8
31.5
32.3
34.5
,36.0
43.7
44.5
48.2
52.2
60.0
35
59.2
70.6
73.5
79.9
86.8
56.9
66.6
66.7
76.0
83.4
45.0
56.8
63.2
64.6
70,0
45.9
46.6
48.4
61.9
64.0
16
52.8
54.Lf.
55.6
60.3
67.4
40.2
42.9
48.9
58.1
60.0
26.3
36.7
37.5
42,8
44.5
33.3
43.4
45.3
54.5
58.6
37
45.4
67.1
75.7
52.9
55.7
67.9
76.7
8:5.8
33.3
49.0
50.5
46.6
59.5
64.4
(,6.1
77.1
38.0
42.2
4?.7
49.9
70.7
30.7
32.6
33.6
48.7
55.9
33.6
36.1
43.5
53.1
40
31.7
51.1
58.4
59.8
77.3
44.1
52.8
59.8
62.3
62,5
43.0
49.1
49.2
51.1
59.1
39.5
41.3
46.7
49.0
2
5
19
21
52.0
56.3
(,3.8
71.0
72.2
35.8
f-5.7
89.6
101.6
36.9
51.4
1;5.8
Al.1
84.9
5~.8
60.3
61.4
92.6
51.7
5~.:;
71.5
73.7
90.6
49.1
51.7
59.4
51• • 7
58.9
61.2
61.2
-13,5
5~.~
55.2
55.3
64.6
41.7
55.2
64.6
76.0
57
APPENDIX II
NORMALITY VS. LOG-NORMALITY OF THE DATA
There are theoretical reasons which support the contention
that a log-normal model might be more suitable than a model based on
the untrasnformed data.
text.
Most of the arguments were presented in the
One further aspect of the data not discussed there regards the
suitability of the normality assumption for either the untransformed
or log-transformed data.
For a brief examination of normality the
following standardized scores were computed:
where k = 1, ... ,4; j = 2,5,. ••• , 40; and i = 1, " ' , n
jk
(= 4 or 5),
and
*
.zijk = standardized score of log(Yijk) in cell (jk).
In each case the resulting ~ =
L n jk
scores were sorted and the
j
Ko1mogorov-Smirnov statistic computed.
The resulting K-S statistics
for each of the four groups for both the untransformed and log-trans-·
formed data are as follows:
Kolmogorov-Smirnov Statistic
Group
Original Scale
Log-transformed
1
2
3
.0603
.0600
.0642
.0968
.0721
.0486
.0527
.0847
4
The critical value (a=0.05) is 0.15; thus, none of the groups
exhibits non-normality regardless of the scale of the data.
Never-
58
theless, the log-scale provides slightly smaller statistics in three
of the four groups.
Also examined were the distributions of plus and minus signs
in the residuals of (Yijk -
Yjk )
each group (k = 1, .•• ,4).
for each day (j
= 2, 5, ... , 40) and
In normality, half of the residuals should
be positive and half negative in each day-group combination.
Let (1, 4)
be interpreted to mean that in 5 observations, 1 was negative and
4 were positive.
These 5 observations constitute the data for one
day x group combination.
The frequency distribution of the signs of
the residuals in the l7x4 day-group combinations are as follows:
Frequency of
signs (minus, plus)
(1,4)
(2,3)
(3,2)
(4,1)
Untransformed
Log-transformed
1
1
7
11
7
3
Untransformed
Log-transformed
2
9
2
10
5
4
Untransformed
Log-transformed
1
1
3
9
4
8
Untransformed
Log-transformed
2
3
10
0
Group/scale
1.
2.
3.
4.
0
9
(2,2)
(3,1)
0
2
0
2
0
0
1
1
0
0
0
0
1
1
2
2
1
1
1
1
4
4
0
0
The log qata has a more symmetric distribution (as measured by signs
only). than the untransformed data in groups 3 and 4 but a less symmetric
distribution in groups 1 and 2.
59
APPENDIX III
POLYNOMIAL FIT TO THE DATA
For purposes of comparison as well as support for the contention
that a polynomial regression would not be useful for these data, the following model was considered:
(A.l)
where Yhik is the observed thyroid weight for the k-th animal in the
= i-th day so that h = 1, 2, 3, 4;
h-th group which is sacrificed on the x.
1
i
= 0,2,
5, •.• ,40; and k
= 1,2,
..• ,
or 4 for i > 0, as specified in Table 1).
~i
(with n
lO
= 10
The parameters
SZh are the intercept and coefficients for the h-th group.
and
~,
~i =
5
Slh' and
In this
regard, the use of ~ in this model was based on inspection of Figure 1
as it reflected the general pattern of initially rapid growth which eventually slowed to little, if any, growth.
The variable xi was included
for its potential ability to improve goodness of fit.
With this back-
ground in mind, the model (A.l) was investigated for each group
separately with the day 0 data being taken into account for each.
The details of these statistical analyses are given in Table 17 and the
plots of the pr~dicted values are given in Figure 7.
All of the goodness
of fit statistics indicate that the model (A.l) does not adequately
describe the relationship under study.
As can be seen from the graphs,
the rapid rate of growth during the early days of study causes prediction
during the latter days to be relatively poor.
Thus, the issues raised
in Section 3.3 as arguments against using the model (A.l) are appropriate concerns.
TABLE 17
ESTIMATED PARAMETERS AND TEST STATISTICS
FOR THE MODEL E(y)
Parameters
(S.D.)
Group
a
B1
1
5.70
11.37
(0.63)
-
2
4.85
8.27
(0.67)
0.42
(0.23)
3
2.98
9.47
B2
(0.89)
4
3.33
9.50
(0.89)
**
-
= a + B1/X + B2x
F statistic
f,or regression
D.F.
F statistic for
lack of fit
D.F.
**
**
(1,91)
18.3
. **
176.8
(2,92)
18.3 **
328.1 **
(1,90)
27.4
**
(16,74)
32502 **
(1,89)
22.3
**
(16,73)
326.7
(16,75)
(15~ 77)
indicates significance at a s .01.
0'\
o
e
e
e
e
e·
e
Figure 7
Predicted Thyroid Weights from Polynomial Models
SG
70
/'
~--4
60-
,
j
I
50·r
....
.
40 -t--+-+
30-
....0'1
Day
62
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~,