ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Split-plot experiments: main plots are independent, subplot measures are correlated since they are taken within
the same plot.
Experiments repeated across years
 Each experimental unit is measured repeatedly across several year.
 Successive measures on the same unit may be correlated.
 Interest in long-term effect of treatments
 Box: assumption is that every pair of subplot times has the same correlation. Randomization of subplot
factors validates this assumption.
Example (Snedecor and Cochran, 1989. Statistical Methods)
Experimental data is from a study on the effect of four cutting treatments on asparagus yield. Cutting began at
Year 2 after planting. There were four block, with 4 plots each. One plot within each block was cut until June 1
in each year; others to June 15, July 1 and July 15. Yields shown are the weights cut to June 1 for each plot on
years 3, 4, 5, and 6. Weight (oz) is a measure of vigor, and objective is to study the relative effectiveness of the
harvesting plans (cuttings).
4  4  4 Factorial Experiment in a RCBD
Experimental Design:
 Blocking factor: Block, Random Effects, j = 1, 2, 3, 4
Treatments:
 Cutting, Fixed Effect Factor , i = 1, 2, 3, 4
 Year,
Fixed Effect Factor, k = 1, 2, 3, 4
DATA: WEIGHT_HARVEST;
BLOCK
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1.
2.
3.
4.
YEAR
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
CUTTING
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
WEIGHT_HARVEST
230
212
183
148
324
415
320
246
512
584
456
304
399
386
255
144
216
190
186
126
317
296
295
201
448
471
387
289
361
280
187
83
BLOCK
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
YEAR
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
CUTTING
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
jun01
jun15
jul01
jul15
WEIGHT_HARVEST
219
151
177
107
357
278
298
192
496
399
427
271
344
254
239
90
200
150
209
168
362
336
328
226
540
485
462
312
381
279
244
168
The linear component of the regression of yield (WEIGHT_HARVEST) on years is used to
analyze time trend and the effect of cutting on this trend
Calculation, on each plot, of the linear effect of time is done through the contrast
Year _ Linearij   3  Weight _ Harvestij1  1 Weight _ Harvestij 2  1 Weight _ Harvestij 3  3  Weight _ Harvestij 4 
20
YEAR_LINEAR measures the average improvement in yield per year.
Alternatively, the calculated slope from the regression of yield on time, for each plot, is used to
analyze the linear trend of time and the effect of cutting on these slopes.
Repeated Measures within each plot, taken at yearly intervals are analyzed in PROC MIXED. We
assume initially that pattern of correlation between timepoints is the same for each plot.
Thursday October 10, 2008 Homework 7
1
ST 524
Homework 7
5.
NCSU - Fall 2008
Due: 11/11/08
Correlation pattern among repeated measures on time is modeled through type = UN, with no
predetermined assumption about correlation between any pair of points in time,
corr  yijk , yijk '   kk '
6.
Correlation pattern among repeated measures on time is modeled through type = CS, which indicates
that any pair of measures on time within the same plot will be equally correlated,
corr  yijk , yijk '   
7.
Correlation pattern among repeated measures on time is modeled through type = AR(1), which
indicates that correlation between a pair of measures on time depends on their distance on
time,
1.


corr yijk , yij  k h    h
Questions
What pattern of correlation best describes the time effect?
-2ResLogLik
AIC
Variance
474.5
480.5
Components
UN1
458.3
478.3
CS1
474.5
480.5
AR(1)1
471.0
477.0
AR(1)2
470.9
478.9
1
Random effects: Block Residual
2
Random effects: Block Block*Cutting Residual
Best pattern: AR(1) with random effects: Block and Residual
Linear Additive Model

AICC
481.1
BIC
478.7
484.3
481.1
477.5
479.9
472.2
478.7
475.1
476.5
Model, RCBD: yijk    bk   i  dik   j   ij  eijk

i
i
0,

j
  
 0,
i, j
j

dik ~ N  0,  d2  eijk ~ N 0,  e2

ij
2
 0 , bk ~ N  0,  b  ,
and
corr  eijk , eijk '   
k k '
, where dik represents
Error (a) and eijk represents Error (b) in a split-plot design. Best model, with
lowest value for AIC and AICC, indicates that the residual effects follow an
autoregressive process of order 1, AR(1), eijk     eij k 1 +vijk , and vijk ~  0,  v2 
The Mixed Procedure
Model Information
Data Set
WORK.A
Dependent Variable
Covariance Structures
Subject Effect
WEIGHT_HARVEST
Variance Components, Autoregressive
BLOCK*CUTTING
Estimation Method
REML
Residual Variance Method
Profile
Fixed Effects SE Method
Model-Based
Degrees of Freedom Method
Containment
Class Level Information
Class
Thursday October 10, 2008 Homework 7
Levels
Values
2
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Class Level Information
Class
Levels
Values
BLOCK
4
1 2 3 4
YEAR
4
0 1 2 3
CUTTING
4
jul01 jul15 jun01 jun15
Dimensions
Covariance Parameters
3
Columns in X
25
Columns in Z
4
Subjects
1
Max Obs Per Subject
64
Number of Observations
Number of Observations Read
64
Number of Observations Used
64
Number of Observations Not Used
0
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
BLOCK
417.31
AR(1)
BLOCK*CUTTING
0.6190
Residual
884.00
Fit Statistics
-2 Res Log Likelihood
471.0
AIC (smaller is better)
477.0
AICC (smaller is better)
477.5
BIC (smaller is better)
475.1
Type 3 Tests of Fixed Effects
Effect
Num DF
Den DF
F Value
Pr > F
CUTTING
3
45
37.45
<.0001
YEAR
3
45
431.64
<.0001
YEAR*CUTTING
9
45
8.46
<.0001
2.
Explain how increasing the cutting time (from june01 to July15) affects the response expressed as the
slope of the regression of yield on Year.
Linear slopes for each plot are from regression of harvest weight on year
Data
Thursday October 10, 2008 Homework 7
3
ST 524
Homework 7
Obs
NCSU - Fall 2008
Due: 11/11/08
BLOCK
CUTTING
_MODEL_
_TYPE_
_DEPVAR_
_RMSE_
Intercept
YEAR
WEIGHT_HARVEST
1
1
jul01
MODEL1
PARMS
WEIGHT_HARVEST
130.778
250.7
35.2
-1
2
1
jul15
MODEL1
PARMS
WEIGHT_HARVEST
95.460
203.6
4.6
-1
3
1
jun01
MODEL1
PARMS
WEIGHT_HARVEST
96.212
262.0
69.5
-1
4
1
jun15
MODEL1
PARMS
WEIGHT_HARVEST
151.236
295.6
69.1
-1
5
2
jul01
MODEL1
PARMS
WEIGHT_HARVEST
117.583
249.5
9.5
-1
6
2
jul15
MODEL1
PARMS
WEIGHT_HARVEST
110.573
180.9
-4.1
-1
7
2
jun01
MODEL1
PARMS
WEIGHT_HARVEST
77.173
250.6
56.6
-1
8
2
jun15
MODEL1
PARMS
WEIGHT_HARVEST
125.526
242.5
44.5
-1
9
3
jul01
MODEL1
PARMS
WEIGHT_HARVEST
120.730
238.0
31.5
-1
10
3
jul15
MODEL1
PARMS
WEIGHT_HARVEST
102.261
160.8
2.8
-1
11
3
jun01
MODEL1
PARMS
WEIGHT_HARVEST
112.446
276.9
51.4
-1
12
3
jun15
MODEL1
PARMS
WEIGHT_HARVEST
104.585
206.0
43.0
-1
13
4
jul01
MODEL1
PARMS
WEIGHT_HARVEST
132.527
274.9
23.9
-1
14
4
jul15
MODEL1
PARMS
WEIGHT_HARVEST
82.247
205.6
8.6
-1
15
4
jun01
MODEL1
PARMS
WEIGHT_HARVEST
126.473
262.6
72.1
-1
16
4
jun15
MODEL1
PARMS
WEIGHT_HARVEST
147.432
232.1
53.6
-1
The Mixed Procedure

Model, RCBD: sij    b j   i  eij , where sij is the linear slope for the change of harvest
weight over years of ith cutting in jth block, i = 1,2,3,4, j=1,2,3,4.
2
2
 i  0 , b j ~ N  0,  b  eij ~ N  0,  e 
i
Model Information
Data Set
WORK.REGDS
Dependent Variable
Covariance Structure
Estimation Method
SLOPE_YR
Variance Components
REML
Residual Variance Method
Profile
Fixed Effects SE Method
Model-Based
Degrees of Freedom Method
Containment
Class Level Information
Class
Levels
Values
BLOCK
4
1 2 3 4
CUTTING
4
jul01 jul15 jun01 jun15
Dimensions
Covariance Parameters
2
Columns in X
5
Thursday October 10, 2008 Homework 7
4
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Dimensions
Columns in Z
4
Subjects
1
Max Obs Per Subject
16
Number of Observations
Number of Observations Read
16
Number of Observations Used
16
Number of Observations Not Used
0
Covariance Parameter Estimates
Cov Parm
Estimate
BLOCK
50.5185
Residual
49.6703
Fit Statistics
-2 Res Log Likelihood
91.3
AIC (smaller is better)
95.3
AICC (smaller is better)
96.7
BIC (smaller is better)
94.1
Type 3 Tests of Fixed Effects
Effect
Num DF
Den DF
F Value
Pr > F
CUTTING
3
9
58.56
<.0001
Orthogonal contrasts to analyze linear, quadratic and deviations-from-quadratic effects
on linear slope over years for harvest weight
Contrasts
Label
Num DF
Den DF
F Value
Pr > F
cutting linear
1
9
170.54
<.0001
cutting quad
1
9
3.00
0.1175
cutting Devquad
1
9
2.16
0.1759
Least Squares Means
Effect
CUTTING
Estimate
Standard Error
DF
t Value
Pr > |t|
CUTTING
jul01
25.0250
5.0047
9
5.00
0.0007
CUTTING
jul15
2.9750
5.0047
9
0.59
0.5669
CUTTING
jun01
62.4000
5.0047
9
12.47
<.0001
CUTTING
jun15
52.5500
5.0047
9
10.50
<.0001
Differences of Least Squares Means
Thursday October 10, 2008 Homework 7
5
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Effect
CUTTING
_CUTTING
Estimate
Standard Error
DF
t Value
Pr > |t|
CUTTING
jul01
jul15
22.0500
4.9835
9
4.42
CUTTING
jul01
jun01
-37.3750
4.9835
9
CUTTING
jul01
jun15
-27.5250
4.9835
CUTTING
jul15
jun01
-59.4250
CUTTING
jul15
jun15
CUTTING
jun01
jun15
Model:
Adjustment
Adj P
0.0017
Tukey-Kramer
0.0074
-7.50
<.0001
Tukey-Kramer
0.0002
9
-5.52
0.0004
Tukey-Kramer
0.0017
4.9835
9
-11.92
<.0001
Tukey-Kramer
<.0001
-49.5750
4.9835
9
-9.95
<.0001
Tukey-Kramer
<.0001
9.8500
4.9835
9
1.98
0.0795
Tukey-Kramer
0.2649
sij  o  bk  1C i 2C 2i 3C 3i eij , where Ci represent the cutting date interval
H o : 1  0 , p-value <0.001, reject Ho
Linear effect of cutting
Quadratic effect of cutting H o :  2  0 , p-value = 0.1175, do not reject Ho
H o :  3  0 , p-value =0.1759, do not reject Ho
Dev-from-linear effect of cutting
There is a significant linear effect of Cutting (cutting interval) on the linear slope of weight at harvest over years.
As the cutting interval increases from June 01 to july 15, the linear slope declines. No differences were found
between linear slopes for cutting intervals June 01 and June 15.
sij  o  bk  1C i eij
3.
Conclusions. Use  = 0.05. Indicate supporting statistical evidence to above Claims when writing up
your conclusions.
i. Claim: Prolonged cutting decreased the vigor.
From Table for AR(1) fitting of residual effects on time
Least Squares Means
Effect
CUTTING
CUTTING
YEAR
Estimate
Standard Error
DF
t Value
Pr > |t|
jun01
356.62
15.4463
45
23.09
<.0001
CUTTING
jun15
322.87
15.4463
45
20.90
<.0001
CUTTING
jul01
290.81
15.4463
45
18.83
<.0001
CUTTING
jul15
192.19
15.4463
45
12.44
<.0001
Cutting LSMean for weight at harvest decline as the cutting frequency increases, 356.62 when cutting
until June 01 to 192.19 when cutting until July 15.
ii. Claim: Annual improvement is greatest for June-1 cutting, and declines linearly with later
cutting times
Linear effect of cutting on linear slopes over time is significant (P<0.0001).
From Tables Least Squares Means and Differences of Least Squares Means for linear Slopes over
time (Yr_Slope)
Effect
CUTTING
Estimate
Standard Error
sign
CUTTING
jun01
62.4000
5.0047
a
CUTTING
jun15
52.5500
5.0047
a
CUTTING
jul01
25.0250
5.0047
CUTTING
jul15
2.9750
5.0047
b
c
Linear slopes changes over time indicates that as the cutting frequency increases, the linear
increments over time decreases from 62.4 for Jun 01 to 2.975 for Jul 15
iii. Claim: Each additional two-week of cutting decreases the annual improvement in yield up to June
1 by the same amount.
Thursday October 10, 2008 Homework 7
6
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Linear effect of cutting on linear slopes over time is significant (P<0.0001). while nor quadratic or dev
from quadratic effects on linear slopes over time are significant (P=0.1175, and P=0.1759 respectively).
Conclusions
Analyzing the effects of cutting on the yearly response, through the use of contrasts, we
have the following,
Random Effects are Block and Modeling residuals as an AR(1) process on time
The Mixed Procedure
Model Information
Data Set
WORK.A
Dependent Variable
WEIGHT_HARVEST
Covariance Structures
Variance Components, Autoregressive
Subject Effects
BLOCK, BLOCK*CUTTING
Estimation Method
REML
Residual Variance Method
Profile
Fixed Effects SE Method
Model-Based
Degrees of Freedom Method
Satterthwaite
Class Level Information
Class
Levels
Values
BLOCK
4
1 2 3 4
YEAR
4
0 1 2 3
CUTTING
4
jul01 jul15 jun01 jun15
Dimensions
Covariance Parameters
3
Columns in X
25
Columns in Z Per Subject
1
Subjects
4
Max Obs Per Subject
16
Number of Observations
Number of Observations Read
64
Number of Observations Used
64
Number of Observations Not Used
0
Estimated R Correlation Matrix for BLOCK*CUTTING
1 jul01
Row
1
Col1
1.0000
Thursday October 10, 2008 Homework 7
Col2
0.6190
Col3
0.3831
Col4
0.2371
7
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Estimated R Correlation Matrix for BLOCK*CUTTING
1 jul01
Row
Col1
Col2
Col3
Col4
2
0.6190
1.0000
0.6190
0.3831
3
0.3831
0.6190
1.0000
0.6190
4
0.2371
0.3831
0.6190
1.0000
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Intercept
BLOCK
417.31
AR(1)
BLOCK*CUTTING
0.6190
Residual
884.00
Fit Statistics
-2 Res Log Likelihood
471.0
AIC (smaller is better)
477.0
AICC (smaller is better)
477.5
BIC (smaller is better)
475.1
Type 3 Tests of Fixed Effects
Effect
Num DF
Den DF
F Value
Pr > F
CUTTING
3
11.2
37.45
<.0001
YEAR
3
34.8
431.64
<.0001
YEAR*CUTTING
9
34.8
8.46
<.0001
Estimates – lsmeans comparisons
Estimates
Label
Estimate
Standard Error
DF
t Value
Pr > |t|
cutting linear
-5.2537
0.5182
11.2
-10.14
<.0001
cutting quad
-3.2438
1.1587
11.2
-2.80
0.0170
cutting Devquad
-0.6825
0.5182
11.2
-1.32
0.2142
year linear
3.5738
0.3094
44.2
11.55
<.0001
year quad
-14.5875
0.4851
33.9
-30.07
<.0001
year Devquad
-3.0838
0.1752
27.3
-17.60
<.0001
Contrasts
Label
Num DF
Den DF
F Value
Pr > F
year linear
1
44.2
133.43
<.0001
year quad
1
33.9
904.15
<.0001
year Devquad
1
27.3
309.88
<.0001
Thursday October 10, 2008 Homework 7
8
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Contrasts
Label
Num DF
Den DF
F Value
Pr > F
cutting linear
1
11.2
102.79
<.0001
cutting quad
1
11.2
7.84
0.0170
cutting Devquad
1
11.2
1.73
0.2142
cutting linear * year linear
1
44.2
55.31
<.0001
cutting linear * year quad
1
33.9
0.04
0.8503
cutting linear * year devq
1
27.3
2.14
0.1553
cutting quad * year linear
1
44.2
0.97
0.3296
cutting quad * year quad
1
33.9
17.10
0.0002
cutting quad * year devq
1
27.3
1.69
0.2049
cutting devq * year linear
1
44.2
0.70
0.4073
cutting devq * year quad
1
33.9
0.36
0.5530
cutting devq * year devq
1
27.3
0.07
0.7980
year linear in cutting Jul01
1
44.2
16.36
0.0002
year linear in cutting Jul15
1
44.2
0.23
0.6330
year linear in cutting Jun01
1
44.2
101.70
<.0001
year linear in cutting Jun15
1
44.2
72.13
<.0001
year quad in cutting Jul01
1
33.9
277.48
<.0001
year quad in cutting Jul15
1
33.9
168.30
<.0001
year quad in cutting Jun01
1
33.9
167.97
<.0001
year quad in cutting Jun15
1
33.9
307.89
<.0001
Residual Checking
Thursday October 10, 2008 Homework 7
9
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Least Squares Means
Effect
CUTTING
CUTTING
YEAR
Estimate
Standard Error
jul01
290.81
15.4463
CUTTING
jul15
192.19
CUTTING
jun01
CUTTING
jun15
DF
t Value
Pr > |t|
8.16
18.83
<.0001
15.4463
8.16
12.44
<.0001
356.63
15.4463
8.16
23.09
<.0001
322.88
15.4463
8.16
20.90
<.0001
YEAR
0
179.50
12.6324
4.07
14.21
0.0001
YEAR
1
299.44
12.6324
4.07
23.70
<.0001
YEAR
2
427.69
12.6324
4.07
33.86
<.0001
YEAR
3
255.88
12.6324
4.07
20.26
<.0001
YEAR*CUTTING
jul01
0
188.75
18.0368
14.3
10.46
<.0001
YEAR*CUTTING
jul15
0
137.25
18.0368
14.3
7.61
<.0001
YEAR*CUTTING
jun01
0
216.25
18.0368
14.3
11.99
<.0001
YEAR*CUTTING
jun15
0
175.75
18.0368
14.3
9.74
<.0001
YEAR*CUTTING
jul01
1
310.25
18.0368
14.3
17.20
<.0001
YEAR*CUTTING
jul15
1
216.25
18.0368
14.3
11.99
<.0001
YEAR*CUTTING
jun01
1
340.00
18.0368
14.3
18.85
<.0001
YEAR*CUTTING
jun15
1
331.25
18.0368
14.3
18.37
<.0001
YEAR*CUTTING
jul01
2
433.00
18.0368
14.3
24.01
<.0001
YEAR*CUTTING
jul15
2
294.00
18.0368
14.3
16.30
<.0001
YEAR*CUTTING
jun01
2
499.00
18.0368
14.3
27.67
<.0001
YEAR*CUTTING
jun15
2
484.75
18.0368
14.3
26.88
<.0001
Thursday October 10, 2008 Homework 7
10
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Least Squares Means
Effect
CUTTING
YEAR
Estimate
Standard Error
YEAR*CUTTING
jul01
3
231.25
18.0368
YEAR*CUTTING
jul15
3
121.25
YEAR*CUTTING
jun01
3
YEAR*CUTTING
jun15
3
DF
t Value
Pr > |t|
14.3
12.82
<.0001
18.0368
14.3
6.72
<.0001
371.25
18.0368
14.3
20.58
<.0001
299.75
18.0368
14.3
16.62
<.0001
Simple effects analysis
year linear in cutting Jul01
25.0250
6.1876
44.2
4.04
0.0002
year linear in cutting Jul15
2.9750
6.1876
44.2
0.48
0.6330
year linear in cutting Jun01
62.4000
6.1876
44.2
10.08
<.0001
year linear in cutting Jun15
52.5500
6.1876
44.2
8.49
<.0001
year quad in cutting Jul01
-80.8125
4.8513
33.9
-16.66
<.0001
year quad in cutting Jul15
-62.9375
4.8513
33.9
-12.97
<.0001
year quad in cutting Jun01
-62.8750
4.8513
33.9
-12.96
<.0001
year quad in cutting Jun15
-85.1250
4.8513
33.9
-17.55
<.0001
lsmeans
Obs
Effect
YEAR
CUTTING
Estimate
StdErr
DF
tValue
Probt
cdate
1
YEAR*CUTTING
0
jun01
216.25
18.0368
14.3
11.99
<.0001
06/01/60
2
YEAR*CUTTING
1
jun01
340.00
18.0368
14.3
18.85
<.0001
06/01/60
3
YEAR*CUTTING
2
jun01
499.00
18.0368
14.3
27.67
<.0001
06/01/60
4
YEAR*CUTTING
3
jun01
371.25
18.0368
14.3
20.58
<.0001
06/01/60
5
YEAR*CUTTING
0
jun15
175.75
18.0368
14.3
9.74
<.0001
06/15/60
6
YEAR*CUTTING
1
jun15
331.25
18.0368
14.3
18.37
<.0001
06/15/60
7
YEAR*CUTTING
2
jun15
484.75
18.0368
14.3
26.88
<.0001
06/15/60
Thursday October 10, 2008 Homework 7
11
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Obs
Effect
YEAR
CUTTING
Estimate
StdErr
DF
tValue
Probt
cdate
8
YEAR*CUTTING
3
jun15
299.75
18.0368
14.3
16.62
<.0001
06/15/60
9
YEAR*CUTTING
0
jul01
188.75
18.0368
14.3
10.46
<.0001
07/01/60
10
YEAR*CUTTING
1
jul01
310.25
18.0368
14.3
17.20
<.0001
07/01/60
11
YEAR*CUTTING
2
jul01
433.00
18.0368
14.3
24.01
<.0001
07/01/60
12
YEAR*CUTTING
3
jul01
231.25
18.0368
14.3
12.82
<.0001
07/01/60
13
YEAR*CUTTING
0
jul15
137.25
18.0368
14.3
7.61
<.0001
07/15/60
14
YEAR*CUTTING
1
jul15
216.25
18.0368
14.3
11.99
<.0001
07/15/60
15
YEAR*CUTTING
2
jul15
294.00
18.0368
14.3
16.30
<.0001
07/15/60
16
YEAR*CUTTING
3
jul15
121.25
18.0368
14.3
6.72
<.0001
07/15/60
Contrast Estimates for linear combination of lsmeans
cutting linear * year linear
-10.2900
1.3836
44.2
-7.44
<.0001
cutting linear * year quad
0.4125
2.1696
33.9
0.19
0.8503
cutting linear * year devq
1.1450
0.7834
27.3
1.46
0.1553
cutting quad * year linear
-3.0500
3.0938
44.2
-0.99
0.3296
cutting quad * year quad
20.0625
4.8513
33.9
4.14
0.0002
cutting quad * year devq
2.2750
1.7518
27.3
1.30
0.2049
cutting devq * year linear
1.1575
1.3836
44.2
0.84
0.4073
cutting devq * year quad
-1.3000
2.1696
33.9
-0.60
0.5530
cutting devq * year devq
0.2025
0.7834
27.3
0.26
0.7980
Quadratic effect of Cutting on weight at harvest is significant (0.0170),
Interaction effects cutting linear * year linear (p<0.0001)
and cutting quad * year quad (p=0.002) are
significant. Weight at harvest show a convex
quadratic response over time that depends on the
cutting interval. For cutting interval (until)
July15, response on time is the lowest, while for
June01 response is the highest, with no significant
difference with respect June 15. Cutting interval
Jl01 show the second lowest response, being
significant from all others on average of years.
There is a significant as the cutting interval increases, the weight at
harvest decreases significantly. There is a significant linear trend over
years with cutting
Thursday October 10, 2008 Homework 7
12
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
2. Strip-Split block
Referencia: Example 7.6.5 A Split Strip-plot Experiment for Soybean Yield
(Schabenberger, O. and F. Pierce, Contemporary Statistical Models for the Plant and Soil Sciences. 2001)
Please refer to handout with above example and to results from statistical analysis of data to support the
following conclusions. You should make reference to relevant parts of the SAS output, indicating p-values.
Frame the answer as the results section in a scientific article.
This exercise looks to the use of Slice option on LSMEANS statement and Pairwise Mean
Comparisons to reach similar conclusions as the one summarized in the example 7.6.5
Conclusions
1. Yield responded to soybean population in a quadratic fashion.
2. Cultivars differed significantly, but no interactions between cultivar and population treatments
were evident,
3. There were no significant differences between row spacing levels.
4. Averaged across population densities, only variety AG3601 shows a significant yield difference
among the row spacing levels.
5. Only for cultivar AG3601 is row spacing of importance for a given population density.
6. For Cultivar AG3601 there is no (row) spacing effect at 60,000 plants per acre, but there are
significant effects for all greater population densities
7. For the other cultivars the row spacing effects are absent with two exceptions: AG4601 and
AG4701 at 120,000 plants per acre.
8. At 9-inch spacing there are significant differences among the cultivars at any population density.
9. For the 18-inch row spacing cultivar effects are mostly absent.
10. Yield is a linearly increasing function of population density for AG3701.
Optional (Bonus points)
11. Cultivar AG4601 shows a slight cubic effect in addition to a linear term
12. AG4701 shows polynomial terms up to the third order.
13. For cultivar AG3601, at 9-inch row spacing, yield depends on population density in quadratic
fashion. At 18-inch row spacing, yield is not responsive to changes in the population density.
Note. Pairwise mean comparisons are at alpha = 0.05 (default value). What is (are) the possible
consequence(s) in the use of this alpha level?
Conclusions – pvalues – statistical evidence
1. Yield responded to soybean population in a quadratic fashion.
pairwise mean comparison for tpop main-effect lsmeans
Effect=tpop Method=LSD(P<.05) Set=1
Obs
cultivar
rowspace
tpop
Estimate
Standard Error
Letter
Group
1
_
60
20.2542
1.0529
D
2
_
120
23.8877
1.0529
C
3
_
180
25.2194
1.0615
BC
4
_
240
26.5046
1.0615
B
5
_
300
28.1906
1.0490
A
As the size of population increases, the increment on yield mean tends to be smaller, 3.6365, 1.3317, 1.2852, 1.686
Thursday October 10, 2008 Homework 7
13
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
2. Cultivars differed significantly, but no interactions between cultivar and population treatments were
evident,
Type2 Test of Fixed Effects, p= 0.2749 for Cultivar*TPOP
3. There were no significant differences between row spacing levels.
Type2 Test of Fixed Effects, p= 0.0795 for Rowspace
4. Averaged across population densities, only variety AG3601 shows a significant yield difference
among the row spacing levels.
Slice =cultivar for lsmeans Cultivar * rowspace, for Cultivar= AG3601, p= 0.0008, indicating that
rowspace effects are significant for this cultivar, lsmean for 9” is 30.1667, while for 18” is 25.5774.
5. Only for cultivar AG3601 is row spacing of importance for a given population density.
Except for tpop=60, row spacing is significant within each population density
cultiva*rowspac*tpop
AG3601
60
1
54.4
0.02
0.8804
cultiva*rowspac*tpop
AG3601
120
1
45.7
7.66
0.0081
cultiva*rowspac*tpop
AG3601
180
1
45.7
11.59
0.0014
cultiva*rowspac*tpop
AG3601
240
1
51.6
13.19
0.0006
cultiva*rowspac*tpop
AG3601
300
1
45.7
14.51
0.0004
6. For Cultivar AG3601 there is no (row) spacing effect at 60,000 plants per acre, but there are
significant effects for all greater population densities
See above. For Cultivar AG3601, row spacing is significant within each population density, except for
tpop=60 (p=0.8804) (see Tukey’s test for row spacing when cultivar is AG3601 at each population
density.)
Effect=cultiva*rowspac*tpop Method=LSD(P<.05) Set=1
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
1
AG3601
9
60
24.4000
1.6645
A
2
AG3601
18
60
24.1336
1.8134
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
3
AG3601
9
120
28.2250
1.6645
A
4
AG3601
18
120
23.7750
1.6645
B
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
5
AG3601
9
180
31.6000
1.6645
A
6
AG3601
18
180
26.1250
1.6645
B
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
7
AG3601
9
240
33.8084
1.8376
A
8
AG3601
18
240
27.1784
1.8380
B
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
AG3601
9
300
32.8000
1.6645
A
Obs
Obs
Obs
Obs
Obs
9
Thursday October 10, 2008 Homework 7
14
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
AG3601
18
300
26.6750
1.6645
B
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
11
AG3701
9
60
20.4250
1.6645
A
12
AG3701
18
60
19.6500
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
13
AG3701
9
120
22.1000
1.6645
A
14
AG3701
18
120
22.4515
1.8134
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
15
AG3701
9
180
23.1669
1.8377
A
16
AG3701
18
180
21.4883
1.8380
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
17
AG3701
9
240
24.7500
1.6645
A
18
AG3701
18
240
26.4500
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
19
AG3701
9
300
26.5000
1.6645
A
20
AG3701
18
300
27.7250
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
21
AG4601
9
60
19.0000
1.6645
A
22
AG4601
18
60
17.6000
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
23
AG4601
9
120
20.8750
1.6645
B
24
AG4601
18
120
25.9250
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
25
AG4601
9
180
23.6000
1.6645
A
26
AG4601
18
180
22.7250
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
27
AG4601
9
240
23.4750
1.6645
A
28
AG4601
18
240
25.7500
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
AG4601
9
300
27.0750
1.6645
A
Obs
10
Obs
Obs
Obs
Obs
Obs
Obs
Obs
Obs
Obs
Obs
29
Thursday October 10, 2008 Homework 7
15
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Obs
30
Obs
cultivar
rowspace
tpop
Estimate
Standard Error
Letter Group
AG4601
18
300
27.7500
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
31
AG4701
9
60
18.6000
1.6645
A
32
AG4701
18
60
18.2250
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
33
AG4701
9
120
25.5250
1.6645
A
34
AG4701
18
120
22.2250
1.6645
B
cultivar
rowspace
tpop
Estimate
Standard Error
35
AG4701
9
180
27.1500
1.6645
A
36
AG4701
18
180
25.9000
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
37
AG4701
9
240
25.4750
1.6645
A
38
AG4701
18
240
25.1500
1.6645
A
cultivar
rowspace
tpop
Estimate
Standard Error
39
AG4701
9
300
27.5000
1.6645
A
40
AG4701
18
300
29.5000
1.6645
A
Obs
Obs
Obs
obs
Letter Group
Letter Group
Letter Group
Letter Group
Letter Group
7. For the other cultivars the row spacing effects are absent with two exceptions: AG4601 and
AG4701 at 120,000 plants per acre.
cultiva*rowspac*tpop
AG3701
60
1
45.7
0.23
0.6321
cultiva*rowspac*tpop
AG3701
120
1
54.4
0.04
0.8426
cultiva*rowspac*tpop
AG3701
180
1
51.6
0.85
0.3622
cultiva*rowspac*tpop
AG3701
240
1
45.7
1.12
0.2960
cultiva*rowspac*tpop
AG3701
300
1
45.7
0.58
0.4501
Thursday October 10, 2008 Homework 7
16
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
cultiva*rowspac*tpop
AG4601
60
1
45.7
0.76
0.3885
cultiva*rowspac*tpop
AG4601
120
1
45.7
9.86
0.0030
cultiva*rowspac*tpop
AG4601
180
1
45.7
0.30
0.5890
cultiva*rowspac*tpop
AG4601
240
1
45.7
2.00
0.1639
cultiva*rowspac*tpop
AG4601
300
1
45.7
0.18
0.6766
cultiva*rowspac*tpop
AG4701
60
1
45.7
0.05
0.8166
cultiva*rowspac*tpop
AG4701
120
1
45.7
4.21
0.0459
cultiva*rowspac*tpop
AG4701
180
1
45.7
0.60
0.4410
cultiva*rowspac*tpop
AG4701
240
1
45.7
0.04
0.8407
cultiva*rowspac*tpop
AG4701
300
1
45.7
1.55
0.2199
8. At 9-inch spacing there are significant differences among the cultivars at any population density.
cultiva*rowspac*tpop
9
60
3
76.5
3.49
0.0198
cultiva*rowspac*tpop
9
120
3
76.5
5.54
0.0017
cultiva*rowspac*tpop
9
180
3
79.2
7.19
0.0003
cultiva*rowspac*tpop
9
240
3
79.1
9.00
<.0001
cultiva*rowspac*tpop
9
300
3
76.5
4.23
0.0081
9. For the 18-inch row spacing cultivar effects are mostly absent.
cultiva*rowspac*tpop
18
60
3
80.1
3.68
0.0154
cultiva*rowspac*tpop
18
120
3
80.3
1.39
0.2530
cultiva*rowspac*tpop
18
180
3
79.2
2.39
0.0750
cultiva*rowspac*tpop
18
240
3
79.1
0.34
0.7982
cultiva*rowspac*tpop
18
300
3
76.5
0.68
0.5660
For a row spacing of 18”, there are significant differences between cultivars only at tpop=60 (p=0.0154),
for tpop greater than 60, p values range from 0.0750 to 0.7982, indicating non significant differences
between row spacing effects.
10. Yield is a linearly increasing function of population density for AG3701.
There is a significant tpop effect (p<0.001) and although there is not a significant cultivar*tpop effect
(p=0.2049), the significant cultivar*rowspac*tpop (p=0.0100) indicates that the effect of tpop within each
cultivar is not the same. In the case of AG3701, tpop is significant (p=0.0115) for spacing 9” and highly
significant (p<.0001) for spacing 18”, which is reflected on the positive increment for yield as a function of
tpop at 9” : 1.675, 1.0669, 1.531, 1.75, when tpop increases from 60 to 120, 120 to 180, 180 to 240, and 240 to
300 respectively. For 18” these increments are more variable, 2.8015, -0.9632, 4.9617, 1.275, when tpop
increases from 60 to 120, 120 to 180, 180 to 240, and 240 to 300 respectively.
cultiva*rowspac*tpop
AG3701
9
4
80
3.47
0.0115
cultiva*rowspac*tpop
AG3701
18
4
82.5
7.08
<.0001
Thursday October 10, 2008 Homework 7
17
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Cultivar*rowspace*tpop LSMEANS for cultivar AG3701
188
cultiva*rowspac*tpop
AG3701
9
60
20.4250
1.6645
25.5
12.27
<.0001
189
cultiva*rowspac*tpop
AG3701
9
120
22.1000
1.6645
25.5
13.28
<.0001
190
cultiva*rowspac*tpop
AG3701
9
180
23.1669
1.8377
34.8
12.61
<.0001
191
cultiva*rowspac*tpop
AG3701
9
240
24.7500
1.6645
25.5
14.87
<.0001
192
cultiva*rowspac*tpop
AG3701
9
300
26.5000
1.6645
25.5
15.92
<.0001
193
cultiva*rowspac*tpop
AG3701
18
60
19.6500
1.6645
25.5
11.81
<.0001
194
cultiva*rowspac*tpop
AG3701
18
120
22.4515
1.8134
33.9
12.38
<.0001
195
cultiva*rowspac*tpop
AG3701
18
180
21.4883
1.8380
34.8
11.69
<.0001
196
cultiva*rowspac*tpop
AG3701
18
240
26.4500
1.6645
25.5
15.89
<.0001
197
cultiva*rowspac*tpop
AG3701
18
300
27.7250
1.6645
25.5
16.66
<.0001
Pairwise differences for rowspacing 9” are 1.675, 1.0669, 1.531, 1.75,
While for 18”, differences are 2.8015, -0.9632, 4.9617, 1.275.
Thursday October 10, 2008 Homework 7
18
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
Optional (Bonus points)
14. Cultivar AG4601 shows a slight cubic effect in addition to a linear term
15.
--- Effect=cultivar*tpop
cultivar
AG4601
AG4601
AG4601
AG4601
AG4601
rowspace
_
_
_
_
_
Method=LSD(P<.05)
tpop
60
120
180
240
300
Estimate
18.3000
23.4000
23.1625
24.6125
27.4125
Set=3 --------
Standard
Error
1.4574
1.4574
1.4574
1.4574
1.4574
Letter
Group
C
B
B
AB
A
Overall, on average of row spacing levels, there are no differences on means for tpop
120, 180, 240, and between 240 and 300. while there are significant differences
between 60 and 120 and between 120 and 300, and 360 and 300.
Cubic trend is more evident at spacing 18”. This cubic effect may be caused by
uncontrolled effects (?)
16. AG4701 shows polynomial terms up to the third order. See above.
Thursday October 10, 2008 Homework 7
19
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
--- Effect=cultivar*tpop
cultivar
AG4701
AG4701
AG4701
AG4701
AG4701
rowspace
_
_
_
_
_
Method=LSD(P<.05)
tpop
60
120
180
240
300
Estimate
Set=3 --------
Standard
Error
18.4125
23.8750
26.5250
25.3125
28.5000
Letter
Group
1.4574
1.4574
1.4574
1.4574
1.4574
C
B
AB
B
A
Overall, on average of row spacing levels, there are no differences on means for tpop
120, 180, 240, and between 180 and 300. while there are significant differences
between 60 and the rest of tpop; and between 120 and 300, and 240 and 300.
Cubic trend is evident at both spacings, 9’ and 18”. This cubic effect may be caused
by uncontrolled effects (??)
17. For cultivar AG3601, at 9-inch row spacing, yield depends on population density in quadratic
fashion. At 18-inch row spacing, yield is not responsive to changes in the population density.
Effect=cultiva*rowspac*tpop
cultivar
rowspace
tpop
AG3601
AG3601
AG3601
AG3601
AG3601
9
9
9
9
9
60
120
180
240
300
- Effect=cultiva*rowspac*tpop
-cultivar
rowspace
tpop
AG3601
AG3601
AG3601
AG3601
AG3601
18
18
18
18
18
60
120
180
240
300
Method=LSD(P<.05)
Set=1 ----Standard
Letter
Estimate
Error
Group
24.4000
28.2250
31.6000
33.8084
32.8000
1.6645
1.6645
1.6645
1.8376
1.6645
Method=LSD(P<.05)
Estimate
24.1336
23.7750
26.1250
27.1784
26.6750
Thursday October 10, 2008 Homework 7
Set=2 ----
Standard
Error
1.8134
1.6645
1.6645
1.8380
1.6645
Note. Pairwise mean comparisons are at alpha = 0.05 (default value). What is (are) the possible
consequence(s) in the use of this alpha level?
Since the number of pairwise mean comparisons is large, we increase the chances of comparisonwise Type I Error,
which was set at =0.05. Tukey test aims to control this Error and set the experimentwise Type I Error at 0.05. thus
greater pairwise mean differences are required to claim significant differences. Note that for AG4701 at tpop=120, Row
spacing was found significant (p=0.0459, Table of Slices for LSMEAN), but Tukey test did not find significance
20
C
B
AB
A
A
Letter
Group
A
A
A
A
A
ST 524
Homework 7
NCSU - Fall 2008
Due: 11/11/08
between these lsmeans. An intermediate empirical solution is to use a = 0.01 instead of 0.05 for the pairwise Student ttest.
Thursday October 10, 2008 Homework 7
21