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 thd
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.
5.
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
Claim: Prolonged cutting decreased the vigor.
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 
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.
Thursday October 10, 2008 Homework 7
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ST 524
Homework 7
6.
7.
8.
9.
NCSU - Fall 2008
Due: 11/11/08
Claim: Annual improvement is greatest for June-1 cutting, and declines linearly with later cutting
times
Claim: Each additional two-week of cutting decreases the annual improvement in yield up to June 1
by the same amount.
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.
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 '
10. 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 '   
11. 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,


corr yijk , yij  k h    h
Questions
12. What pattern of correlation best describes the time effect?
13. Explain how increasing the cutting time (from june01 to July15) affects the response expressed as
the slope of the regression of yield on Year.
14. Conclusions. Use  = 0.05. Indicate supporting statistical evidence to above Claims when writing up
your conclusions.
Thursday October 10, 2008 Homework 7
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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. G701 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, yields 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?
Thursday October 10, 2008 Homework 7
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