Consider the following study to investigate the effect of temperature and wetness period on the appearance of disease lesions. This experiment was done in growth cabinets, and only one temperature could be applied to each whole cabinet. The temperatures considered are 10°C, 15°C and 20°C. As only three cabinets were available at any one time, the experiment had to be repeated in order to get replication and so it was done three times, using different temperatures in each cabinet every time. Each run is considered as a separate replicate, labeled Block, and the cabinets within each experiment act as the whole plots. Within each cabinet, four wetness periods (16, 24, 48 or 72 hours) were applied to trays each containing 12 small plants. The allocation of temperatures (whole plot treatments) to cabinets was randomized for each experiment, and the position of each treatment within each cabinet (sub-plot treatments) was also randomized to avoid any bias due to systematic within-cabinet effects. For this study we will consider the temperature and wetness period as factors. The number of disease lesions per tray, transformed to log10(lesions + 1), recorded for this experiment are presented in the table below.
|
Block 1
|
Cabinet 1
|
Cabinet 2
|
Cabinet 3
|
|
20°C
16h
0.903
|
20°C
48h
1.716
|
15°C
24h
1.176
|
15°C
72h
1.851
|
10°C
24h
1.230
|
10°C
48h
1.964
|
|
20°C
72h
1.771
|
20°C
24h
1.176
|
15°C
16h
0.778
|
15°C
48h
1.204
|
10°C
16h
0.954
|
10°C
72h
2.130
|
|
Block 2
|
15°C
16h
1.000
|
15°C
72h
2.619
|
20°C
72h
1.672
|
20°C
24h
0.954
|
10°C
24h
1.255
|
10°C
16h
0.301
|
|
15°C
24h
1.447
|
15°C
48h
1.878
|
20°C
16h
0.845
|
20°C
48h
0.954
|
10°C
72h
1.477
|
10°C
48h
1.380
|
|
Block 3
|
10°C
24h
1.204
|
10°C
16h
0.778
|
15°C
24h
1.041
|
15°C
72h
2.338
|
20°C
72h
2.375
|
20°C
24h
1.362
|
|
10°C
48h
1.886
|
10°C
72h
2.484
|
15°C
48h
1.851
|
15°C
16h
0.602
|
20°C
16h
1.114
|
20°C
48h
1.982
|
|
|
a) Specify the design and treatment structure of this experiment. What is the number of replications for each of the main treatment factors and their interaction?
b) Write the linear model, detailing each of its components. Also, construct a partial ANOVA table identifying each model term and its corresponding degrees of freedom (but no SSs, MSs, Fs, or p-values).
c) Fit the model specified in part (b). (Partial SAS code is can be found at the end of this document.) Make sure to report the ANOVA table and the residual plots. Is it appropriate to make conclusions in this case? Why or why not? If so, what do you conclude from the overall ANOVA analysis?
d) Regardless of your responses to part (c), report treatment means and SEMs for a 5% level for each of the treatment factors (i.e. temperature, wetness period and the sliced interaction). Also report the differences between treatment means and their corresponding SEDs, Fisher's LSDs, and Tukey's HSDs.