Split Plot Homework

* (1) Read the description of the forest soil study,Example 7.4, pg. 194 of 
G&G.  While this is really a repeated measures design we will, as 
suggested, treat it as a 3x2 factorial in blocks for the whole plot
part with depth at 2 levels as the split plot factor. Using this idea, 
work problems 7.4 a and b on page 196.

* (2) Explain how the random assignment of treatments here differs from 
that of a true split plot (this is the reason for the "repeated measures" 
comment above.  Hint:  What is the split plot treatment?  How, if at all, 
was it randomized. 

* (3) Now we will do part C, but with a bit more detail for understanding.
Folow these steps: 

Suppose you call the sources Block, O (Organic matter), C (Compaction), 
and Depth in your SAS dataset. Assuming blocks random and everything else 
fixed, run the model in PROC GLM calling for Compaction LSMEANS and their 
standard errors ( lsmeans C/E=block*O*C pdiff stderr;).  Note that using 
Block*O*C in SAS is the way to specify the use of the whole plot error 
term, error(1) in the book.  Show the proper ANOVA table for the split 
plot and give the F test for Compaction. 

Do the same LSMEAN computation within PROC MIXED (lsmeans C /diff;).  
Writing down the split plot model, what does statistical theory say about 
these standard errors?  In other words, according to the model, what is 
the standard error for the mean of all data at a given compaction level? 
Use the variance component symbols in your formula first, then plug in 
the estimates of the variance components to get a standard error. 
Does your number match the GLM or MIXED results? 
(notice that the LSMEANS are really just means in this balanced case and 
I am using LSMEANS in SAS to produce model based standard errors). 


Explain why the GLM standard errors for LSMEANS differ from the MIXED standard 
errors even though we properly specified error(1). (hint: redo your standard error 
calculation without the block variance component's contribution)  Also explain 
why the p-value for the difference in the two compaction levels 
matches in GLM and MIXED (hint: when you subtract, what happens to 
the random Block effects?  How does the Block variation come into 
play in the two procedures?)  Here are the data as shown on page 196;

  5.9249    4.3280
  5.7004    4.9441
  3.3863    3.4636
 10.9314    9.5983
  5.5351    3.7102
  8.5418    6.5739
  7.2951    5.5796
  7.8114    3.9977
  3.8361    2.8936
  3.2403    3.8276
  3.1920    4.1234
  3.8662    4.6499
  3.1194    5.6948
  6.7300    9.6583
  4.0993    2.8230
  4.7151    5.3373
  2.5725    2.8753
  3.3117    5.6909