Comments on HW#2
13 Jan 2010



QUESTION #1 --

(*) For fitting the proportional odds model there are two options:  load the 
needed files after downloading from the web page; or use "library(MASS)" if
the entire MASS library is installed on your computer.

(*) Interpretation of POM:  like shown in the lecture notes, the POM fits
a logistic regression for the "high" versus "low" derived binary outcomes.
The fitted log odds ratio describes the log of the odds of a "high outcome" 
among the "exposed" (e.g. X=1) as compared to the odds of a "high outcome"
among the non-exposed (e.g. X=0) -- the wrinkle is that the odds ratio is
assumed to apply for any possible dichotomization of the outcome into
"high" and "low" levels.

(*) To check the POM assumption one can simply comment qualitatively 
regarding the plausibility that the odds ratio is independent of the
choice of dichotomization -- that is, can be applied to any 2x2 table 
formed by taking the ordinal outcome and converting to "high"/"low".
Formal testing could be done using a likelihood ratio test, but may 
not be easily obtained from standard software...

(*) COMPUTING:  in the "handouts" section of the web page see the file
"wisc_newCDA.q" for guidance on using the function "polr()" to fit the
POM.



QUESTION #2

(*) For part (a) it may be best to think of an approach that uses "forward
induction" to figure out the elements of the matrix B.  That is, for the
first element of the vector Y, call this Y_1, what is needed to convert
      ( Y - pi )
into the CRM model residual with conditional expectation given by mu*H?
      ( Y - mu*H )

Actually for Y_1 there is little to do to get ( Y_1 - mu_1 * H_1 ).

Now consider Y_2 - mu_2 * H_2 -- look at the vector Y and see what 
coefficients would be needed to get the linear contrast with Y_2 and
with H_2 = (1 - Y_1).

If you see this then move to Y_3 and the rest...

(*) COMPUTING:  to fit the CRM model one can simply use logistic regression
once the data has been properly structured -- so "setting up the data" is the
major hurdle for implementation -- see the notes pp. 123-1 to 123-6, 
and the corresponding file "wisc_CRM.q" that has been recently added to 
the "handouts" section (week 2 Examples).