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Re: st: Help please re Ocratio


From   "Anders Alexandersson" <andersalex@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Help please re Ocratio
Date   Wed, 31 Oct 2007 11:15:29 -0400

Jon Heron (ALSPAC) <Jon.Heron@bristol.ac.uk> would like an extra bit
of clarification about the continuation ratio model:

>  It sounds like, if I want to fit an ocratio model and then examine the
>  individual pairs of effects I either fit two logistics
>
>     1 [=1] versus 2 & 3[=0]
>     2 [=1] versus 3 [=0]
>
>  and compare with an ocratio on an outcome ordered 1,2,3
>
>
>  or fit two logits
>
>     1 [=0] versus 2 & 3[=1]
>     2 [=0] versus 3 [=1]
>
>  and compare with an ocratio with outcome ordered the other way i.e.
>  3,2,1
>
>
>  If this is the case, and a quick experiment with Stata suggests it is,
>  and since the continuation ratio model is not reversible (unlike
>  proportional odds), how do I decide which one to do?
>
>  I guess the former

I think that the answer depends on what you want to compare. Bender and
Benner (2000, 681) suggest that a "forward" continuation ratio model
(your option 1) seems useful if the response Y represents "survival
times". For example, you want to compare short survival times versus
high survival
times. In contrast, a "backward" (or reverse) continuation ratio model
(your option 2) seems useful if Y represents "disease status". For example,
you want to compare bad disease vs mild disease. Bender and Benner
illustrate the backward continuation ratio model.

It is also important to distinguish between "stopping" and "continuation"
continuation ratio models. I find the terminology in the documentation for
R's function vgam() by Yee helpful, see
http://www.stat.auckland.ac.nz/~yee/VGAM/. I try to summarize the four
models here:

   continuation ratio      Probability, j = 1,2
   --------------------    --------------------
a. forward stopping        P[Y=j|Y>=j]
b. reverse stopping        P[Y=j+1|Y<=j+1]
c. forward continuation    P[Y>j|Y>=j]      = a with reversed signs
d. reverse continuation    P[Y<j+1|Y<=j+1]  = b with reversed signs

It seems to me that logistic regression and Rory Wolfe's -ocratio-
estimates (a). Logistic regression assumes equal slopes on original
data, or unequal slopes on expanded data. In comparison, -ocratio-
assumes equal slopes only but you can test this assumption, and the
sign of the slope is opposite but that is easy to handle. It seems
that Maarten Buis's -seqlogit- estimates (c).

How to estimate version (b) or (d) of the continuation ratio model in
Stata, or how to extend the continuation ratio model(s) as a
mixed-model? For mixed-model version, the literature (e.g, Grilli
2005) suggests the use of logistic regression on expanded data, but I
have not had a chance to try this nor have I found an example to
replicate. Any suggestions?

References
----------
Bender, R., Benner, A. 2000. Calculating ordinal regression models in SAS
    and S-Plus. Biometrical Journal 42(6): 677-699.
Grilli, L. 2005. The random-effects proportional hazards model with
    grouped survival data: a comparison between the grouped continuous and
    continuation ratio versions. Journal of the Royal Statistical Society,
    Series A. 168(1): 83-94

Anders Alexandersson
andersalex@gmail.com
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