Re: st: Can Y be a predicted variable?

[Roger Newson <roger.newson@kcl.ac.uk>]

At 18:31 09/09/2005, Tina wrote:
> Dear statalisters,
>
> I have a dependent variable in 5 levels (Self-Assessed Health Status
> from very good to very poor). I am currently assuming a latent
> continuous variable, but that is problematic for some of my analysis.
> I have some other measures of health in my data and was wondering if
> it was appropriate to create a new one that would be continuous. My
> suggestion would be:
>
> 1. regress SAHS on other health variables.
> 2. Predict SAHS (lets call it SAHShat) based on the previous regression.
> 3. The new measure would be calculated as an average of SAHS and SAHShat
>
> This looks like a good idea to me, but I wonder why I don't see anyone
> else doing this if it is OK. Those of you that fell of your office
> chairs in laughter could maybe get back on and explain why not,
> because it seems fine idea to me right now.
In general, categorical ordinal outcomes like SAHS are a problem, 
especially if you want to have a parameter estimate that can be understood 
by non-statisticians.

A possible solution is to use Somers' D, which can be estimated (with 
confidence limits) using the -somersd- package (downloadable from SSC using 
the -ssc- command). Somers' D is defined in terms of Kendall's tau-a, which 
is defined as

tau(X,Y) = E[sign(X1-X2)sign(Y1-Y2)]

where (X1,Y1) and (X2,Y2) are sampled independently from the same 
population. Somers' D is defined as

D(Y|X) = tau(X,Y)/tau(X,X)

Therefore, Kendall's tau-a is the difference between 2 probabilities, 
namely the probability that the larger of 2 randomly-sampled X-values is 
associated with the larger of the 2 corresponding Y-values and the 
probability that the larger of the 2 X-values is associated with the 
smaller of the 2 Y-values. Somers' D is the difference between the 2 
corresponding conditional probabilities, given that the 2 X-values are not 
equal. Somers' D and Kendall's tau-a are discussed in the manual 
-somersd.pdf-, distributed on SSC with the -somersd- package, and also in 
Newson (2002).

Tina does not mention the proposed predictor variables in the proposed 
regression model. However, in a multivariate regression model, there is 
usually one predictor X that is really interesting and other predictors 
that are confounders. For instance, we might want to know how daily 
cigarette consumption predicts SAHS, adjusting for confounders such as 
income, access to a car. and other indicators of general standard of 
living. To estimate a Somers' D of SAHS with respect to cigarettes adjusted 
for the confounders, the first step is to define a propensity score for 
cigarette consumption by regressing cigarette consumption with respect to 
confounders and using the predicted level of cigarette consumption 
(calculated using -predict-) as the cigarette propensity score. We can then 
use -xtile- to define a number of cigarette propensity groups from the 
propensity score, and use -somersd- with the -wstrata()- option to estimate 
a Somers' D of SAHS with respect to cigarette consumption stratified by 
cigarette propensity group. This Somers' D measures association between 
SAHS and cigarette consumption in pairs of patients in the same cigarette 
propensity group. If it is high, then we can say that higher cigarette 
consumers have poorer health than lower cigarette smokers with similar 
"cigarette propensity" based on the confounders. In other words, if the 
stratified Somers' D is high, then the poorer health of cigarette smokers 
is not caused by the fact that cigarette smokers are cigarette-prone 
because of their low general standard of living. Some references about 
propensity scores are given on the manual -somersd.pdf-.

I hope this helps.

Roger


References

Newson R. Parameters behind "nonparametric" statistics: Kendall's tau, 
Somers' D and median differences. The Stata Journal 2002; 2 (1): 45-64. 
Also downloadable from my website at 
http://phs.kcl.ac.uk/rogernewson/papers.htm



--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
Division of Asthma, Allergy and Lung Biology
King's College London

5th Floor, Capital House
42 Weston Street
London SE1 3QD
United Kingdom

Tel: 020 7848 6648 International +44 20 7848 6648
Fax: 020 7848 6620 International +44 20 7848 6620
  or 020 7848 6605 International +44 20 7848 6605
Email: roger.newson@kcl.ac.uk
Website: http://phs.kcl.ac.uk/rogernewson/

Opinions expressed are those of the author, not the institution.

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