Re: st: proportion as a dependent variable

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

At 14:06 14/07/03 +0200, Ronnie Babigumira wrote:
> I was attending a workshop in which one of the presenters had a regression
> in which a dependent variable was a proportion. One of the participants
> noted that it was wrong but didnt follow it up with a clear explanation.
I assume that Ronnie is talking about a homoskedastic linear regression, 
not a logistic regression. A homoskedastic linear regression model should 
not be used for binary data, because the conditional variance of a binary 
dependent variable is dependent on the conditional mean, ie the conditional 
probability of a success, given the value of the independent variables. The 
standard errors for a homoskedastic linear regression are computed assuming 
that the conditional variance is constant, and therefore will either be too 
laarge or too small.

If you use Huber variances by using the -robust- option of -regress-, then 
the standard errors will be correctly estimated, so it would not be 
actually wrong. However, the regression coefficient would be calculated by 
an inefficient formula, so the confidence intervals would be wider than 
necessary. The best way to do linear regression with a binary dependent 
variable is probably to use the -glm- command with the options

family(binomial) link(identity) robust

which will calculate regression coefficients equal to binomial 
probabilities and their differences. For instance, if you type

glm y x,family(binomial) link(identity) robust

then the intercept will be a baseline probability that y==1 if x is zero, 
and the slope will be an incremental probability that y==1, given a unit 
increase in x. These parameters will be calculated by an efficient formula, 
so the confidence limits will probably be narrower than those created if 
you type

regress y x,robust

which is not actually wrong for a binary y, but is inefficient.

If you want the parameters to be probabilities and their ratios, instead of 
probabilities and their differences, then you can use -glm- with the log 
link. For instance, if you type

gene byte baseline=1
glm y x baseline,family(binomial) ling(log) robust eform noconst

then the intercept parameter will correspond to the variable -baseline-, 
and represent a baseline probability that y==1 if x is zero, and the slope 
parameter will  correspond to x, and be a relative risk per unit x, 
assuming that the probability increases exponentially with x. 
Alternatively, if you want the parameters to be baseline odds and odds 
ratios, then you can use the -logit- command.

I hope this helps.

Roger


--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
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Tel: 020 7848 6648 International +44 20 7848 6648
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Email: roger.newson@kcl.ac.uk
Website: http://www.kcl-phs.org.uk/rogernewson

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

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