6
$\begingroup$

Suppose we have case-control data, where cases have some disease ($Y$) and controls don't and we are interested in the association of some other variable(s) ($X$). I know that in this scenario we cannot use the disease as the response variable because of the experimental design (the marginal distribution of disease is fixed by sampling).

I also know that the odds ratio however can be calculated in such designs because it takes the same value when using the conditional distribution of $X|Y$ or $Y|X$.

My question is: is it appropriate to use logistic regression in this case to model the odds of disease? i.e. $\text{logit}\left\{\dfrac{P(Y=1)}{1-P(Y=1)}\right\} = \mathbf{\beta} X $

context: GWAS (Genome Wide Association Studies) are typically case-control studies, where one wants to assess the association between disease and number of minor alleles of a particular SNP. $P$-values are typically obtained from a chi-squared test of independence. However, this doesn't allow you to add in other covariates. A lot of the packages that offer GWAS analysis also allow you to do logistic regression. I just wanted to verify that it was valid to do such an analysis.

$\endgroup$
8
$\begingroup$

Logistic regression is a valid inferential method, because, as you've noted you're modeling the odds. The coefficients on explanatory variables $X$ will also be valid. However, the intercept term $\beta_0$ will not be; this is because the number of positive and negative outcomes are fixed by the case-control design. So the intercept term will be meaningless, but your other estimates are fine. More information is in Agresti, An Introduction to Categorical Data Analysis.

$\endgroup$
  • 3
    $\begingroup$ +1 - Also see this response about how to estimate the correct intercept plus other references. $\endgroup$ – Andy W Sep 9 '13 at 14:05
  • $\begingroup$ Thank you sir. This is what I believed. I was also reading Agresti, and just wanted to verify I was interpreting what he was saying correctly. $\endgroup$ – bdeonovic Sep 9 '13 at 14:15
  • $\begingroup$ Just found the page. In the second edition (2007), the logistic regression in case-control studies is discussed on p. 105. $\endgroup$ – Sycorax says Reinstate Monica Sep 9 '13 at 14:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.