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I am doing a case-control study analysis with 2500 cases and 2500 controls. I am interested in finding out if the cases have higher odds of having a particular disease than the controls, so I am calculating odds ratios for each of the 1000 diseases using a logistic regression model.

I want to know which odds ratios > 1 are actually significant enough to the point where I can say that the case is at higher risk for the disease than the control. To do so, I employed Efron's method of estimating the empirical null distribution and local FDR values (link: http://www.uni-leipzig.de/~strimmer/lab/courses/ss06/seminar/papers/B/efron2004.pdf).

Most of the odds ratios should be 1 (betas are 0), corresponding to z-value of 0. When I estimate the empirical null distribution, it is not distributed ~ N(0, 1) but rather N(mu, sigma). So this allows me to identify significant odds ratios. However, I want to know if I can transform the odds ratios using the empirical null distribution as such:

  • z-value for beta coefficients is equal to coefficient/standard error.

  • z-values are distributed N(mu, sigma).

  • Normalize the z-values: (z - mu)/sigma.

  • Multiply the new z-values by the standard error.

Now, you have new coefficients and odds ratios, which have a more accurate distribution of odds ratios (most of them are around 1).

Is this a valid approach?

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  • $\begingroup$ The Efron paper link do not work now ... $\endgroup$
    – kjetil b halvorsen ♦
    Nov 28, 2019 at 11:37

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I don't think it works well in general to assume the null hypothesis when forming an estimator other than occasionally an estimate of a variance. More likely to work is the bootstrap bias estimator akin to the Efron-Gong optimism estimator. Here's an example. Suppose you are interested in more unbiasedly estimating the true odds ratio for the largest or most significant odds ratio from among the 1000. For B repeats (B may be typically 400 to 2000) find the largest or most significant odds ratio from among the 1000 in a bootstrap sample (sample with replacement). Then estimate the same odds ratio for the same feature in the original sample. Record the drop in log odds ratio. Average this over the B simulations to estimate the average optimism (bias) in log-odds ratio. Subtract this from the original most impressive log odds ratio and anti-log.

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  • $\begingroup$ Would I be able to translate this approach to not only the most significant odds ratio but all significant odds ratios as deemed by their local FDRs? $\endgroup$
    – user3821273
    Jul 28, 2014 at 15:53
  • $\begingroup$ Yes. You can find features using any algorithm, then estimate the bias by contrasting the apparent log odds ratio with the log odds ratio for that feature evaluated on the original sample. You could say that with sample with replacement the bootstrap sample is "super-overfitted" and the original sample us just "overfitted". The difference between super overfitted and overfitted mimics the difference between overfitted and non-overfitted, hence the justification for the bias correction. $\endgroup$
    – Frank Harrell
    Jul 28, 2014 at 16:49

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