# How can I calculate power for a conditional logistic regression?

A client wants to perform a 1:1 matched case control study, matching on 3 risk factors, and analyze results using conditional logistic regression. A sample size calculation is required in order to plan the study. How can I estimate the sample size?

Some thoughts:

• I could use sample size estimates for regular logistic regression since we are not matching on the exposure of interest.

• Conditional logistic regression is a form of Cox regression, so in principle, I suppose I could use sample size estimates for Cox regression

• If neither of the above work, and there is no principled way to compute sample size, I can just simulate the power curve.

If the odds ratio is homogeneous in each of the strata, the conditional logistic regression model is asymptotically equivalent to the Cochrane Mantel Haenszel statistic where

$$\widehat{cOR} = \sum_{i=1}^K R_i / \sum_{i=1}^K S_i = \left( \sum_{i=1}^{K} \frac{a_i d_i}{N_i} \right) / \left( \sum_{i=1}^{K}\frac{b_i c_i}{N_i} \right)$$

where $$a_i$$ is the number of exposed cases and $$d_i$$ the number of unexposed controls (concordant observations) in the $$i$$-th stratum and $$b_i$$, $$c_i$$ the discordant observations respectively.

The variance due to Fleiss is asymptotically correct:

let $$W_i$$ be the precision (inverse variance) of the log of the $$i$$-th stratum specific odds ratio:

$$W_i = (a_i ^{-1} + b_i^{-1} + c_i ^{-1} + d_i^{-1})^{-1}$$

then

$$var (\widehat{cOR}) = \widehat{cOR}^2 \left( \frac{\sum_{i=1}^K S_i^2 / W_i }{(\sum_{i=1}^K S_i)^2}\right)$$

or the normal approximation to the log-cOR has variance

$$var (\log \widehat{cOR}) = \left( \frac{\sum_{i=1}^K S_i^2 / W_i }{(\sum_{i=1}^K S_i)^2}\right)$$

So you can calculate critical values and probabilities under various alternatives using either simulation or numeric integration. Ideally compare the two.

The general expression in terms of the prevalence of the outcome, the number of strata, or a possible imbalance in the number of cases is reflective of the general type(s) of designs that can be applied in this scenario and/or the population frequency of these traits/outcomes. Be explicit in spelling out the strategies in sampling.

https://www.annualreviews.org/doi/pdf/10.1146/annurev.pu.09.050188.001011