I conducted a logistic regression with a 3-levels categorical predictor in SPSS using the GENLIN procedure. As I am interested in the pairwise comparisons between all three predictor levels, I used the /EMMEANS subcommand, SCALE=transformed, and obtained log OR which I could then transform into OR.

However, I am wondering how to to a proper a-priori sample size planning/power analysis. Let's say the minimum effect size between two levels that I am interested in is OR = 2.5 - how can I derive the required sample size in the next step?

Any input is highly appreciated.


Think about equal sample sizes $n$, and suppose that two proportions being compared, $p_1$ and $p_2$, are near $0.5$. I determined by trial and error that the odds ratio is about 2.5 when $p_j = \frac12 \pm 0.112$. The $z$ statistic for comparing these is $$ z \approx \frac{\hat p_1- \hat p_2}{\sqrt{2\times(\frac12\cdot\frac12)/n}} = \sqrt{2n}(\hat p_1 - \hat p_2) $$ Now, if we want a power of $\beta$ when the comparison is tested at a significance level of $\alpha$, we need to set a target value of $z$ at $\sqrt{2n}\times(2\times0.112) = 0.318\sqrt n$, and equate that with $(z_{\alpha/2}+z_\beta)$. But in this case, let's throw in a Bonferroni correction and really use $\alpha/3$, because we'll be doing 3 such comparisons. Thus, for significance 0.05 and power 0.8, we have $$ n \ge \left(\frac{z_{0.025/3} + z_{0.2}}{0.318}\right)^2 = \left(\frac{2.394 + 0.842}{0.318}\right)^2 \approx 104 $$ You'll need 104 observations at each level of the categorical predictor. You may get by with less if in fact the $p_j$ are not near $\frac12$, which is the worst-case scenario.

PS -- I confirmed this result using the Java app available at https://homepage.divms.uiowa.edu/~rlenth/Power/index.html (The "test comparing two proportions" dialog, with p1 = .388 and p2 = .612). If you incorporate the continuity correction, however, the required sample size becomes $n=112$.

| cite | improve this answer | |

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.