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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.

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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 $n=112$.

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