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I've just been introduced to G*Power. The only option I could find for analysing logistic regression is described as:

Options: Large sample z-Test, Demidenko (2007) with var corr

That seems to refer to this paper, which uses the Wald chi-squared test. It does not discuss the Hauck-Donner effect.

Is G*Power a reliable way to calculate power for logistic regression? (The paper has 500 citations, so I feel I must be missing something...)


Demidenko, E. (2007), Sample size determination for logistic regression revisited. Statist. Med., 26: 3385-3397. https://doi.org/10.1002/sim.2771

Hauck, W. W., & Donner, A. (1977). Wald’s Test as Applied to Hypotheses in Logit Analysis. Journal of the American Statistical Association, 72(360), 851–853. https://doi.org/10.2307/2286473

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You are correct that G*Power does not account for Hauck-Donner effect. This does not mean that G*Power is unreliable for a logistic regression analysis. You access the documentation here:

https://www.psychologie.hhu.de/fileadmin/redaktion/Fakultaeten/Mathematisch-Naturwissenschaftliche_Fakultaet/Psychologie/AAP/gpower/GPower31-BRM-Paper.pdf

which says:

5.1. Logistic Regression
Logistic regression models address the relationship between a binary dependent variable (or criterion) $Y$ and one or more independent variables (or predictors) $X_j$, with discrete or continuous probability distributions. In contrast to linear regression models, the logit transform of $Y$, rather than $Y$ itself, serves as the criterion to be predicted by a linear combination of the independent variables. More precisely, if $y=1$ and $y=0$ denote the two possible values of $Y$, with probabilities $p(y=1)$ and $p(y=0)$, respectively, so that $p(y=1) + p(y=0) = 1$, then $\text{logit}(Y) := \ln[p(y=1)/p(y=0)]$ is modeled as $\text{logit}(Y) = \beta_0 + \beta_1\cdot X_1 + \dots + \beta_m \cdot X_m$. A logistic regression model is called simple if $m = 1$. If $m > 1$, we have a multiple logistic regression model. The implemented procedures provide power analyses for the Wald test $z = \widehat{\beta}_j/\text{se}(\widehat{\beta}_j)$ assessing the effect of a specific predictor $X_j$ (e.g., $\text{H}_0\text{: }\beta_j = 0\text{ vs. H}_1\text{: }\beta_j \ne 0\text{, or H}_0\text{: }\beta_j \le 0\text{ vs. H}_1\text{: }\beta_j > 0$) in both simple and multiple logistic regression models. In addition, the procedure of Lyles et al. (2007) also supports power analyses for likelihood ratio tests. In the case of multiple logistic regression models, a simple approximation proposed by Hsieh et al. (1998) is used: The sample size $N$ is multiplied by ($1-R^2$), where $R^2$ is the squared multiple correlation coefficient when the predictor of interest is regressed on the other predictors. The following paragraphs refer to the simple model $\text{logit}(Y) = \beta_0 + \beta_1 \cdot X$.

Hsieh's tables and expressions seem to suggest a monotonic relationship between the effect size and the power, when other study attributes are fixed: http://www.statpower.net/Content/312/Handout/Hsieh%281989%29.pdf

Although these expressions are based on the Wald statistic, under normal conditions - such as the parameter not lying on or close to the boundary of the parameter space - the Wald, Likelihood Ratio, and Score test are asymptotically consistent.

In that case, you'd be justified to calculate your power using these tools, and then perform the likelihood ratio test which is slightly less sensitive to the issues described in Hauck and Donner.

My experience with power calculator tools like G*Power, EAST, and so on is that, despite every attempt to handle every possible case, you don't need to think long or hard about a scenario in which the software is completely incapable of modeling that specific power. In that case, you just have to use simulation to find the answer to the question. I suspect Hauck Donner is no different in that regard.

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