PowerGraph: Using neural networks and principal components to multivariate statistical power trade-offs

It is increasingly acknowledged that a priori statistical power estimation for planned studies with multiple model parameters is inherently a multivariate problem. Power for individual parameters of interest cannot be reliably estimated univariately because sampling variably in, correlation with, and variance explained relative to one parameter will impact the power for another parameter, all usual univariate considerations being equal. Explicit solutions in such cases, especially for models with many parameters, are either impractical or impossible to solve, leaving researchers with the prevailing method of simulating power. However, point estimates for a vector of model parameters are uncertain, and the impact of inaccuracy is unknown. In such cases, sensitivity analysis is recommended such that multiple combinations of possible observable parameter vectors are simulated to understand power trade-offs. A limitation to this approach is that it is computationally expensive to generate sufficient sensitivity combinations to accurately map the power trade-off function in increasingly high dimensional spaces for the models that social scientists estimate. This paper explores the efficient estimation and graphing of statistical power for a study over varying model parameter combinations. Optimally powering a study is crucial to ensure a minimum probability of finding the hypothesized effect. We first demonstrate the impact of varying parameter values on power for specific hypotheses of interest and quantify the computational intensity of computing such a graph for a given level of precision. Finally, we propose a simple and generalizable machine learning inspired solution to cut the computational cost to less than 7\% of what could be called a brute force approach. [abridged]