Predictive Power and Efficient Sample Size in Linear Regression Models

Ma, Yifan

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Predictive Power and Efficient Sample Size in Linear Regression Models

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This work explores the relationship between prediction accuracy, the impact of additional predictors, and sample size in the context of multiple linear regression models. The objective is to facilitate sample size calculations for study designs that directly target predictive power (i.e., prediction accuracy) in various applicational studies. To achieve this goal, we analyze the functional relationship between prediction mean square error (PMSE) and factors such as the number, effect sizes, and correlations among predictors, as well as sample size. Building on this analysis, we introduce a metric referred to as the percentage of PMSE reduction (pPMSEr) to quantify the improvement in prediction accuracy when sample size is increased and/or new important predictors are added to a model. Given a set of predictors, we can compute an efficient sample size, defined as the smallest sample size that achieves, for example, 90% of the prediction accuracy ever achievable at an infinite sample size. Beyond this efficient sample size, increasing the sample size does not significantly improve prediction accuracy unless more important predictors are incorporated into the model. We validate these calculations through computations and simulations based on a pain study, demonstrating a practical application and interpretation of the proposed measures in planning prediction-related studies.

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