The out-of-sample $R^2$: estimation and inference

• URL: http://arxiv.org/abs/2302.05131v1
• Date: Fri, 10 Feb 2023 09:29:57 GMT
• Title: The out-of-sample $R^2$: estimation and inference
• Authors: Stijn Hawinkel, Willem Waegeman, Steven Maere
• Abstract summary: We define the out-of-sample $R2$ as a comparison of two predictive models. We exploit recent theoretical advances on uncertainty of data splitting estimates to provide a standard error for the $hatR2$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Out-of-sample prediction is the acid test of predictive models, yet an independent test dataset is often not available for assessment of the prediction error. For this reason, out-of-sample performance is commonly estimated using data splitting algorithms such as cross-validation or the bootstrap. For quantitative outcomes, the ratio of variance explained to total variance can be summarized by the coefficient of determination or in-sample $R^2$, which is easy to interpret and to compare across different outcome variables. As opposed to the in-sample $R^2$, the out-of-sample $R^2$ has not been well defined and the variability on the out-of-sample $\hat{R}^2$ has been largely ignored. Usually only its point estimate is reported, hampering formal comparison of predictability of different outcome variables. Here we explicitly define the out-of-sample $R^2$ as a comparison of two predictive models, provide an unbiased estimator and exploit recent theoretical advances on uncertainty of data splitting estimates to provide a standard error for the $\hat{R}^2$. The performance of the estimators for the $R^2$ and its standard error are investigated in a simulation study. We demonstrate our new method by constructing confidence intervals and comparing models for prediction of quantitative $\text{Brassica napus}$ and $\text{Zea mays}$ phenotypes based on gene expression data.

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