Improving prediction of linear regression models by integrating external information from heterogeneous populations: James-Stein estimators

Abstract

We consider the setting where (1) an internal study builds a linear regression model for prediction based on individual-level data, (2) some external studies have fitted similar linear regression models that use only subsets of the covariates and provide coefficient estimates for the reduced models without individual-level data, and (3) there is heterogeneity across these study populations. The goal is to integrate the external model summary information into fitting the internal model to improve prediction accuracy. We adapt the James-Stein shrinkage method to propose estimators that are no worse and are oftentimes better in the prediction mean squared error after information integration, regardless of the degree of study population heterogeneity. We conduct comprehensive simulation studies to investigate the numerical performance of the proposed estimators. We also apply the method to enhance a prediction model for patella bone lead level in terms of blood lead level and other covariates by integrating summary information from published literature.

Keywords: James–Stein shrinkage; data integration; external summary information; meta-analysis; population heterogeneity; prediction mean squared error.

FIGURE 1

Comparison of different estimators in prediction mean squared error (PMSE) relative to the PMSE of OLS estimator when using different external studies by varying the value of of the external studies while holding the other parameters at the internal study values. for the internal study and for all 3 external studies. OLS: ordinary least square estimator; JS, JS+: JS estimator and the positive part JS estimator; CLS: constrained least square estimator; EB: empirical Bayes estimator; RR: ridge regression estimator; MJS, MJS+: multiple shrinkage JS estimator and the positive part multiple shrinkage JS estimator; OCW-CLS: optimal-covariance-weighted CLS estimator; OCW-EB: optimal-covariance-weighted EB estimator.

FIGURE 2

Comparison of different estimators in prediction mean squared error (PMSE) relative to the PMSE of OLS estimator, when using all 3 external studies together by varying the value of each parameter of the external studies while holding the other parameters at the internal study values. for the internal study and for all 3 external studies. OLS: ordinary least square estimator; MJS, MJS+: multiple shrinkage JS estimator and the positive part multiple shrinkage JS estimator; OCW-CLS: optimal-covariance-weighted CLS estimator; OCW-EB: optimal-covariance-weighted EB estimator.

FIGURE 3

Performance of using external study 3 under 4 combinations of sample sizes, by varying the value of of external study 3 while holding other parameters at the internal study values. For each value of , the lower curve and higher curve represent the minimum and maximum of the prediction mean squared error ratios corresponding to 100 regenerated external study datasets, respectively, and the dot represents the mean.

FIGURE 4

The left plot is for when using external study 3 only with and varying . The right plot is for when using all 3 external studies with , and varying . All external studies have the same parameter values as the internal study. For each value of , the lower curve and higher curve represent the minimum and maximum of the prediction mean squared error (PMSE) ratios corresponding to 100 regenerated external study datasets, respectively, and the dot represents the mean. In the right plot, the bar positioned at represents the minimum and maximum of the PMSE ratios for using only external studies 1 and 2, and the dot represents the mean.