Empirical Bayes Estimates as Covariate Screening Tools

Why Examine Empirical Bayes Estimates?

A covariate may explain why individuals differ in a pharmacokinetic parameter. For example, if a model omits body size, the individual random effects for clearance may partly reflect body-size differences. Plotting or correlating those estimates with body size can therefore help identify a relationship worth investigating. This is an exploratory use: the pattern suggests a candidate covariate model, but it does not establish that the covariate effect is real or quantify its uncertainty correctly.

An empirical Bayes estimate (EBE) is an individual-specific estimate of a random effect, obtained conditionally on the individual’s observations and the fitted population model. In a typical NLME model, the individual parameter is represented as a population parameter modified by covariates and a random effect. For example, under a model without a body-size effect on clearance, one might write \(CL_i=CL_{\mathrm{pop}}\exp(\eta_{CL,i})\), where \(\eta_{CL,i}\) is a subject-specific random effect. The EBE \(\hat{\eta}_{CL,i}\) is an estimate of that effect, not a direct measurement of the subject’s true deviation.

For a continuous covariate \(x_i\), an exploratory screen may examine a scatterplot of \(\hat{\eta}_{i}\) against \(x_i\), or calculate their sample correlation. A trend can indicate that the random effect left unexplained by the current model varies systematically with the covariate. The interpretation depends on the parameter scale: if clearance is modeled exponentially, a positive relationship between \(x_i\) and \(\hat{\eta}_{CL,i}\) suggests that the omitted covariate effect may increase log-clearance as \(x_i\) increases.

What a Correlation Can—and Cannot—Show

For paired observations \((x_i,\hat{\eta}_i)\), \(i=1,\ldots,n\), the Pearson sample correlation is

The value \(r_{x,\hat{\eta}}\) describes the direction and strength of a linear association in the estimated random effects in this dataset. It is not itself an estimate of the covariate coefficient in a specified NLME model. Nor is the usual correlation-test \(p\)-value automatically a valid test of that coefficient: EBEs are estimated quantities, often with different precision across individuals, and they depend on the model used to obtain them.

Why Shrinkage and Estimation Noise Matter

An EBE is informed by both the individual’s observations and the population distribution of random effects. When an individual’s data provide little information about a parameter, the EBE is generally pulled more strongly toward zero. The degree of shrinkage can differ across individuals because their sampling times, number of observations, residual variability, or information about the parameter differ. Consequently, the spread and pattern of EBEs need not reproduce the spread and pattern of the underlying individual effects.

A simple approximation makes the issue clear. Suppose the true random effect is \(\eta_i=\beta x_i+u_i\), where \(u_i\) has mean zero and is independent of \(x_i\). Approximate the EBE as \(\hat{\eta}_i=\lambda\eta_i+\epsilon_i\), where \(0\leq\lambda\leq1\) represents a common degree of shrinkage and \(\epsilon_i\) is estimation noise independent of \(x_i\) and \(\eta_i\). Under these simplifying assumptions, the covariance and variance are

Thus the correlation is

This is a justification under the stated approximation, not a universal correction formula. Shrinkage reduces the covariance signal, while estimation noise adds to the variance of the estimates. Both can make the observed correlation smaller than the association between \(x_i\) and the underlying effects. In real analyses, shrinkage may vary across individuals, and estimation error may not be independent of the covariate; distortions can then be more complicated than simple attenuation.

Use the Screen as a Hypothesis-Generating Step

The practical role of EBE screening is to help prioritize plausible model relationships, particularly when many covariate hypotheses have already been assembled. It should be interpreted alongside the pharmacological rationale, covariate support, and exploratory plots. A correlation does not resolve whether the apparent pattern reflects a covariate effect, chance variation, a design feature, an influential individual, or some combination of these.

For a binary covariate, a plot of the EBE distributions by group can serve the same exploratory purpose; the difference between group means summarizes the group difference, whereas the point-biserial correlation is the Pearson correlation between the binary group indicator and the EBE. For a categorical covariate with more than two levels, plots by level are often more informative than forcing the pattern into a single correlation. The scale and parameterization must remain clear: correlations with EBEs for a log-parameter model concern deviations on the log scale, not directly a percentage change in the original parameter.

Diagnostics, Alternatives, and Limitations

Several warning signs call for caution. Strong shrinkage means EBEs may contain little individual-level information, so apparent absence of a trend is weak evidence against a covariate relationship. A handful of extreme EBEs can dominate a correlation; inspect the scatterplot and consider whether the pattern persists without a single unusual point, while avoiding data-driven deletion. Curvature can produce a small Pearson correlation despite a systematic relationship. Finally, if EBE precision is related to the covariate—for example, one covariate group has systematically fewer informative observations—the estimates may be comparably biased or noisy in a way that changes the observed pattern.

EBE screening differs from formal covariate assessment in a full model. A Wald statistic evaluates a coefficient estimate relative to its estimated standard error in the specified model; likelihood-based comparisons assess fit between specified models under their required conditions. Those approaches target model parameters and account for model fitting in ways a raw EBE-covariate correlation does not. They also have assumptions and limitations, but an EBE correlation should not be substituted for them or assigned their inferential interpretation.

A further limitation is model dependence: EBEs change when the structural model, residual model, random-effects structure, or included covariates change. Repeatedly screening many EBE-covariate pairs can also generate striking patterns by chance. Screening results are therefore best recorded as exploratory evidence, not converted into unadjusted claims of significance. The later model-building decision should use the prespecified covariate hypotheses and an appropriate evaluation strategy.

Conversely, a large correlation is not proof of a covariate effect. It is a pattern among estimates produced by a particular model and dataset. The main value of the screen is to make candidate relationships visible and guide the next, model-based assessment.