Population-Average and Subject-Specific Predictions

Two Prediction Targets From One Model

A longitudinal model can describe both the average biomarker trajectory in a population and how an individual subject’s trajectory differs from that average. Those are related, but not identical, prediction targets. For a population-level question, the relevant prediction averages over subject-to-subject random variation. For a question about a particular subject, the prediction can incorporate that subject’s estimated random effects.

The distinction matters whenever a prediction is reported: the same fixed-effects estimates may produce different answers depending on whether subject effects are included. A prediction for an average member of a population is not necessarily a good prediction for a particular person whose observed biomarker history indicates a departure from the population trajectory.

Model and Assumptions

Write the repeated biomarker measurements for subject \(i\) as a vector \(Y_i\). A Gaussian linear mixed model is

Here \(X_i\) contains the fixed-effect covariates, \(\beta\) is the vector of fixed-effect coefficients, \(Z_i\) contains the random-effect covariates, \(b_i\) is the subject’s random-effect vector, and \(\varepsilon_i\) is the residual vector. The usual assumptions for this prediction interpretation are that the random effects have mean zero, the residuals have mean zero, and random effects and residuals are independent. The covariance matrices \(D\) and \(R_i\) describe their respective variation. Predictions also rely on the specified mean structure being suitable for the covariate profile of interest.

For one measurement at a specified profile, let \(x\) and \(z\) be the corresponding fixed- and random-effect design vectors. Conditional on the random effect, the expected response is \(x^\mathsf{T}\beta+z^\mathsf{T}b_i\). The population prediction averages this expression over the random-effect distribution. In applications, \(\beta\) is replaced by its estimate \(\hat\beta\); for a subject-specific prediction, \(b_i\) is replaced by its estimated value \(\hat b_i\), often called the empirical BLUP.

Why Averaging Removes the Random-Effect Term

The marginal prediction follows directly from the law of total expectation. First take the conditional expectation of a measurement given the subject effect. Then average that conditional expectation over the random-effect distribution:

The last equality uses \(E(b_i)=0\). Thus the marginal and subject-specific predictions share the same fixed-effects component, but only the subject-specific prediction adds the estimated deviation \(z^\mathsf{T}\hat b_i\). This result depends on the linear mean and identity link. It should not be transferred without adjustment to models with nonlinear links, where averaging over random effects generally does not leave the fixed-effects expression unchanged.

Calculating Predictions for a Profile

To calculate either prediction, first specify the profile completely: time, treatment group, and any other fixed covariates in the model. Then calculate the fixed-effects part. For the marginal prediction, stop there. For the subject-specific prediction, calculate the random-effects contribution using that profile’s random-effect design vector and the subject’s estimated random effects, then add it to the fixed-effects part.

Predicted Mean Versus Future Measurement

A predicted mean response and a prediction of a future observed measurement are different targets. Under the stated model, the future residual has conditional expectation zero. Therefore, the point prediction for a future observation has the same mean as the corresponding predicted trajectory: \(x^\mathsf{T}\hat\beta\) marginally or \(x^\mathsf{T}\hat\beta+z^\mathsf{T}\hat b_i\) for a subject-specific prediction. But an actual future measurement also includes residual variation, so it will not generally equal that mean.

Diagnostics, Comparisons, and Limitations

The main alternative to a subject-specific prediction is the marginal prediction, not a competing fitted model: the two answer different questions using the same model. A fixed-effects-only calculation is appropriate for the population-average target under this Gaussian linear model. Adding estimated random effects is appropriate when the target is a particular subject and those effects have been estimated from that subject’s data. A raw subject average is a different alternative, but it may ignore time, treatment, and shrinkage, and may be unstable with few measurements.

Before interpreting either prediction, check that the profile is represented by the model’s covariates and that the prediction is not an unsupported extrapolation in time or covariate values. Compare the subject-specific estimate with the fixed-effects prediction at the same profile, rather than comparing values at different times or treatment groups. Large subject deviations can be meaningful, but may also prompt examination of data quality, sparse follow-up, or a poorly fitting trajectory. The earlier material on reading shrunk subject trajectories explains why estimated random effects tend to be pulled toward zero, particularly when a subject has limited information.

A common mistake is to add the subject’s estimated random intercept but omit a random slope contribution at a nonzero time, or to add random effects when reporting the population mean. The correction is to use the full random-effects design at the requested profile, \(z^\mathsf{T}\hat b_i\), only for a subject-specific target. Another mistake is to call that plug-in value the subject’s known true trajectory. It is an estimate, and its precision depends on the data and model. Predictions may also be unreliable if the mean structure, random-effects distribution, or residual assumptions are inadequate; conditional predictions do not repair model misspecification.