Worked Subject-Specific Biomarker Prediction

From a Population Trend to One Subject’s Trajectory

A fitted longitudinal model describes both an average biomarker pattern and how individual subjects may depart from that pattern. A subject-specific prediction combines these pieces: the fixed effects give the model’s population-level trend, while the subject’s predicted random effects shift or reshape that trend. In joint modeling, this trajectory can later be used alongside event-time information, but the calculation here concerns the longitudinal model alone.

The central distinction is between predicting a subject’s latent mean trajectory and predicting a future observed biomarker value. The latent trajectory excludes measurement-level residual variation. A future observed value includes that variation, so its prediction interval is generally wider.

For example, with a linear time effect and a random intercept and slope, the model is

The subject-specific intercept is \(\beta_0+b_{0i}\), and the subject-specific slope is \(\beta_1+b_{1i}\). Thus, a positive predicted random intercept shifts the trajectory upward, while a positive predicted random slope makes it increase faster over time. If time has been centered, the intercept refers to the biomarker level at the centering time, not necessarily at time zero on the original scale.

How the Random-Effect Prediction Is Obtained

Assume the observed biomarker vector for subject \(i\) follows \(y_i=X_i\beta+Z_i b_i+\epsilon_i\). The standard Gaussian mixed-model assumptions for the calculation below are \(b_i\sim N(0,D)\), \(\epsilon_i\sim N(0,R_i)\), and independence of \(b_i\) and \(\epsilon_i\). Subjects’ random effects and residuals are independent across subjects. The covariance matrices \(D\) and \(R_i\) are assumed known for the derivation; in practice they are estimated from the data and substituted. The marginal covariance of \(y_i\) is \(V_i=Z_iDZ_i^\mathsf{T}+R_i\).

If \(\beta\) is treated as known, the conditional distribution of \(b_i\) given \(y_i\) is normal. Its mean is the BLUP and its covariance quantifies remaining uncertainty about the random effects:

In applications, \(\beta,D,\) and \(R_i\) are estimated. Replacing them with fitted estimates gives the empirical BLUP and a plug-in estimate of \(C_i\). The prediction is shrunk toward zero relative to the raw subject-specific residual pattern: subjects with little information, or whose measurements are noisy, tend to have more strongly shrunk random effects.

Calculating the Trajectory

The calculation uses the model’s chosen time basis and covariate values. With polynomial time effects, for instance, the appropriate powers of time must be included in \(x_i(t)\) and \(z_i(t)\). A categorical treatment or baseline covariate must also be set to the subject’s value. Substituting a time value into the wrong design vector, or mixing centered time with uncentered coefficients, changes the prediction.

Uncertainty at a Selected Time

Under the Gaussian assumptions and treating fitted parameters and \(\beta\) as known, the conditional distribution of the latent mean at a selected time is normal, with mean \(\hat{\eta}_i(t)\) and variance \(z_i(t)^\mathsf{T}C_i z_i(t)\). A plug-in standard error is therefore the square root of that variance. A conditional interval can be formed as the predicted mean plus or minus 1.96 standard errors. Because parameters are estimated in real analyses, such intervals are usually approximate unless parameter-estimation uncertainty is also incorporated.

For a future observed biomarker value, add the future residual variation when it is independent of the existing observations. With residual variance \(\sigma^2\), the prediction variance is \(z_i(t)^\mathsf{T}C_i z_i(t)+\sigma^2\). If residuals are correlated over time, the prediction must instead account for the covariance between the future residual and the observed residuals; simply adding \(\sigma^2\) may not be correct.

The preceding intervals condition on fixed-effect estimates as though they were known. To include their estimation uncertainty, a common plug-in prediction-error variance for the latent mean can be written using the full observation vector \(y\). Let \(V\) be its marginal covariance, \(X\) its fixed-effect design matrix, \(A=X^\mathsf{T}V^{-1}X\), \(c=\operatorname{Cov}(y,z_i(t)^\mathsf{T}b_i)\), \(v_{00}=\operatorname{Var}(z_i(t)^\mathsf{T}b_i)\), and \(a=x_i(t)^\mathsf{T}-c^\mathsf{T}V^{-1}X\). With known covariance parameters and generalized least squares for \(\hat{\beta}\), the prediction-error variance is

The first two terms are the conditional random-effect uncertainty; the final term accounts for estimating the fixed effects. In practice, covariance-parameter estimation adds further uncertainty. A parametric bootstrap that refits the model in each simulated data set is one way to capture these sources, provided the fitted model is an adequate description of the data.

Diagnostics, Alternatives, and Limitations

A subject-specific prediction is useful when the scientific question concerns a particular subject’s biomarker course, rather than only the average course. It is also a model-based estimate, not a direct observation of a person’s true trajectory. Its reliability depends on the model’s mean structure, covariance assumptions, and the amount and timing of that subject’s data.

• Check the fitted model first. Residual patterns, influential subjects, weakly identified random slopes, or a singular fit can undermine predicted trajectories. Predictions far beyond a subject’s observed follow-up are especially sensitive to the assumed time trend.
• Check the scale and design vectors. Confirm time units, centering, covariate values, and the correct fixed- and random-effect terms before substituting into the prediction formula.
• Do not treat the predicted random effect as known. It is estimated with shrinkage. A trajectory drawn through observed values without accounting for residual noise is not the same as a fitted subject-specific mean.
• Separate the target quantities. A latent-mean interval excludes residual measurement noise; a future-observation prediction interval includes it. Calling the first a prediction interval for a future measurement understates uncertainty.
• Do not confuse empirical BLUPs with unshrunk individual fits. Estimating a separate intercept and slope from each subject’s data alone can be unstable when that subject has few observations. The mixed-model prediction uses information from the population distribution to stabilize those estimates.

A population-average prediction uses the fixed effects alone, \(x_i(t)^\mathsf{T}\hat{\beta}\), and answers a different question from the subject-specific prediction, which adds \(z_i(t)^\mathsf{T}\hat{b}_i\). Neither is universally preferable: the appropriate choice depends on whether inference concerns an average pattern or an individual trajectory. Both remain limited by the fitted model; a smooth-looking prediction does not verify that the assumed trajectory or residual structure is correct.