Population And Individual Fit Targets

Why Fit Has More Than One Target

An NLME model describes both a population and the differences among individuals in that population. Evaluating its fit therefore requires asking which target is being evaluated. A prediction for a typical individual asks whether the model’s central parameter values describe the centre of the population. A prediction for a particular individual asks whether the model, after using information from that individual, describes that individual’s observations. These are related questions, but they are not interchangeable.

A third target is the marginal population mean: the expected response after averaging over the distribution of individual random effects. This target matters when predicting an average response across the population. Because an NLME model may be nonlinear, this marginal mean need not equal the prediction for a typical individual. A sound fit assessment names the target before interpreting any comparison.

Three Prediction Targets

Let \(Y_{ij}\) be the response for individual \(i\) at observation time \(t_{ij}\). A general NLME model can be written as

Y_{ij}=f(t_{ij},\theta_i)+\epsilon_{ij}, \qquad \theta_i=g(x_i,\beta,\eta_i).

Here \(f\) is the structural model, \(\theta_i\) is the parameter vector for individual \(i\), \(x_i\) denotes that individual’s covariates, \(\beta\) denotes population parameters, and \(\eta_i\) denotes individual random effects. The observation error \(\epsilon_{ij}\) describes variation between the model-predicted response and the observed response. For the definitions below, assume the random effects have a specified population distribution with mean zero, and that the observation error has conditional mean zero. These assumptions make \(f\) the conditional expected response given the individual parameters.

Definition: The typical-subject prediction (often called the population prediction in diagnostic practice) sets the random effects to zero: \(f(t_{ij},g(x_i,\beta,0))\). The marginal population mean prediction averages the model prediction over the random-effects distribution: \(E_{\eta_i}[f(t_{ij},g(x_i,\beta,\eta_i))]\). An individual prediction plugs in an estimate of the random effects for that individual, commonly an empirical Bayes estimate \(\hat{\eta}_i\): \(f(t_{ij},g(x_i,\beta,\hat{\eta}_i))\).

The phrase “population prediction” is used differently across software and settings. It may mean the zero-random-effect typical-subject prediction or, more precisely, the marginal mean obtained by integrating over random effects. Always check how the prediction was constructed. In this tutorial, “typical-subject prediction” refers specifically to setting \(\eta_i=0\); “marginal population mean” refers to integration over the random-effects distribution.

An individual prediction is conditional on information from that individual when \(\hat{\eta}_i\) is estimated using their observations. It is therefore not a prediction made without access to those observations. The target for a new individual with no individual data is generally the population model, not a subject-specific prediction based on an estimate that cannot yet be obtained.

Why Typical and Marginal Predictions Can Differ

The typical-subject prediction evaluates the model at the average random effect. The marginal population mean averages the model’s responses across random effects. For a nonlinear function, evaluating at an average input is generally not the same as averaging the function’s output. This distinction can be seen with a simple derivation.

\eta_i\sim N(0,\omega^2), \qquad h(\eta_i)=\exp(\eta_i).

The typical-subject value is \(h(0)=1\). To find the marginal mean, integrate over the normal density and complete the square:

\begin{aligned} E[\exp(\eta_i)] &=\frac{1}{\sqrt{2\pi\omega^2}} \int_{-\infty}^{\infty} \exp\left(\eta-\frac{\eta^2}{2\omega^2}\right)\,d\eta\\ &=\frac{1}{\sqrt{2\pi\omega^2}} \int_{-\infty}^{\infty} \exp\left(-\frac{(\eta-\omega^2)^2}{2\omega^2} +\frac{\omega^2}{2}\right)\,d\eta\\ &=\exp\left(\frac{\omega^2}{2}\right). \end{aligned}

The final equality follows because the remaining integral is the integral of a normal density with mean \(\omega^2\) and variance \(\omega^2\), so it equals one after accounting for its normalizing constant. If \(\omega^2=0.25\), the typical-subject value is \(1\), whereas the marginal mean is \(\exp(0.125)=1.133\). The difference is not a calculation error: it arises from nonlinearity and between-individual variability.

In an NLME application, the exact difference depends on the model, the random-effects distribution, and the time or covariate values. It may vary across the response curve. Thus a typical-subject curve should not automatically be described as the expected population-average curve.

Worked Example: Typical and Individual Concentration Predictions

Consider a hypothetical one-compartment decline model with concentration \(C_i(t)=10\exp(-k_i t)\), where the initial concentration is \(10\) mg/L. Let the typical elimination rate be \(0.20\) h\(^{-1}\), and suppose an individual’s estimated random effect implies an elimination rate of \(0.30\) h\(^{-1}\). Compare predictions at \(t=2\) h with that individual’s observed concentration of \(5.60\) mg/L.

Typical-subject prediction. Set the random effect to zero, giving \(k=0.20\) h\(^{-1}\):

C_{\mathrm{typ}}(2)=10\exp[-(0.20\ \mathrm{h}^{-1})(2\ \mathrm{h})] =10\exp(-0.40)=6.703\ \mathrm{mg/L}.

Individual prediction. Use the estimated individual rate \(k_i=0.30\) h\(^{-1}\):

C_{\mathrm{ind}}(2)=10\exp[-(0.30\ \mathrm{h}^{-1})(2\ \mathrm{h})] =10\exp(-0.60)=5.488\ \mathrm{mg/L}.

The individual prediction is \(5.60-5.488=0.112\) mg/L below the observation. The typical-subject prediction is \(6.703-5.60=1.103\) mg/L above it. These comparisons answer different questions: the first asks how the typical parameters perform for this observation, while the second asks how well the model performs after allowing for this individual’s estimated deviation. The example does not calculate the marginal population mean, which would require averaging the nonlinear concentration curve over the random-effects distribution.

Reading Population-Level and Individual Fit Together

Population-level fit concerns whether the population parameters and structural model represent the central or average behaviour of the intended population. Typical-subject predictions are useful for examining the model’s central trajectory. Marginal population predictions are more directly aligned with expectations averaged over the modelled distribution of random effects. Which one is relevant depends on the scientific question: describing a typical patient and predicting the average response are not always the same task.

Individual-level fit concerns how well the model describes observations for particular individuals after incorporating estimated individual effects. It can reveal whether the model accommodates subject-specific trajectories. However, an individual prediction uses information from the observations that helped estimate \(\hat{\eta}_i\). It is therefore expected to fit those data more closely than a prediction that does not use individual information. A close individual fit is not, by itself, evidence that the population structure or covariate relationships are adequate.

Worked Example: Opposing Individual Deviations Cancel in the Pooled Average

Suppose two hypothetical individuals have observed concentrations of \(8\) and \(12\) mg/L at the same time. The typical-subject prediction is \(10\) mg/L for each. For illustration, suppose the fitted individual predictions are \(8\) and \(12\) mg/L.

The observed mean is

\bar{Y}=\frac{8+12}{2}=10\ \mathrm{mg/L}.

The mean typical-subject prediction is also \((10+10)/2=10\) mg/L. Consequently, the pooled observed mean and pooled typical-subject prediction agree exactly. Yet the individual differences from the typical prediction are \(8-10=-2\) mg/L and \(12-10=2\) mg/L. The agreement of the pooled means conceals these opposing deviations.

The individual predictions match both observations in this constructed example. That close match describes subject-specific fit, not proof that the population model explains why one individual is lower and the other higher. The random effects may accommodate differences without establishing that the model captures their cause or distribution correctly.

Worked Example: A Pooled Fit Can Conceal a Covariate Pattern

Suppose a hypothetical study includes two covariate groups, A and B, with two individuals per group. At a specified time, all four typical-subject predictions are \(10\) mg/L. The observed concentrations are \(8\) and \(8\) mg/L in group A, and \(12\) and \(12\) mg/L in group B.

The pooled observed mean is

\bar{Y}=\frac{8+8+12+12}{4}=10\ \mathrm{mg/L}.

It matches the pooled typical-subject prediction of \(10\) mg/L. But the group means are \(8\) mg/L for A and \(12\) mg/L for B, each \(2\) mg/L from the prediction in opposite directions. The overall average therefore hides a systematic pattern across the groups. A population-level assessment should examine relevant covariate groups rather than rely only on a pooled average.

If individual predictions use estimated random effects and move toward the observed values, they may describe these four subjects more closely. That does not establish that the model’s population covariate structure is correct: subject-specific deviations can absorb patterns that a population model should explain. The group pattern is a reason to investigate the population model, not by itself proof of a particular omitted effect.

Assumptions, Diagnostics, and Limits

The interpretations above rely on a correctly specified prediction target and on the model’s assumptions being plausible for the question at hand. In particular, the random-effects distribution must represent the population being described, the structural model must relate parameters to responses appropriately, and the observation model must represent conditional response variation. A typical-subject prediction additionally requires a meaningful zero-random-effect reference, which follows the model’s centring convention. If covariates are included, predictions must use the covariate values appropriate to the intended individual or population.

Several checks help prevent overinterpretation:

State the construction. Record whether “population prediction” means \(\eta_i=0\) or integration over the random-effects distribution, and whether individual predictions use estimated random effects.
Match the prediction to the question. Use a typical-subject target to discuss central parameter values; use a marginal mean to discuss average response across the modelled population; use individual predictions to discuss subject-specific fit.
Look beyond pooled agreement. Summaries across all observations can conceal opposing deviations. Consider whether patterns differ by individual, time, regimen, or relevant covariate groups.
Interpret individual predictions cautiously. Since estimated individual effects use that subject’s observations, close agreement may partly reflect this conditioning. It does not independently validate population parameters or out-of-sample prediction.

These comparisons have limits. A good typical-subject prediction does not ensure that variability is described correctly. Good individual predictions do not ensure that the random-effects distribution, covariate effects, or population structure is correct. Conversely, a typical-subject prediction can differ from many observations because individuals vary, even when the model is appropriate. Prediction-target comparisons are therefore complementary evidence, not a single pass-or-fail test.

Key takeaway: Always identify what was predicted and for whom. Typical-subject, marginal population, and individual predictions target different features of an NLME model; agreement for one target cannot substitute for evaluating the others.

Check Your Understanding

Answer each question, then compare with the solutions below.

What is held fixed when calculating a typical-subject prediction, and how does that differ from a marginal population mean prediction?
In a nonlinear model, why can the typical-subject prediction differ from the marginal population mean?
Two individuals have observations of \(6\) and \(14\) mg/L, while both typical-subject predictions are \(10\) mg/L. What is the pooled observed mean, and what does it fail to show?
Why does close agreement between individual predictions and the observations used to estimate random effects not, by itself, validate the population model?

Solutions. (1) A typical-subject prediction sets \(\eta_i=0\); a marginal mean averages predictions over the random-effects distribution. (2) In general, a nonlinear function evaluated at the mean input is not equal to the mean of its outputs. (3) The pooled mean is \((6+14)/2=10\) mg/L. It conceals deviations of \(-4\) and \(+4\) mg/L for the two individuals. (4) The individual predictions use information from those same observations through the estimated random effects, so they can fit better without demonstrating that the population structure, covariate effects, or random-effects distribution is adequate.