What an Individual Fit Display Can and Cannot Show
An individual fit display commonly places one subject’s observed measurements against time and overlays a curve predicted using that subject’s estimated parameters. When measurements are frequent and informative, the display can help reveal discrepancies between the model and the observed profile. With sparse sampling, however, a smooth curve may look persuasive even though only a few points constrain it. The distinction is important: the curve is a model-based prediction, not a set of additional observations.
In a nonlinear mixed-effects model, write the observation for subject \(i\) at time \(t_{ij}\) as \(y_{ij}\). The structural prediction is \(f(t_{ij},\theta_i)\), where \(\theta_i\) is the subject-specific parameter vector. A common specification links it to population parameters \(\theta\) and random effects \(\eta_i\), and represents residual error by \(\epsilon_{ij}\).
For the discussion here, assume the random effects follow a specified population distribution, often \(\eta_i\sim N(0,\Omega)\), and that residuals have mean zero conditional on the model and are independent across observations given the subject effects. The error variance may depend on the prediction if the model specifies that. Population parameters and variance parameters are estimated from the study data; they are treated as fixed when describing the basic empirical Bayes calculation below. In practice, uncertainty in those estimates adds uncertainty to the display.
An empirical Bayes estimate combines the subject’s observed data with the estimated population distribution of subject effects. It is not an independent measurement of that person’s parameters. When the subject contributes little information, the estimate is pulled toward the population distribution; the earlier material on eta distributions and shrinkage describes that general phenomenon. The practical consequence for a plot is that a subject-specific curve can reflect substantial population information even when few observations are available for that subject.
Why Sparse Sampling Weakens Individual-Level Evidence
The number of observations is only part of the problem. Their timing and the parameters of interest matter too. If two parameters affect a profile in similar ways over the observed times, even several observations may not distinguish them. Conversely, a small number of measurements at particularly informative times may constrain some features of a profile. In all cases, a smooth curve between observed times is generated by the structural model and estimated parameters; it does not mean the profile was observed continuously.
Worked Example: One Concentration, Multiple Plausible Parameter Pairs
Hypothetical data. After a single intravenous bolus dose of \(D=100\) mg, one subject has a concentration of \(2.00\) mg/L at \(t=4\) h. Consider the one-compartment prediction \(C(t)=D/V\exp(-CL\,t/V)\), where \(V\) is volume in L and \(CL\) is clearance in L/h. Assume this structural model for the illustration.
With \(V=40\) L, solve for the clearance that gives the observed concentration:
A different pair, \(V=25\) L and \(CL=4.332\) L/h, also gives the observation: \(C(4)=4\exp(-4(4.332)/25)\approx 2.00\) mg/L. Both pairs fit the single measured value, yet their predicted concentrations at time zero are \(100/40=2.50\) mg/L and \(100/25=4.00\) mg/L, respectively. The one observation cannot select between these profiles. A smooth individual curve based on one estimated pair may therefore look definite while the individual data provide little evidence about the parameters that determine its shape.
This is an identifiability issue at the individual level, not proof that the population model is wrong. Information from the population distribution can select or regularize an estimate, but the resulting subject-specific curve should not be mistaken for a profile independently established by that subject’s measurements.
Empirical Bayes Estimates and the Pull Toward the Population
A simple normal random-intercept model makes the role of sparse data explicit. Suppose \(y_{ij}=\mu+\eta_i+\epsilon_{ij}\), with \(\eta_i\sim N(0,\tau^2)\), independent \(\epsilon_{ij}\sim N(0,\sigma^2)\), and known \(\mu,\tau^2,\sigma^2\). The conditional distribution of \(\eta_i\) given the subject’s observations is normal. Its mean, which is also its mode in this model, is the empirical Bayes estimate.
The multiplier is between zero and one. With few observations or relatively large residual variance, the subject’s data are less influential and the estimate lies closer to zero, the population mean of the random effect. This formula is a simple illustration; in nonlinear models the conditional estimate is generally obtained numerically, and its behavior depends on the model, design, and data.
Worked Example: Calculating Shrinkage with Two Observations
Hypothetical data. In the random-intercept model, let the population mean be \(\mu=10\) concentration units, the random-effect standard deviation be \(\tau=1\) unit, and the residual standard deviation be \(\sigma=2\) units. One subject has two measurements, \(13\) and \(9\) units. Assume the independent normal distributions and known parameters specified above.
The subject mean is \(\bar{y}_i=(13+9)/2=11\) units, so the deviation from the population mean is \(1\) unit. The shrinkage multiplier and empirical Bayes estimate are
The estimated individual mean is \(10+0.333=10.333\) units, closer to the population mean than the observed subject mean of \(11\) units. The conditional variance of \(\eta_i\) is \(\tau^2\sigma^2/(\sigma^2+n_i\tau^2)=4/6=0.667\) square units, corresponding to a conditional standard deviation of \(0.816\) units. Thus, even this simple conditional calculation indicates uncertainty that a single fitted line would not show.
At the estimated individual mean, the observed-minus-predicted residuals are \(13-10.333=2.667\) units and \(9-10.333=-1.333\) units. The conditional fit is not required to pass through both measurements. A line or curve through the fitted mean does not erase those residuals or make the individual effect precisely known.
In nonlinear mixed-effects models, the same caution applies to individual predictions \(f(t,\widehat{\theta}_i)\). The estimate \(\widehat{\theta}_i\) is informed by the population model as well as by that subject’s data. A fitted curve can look smooth because the structural function is smooth, even if the measurements are too sparse to verify its behavior between sample times.
Reading an Overlay and Its Residuals
A useful display distinguishes the observed points from the population and individual predictions, labels the time and response units, and makes clear which curve is which. At observed times, an individual residual can be written as \(y_{ij}-f(t_{ij},\widehat{\theta}_i)\). Inspecting these residuals alongside the points prevents the curve from becoming the sole focus. The display should also make gaps in sampling apparent; a long uninterrupted line across such a gap is still a model prediction.
Worked Example: A Smooth Overlay with Only Three Measurements
Hypothetical data. A subject has concentrations of \(8.4\), \(3.1\), and \(1.4\) mg/L at \(1\), \(4\), and \(8\) h. Suppose the population predictions at those times are \(7.0\), \(3.8\), and \(1.5\) mg/L, and the fitted individual predictions are \(8.2\), \(3.0\), and \(1.5\) mg/L. Assume the displayed individual predictions come from the fitted model and empirical Bayes estimates.
The observed-minus-individual-prediction residuals are \(8.4-8.2=0.2\), \(3.1-3.0=0.1\), and \(1.4-1.5=-0.1\) mg/L. The corresponding observed-minus-population-prediction differences are \(1.4\), \(-0.7\), and \(-0.1\) mg/L. The individual predictions are closer to the observations at the first two times, as expected for estimates conditioned on those observations.
An overlay connecting the individual predictions smoothly from \(1\) to \(8\) h may appear to describe this subject’s concentration trajectory closely. But there are only three measured values, and the between-time shape is supplied by the structural model. The small individual residuals are in-sample quantities, not independent confirmation that the model captures this person’s unobserved trajectory.
The comparison between population and individual predictions is descriptive, not a test of individual truth. A close individual fit can arise partly because the same subject’s measurements helped estimate the individual parameters. Conversely, visible residual patterns may be especially informative when the number of observations is sufficient to reveal a pattern, but sparse points generally provide limited ability to distinguish systematic structure from chance variation.
Diagnostics, Alternatives, and Limitations
Before interpreting an individual-fit display, check whether the model and display answer the question being asked. Confirm that the plotted points are the observations actually used for the individual estimate, identify whether the curve is population-based or individual-based, and inspect residuals at each observed time. Note the number and timing of observations and whether important phases of the profile are represented. A subject-specific curve that changes substantially under modest changes in the data or estimation setup is not strong individual-level evidence.
Warning signs include treating the curve between observations as measured data, describing a fitted curve as proof of a subject’s parameter values, and overlooking large residuals because the line looks smooth. If few data points are present, a saturated-looking overlay may mainly reflect the model’s functional form and the population distribution. Lack of an obvious residual pattern is also weak reassurance when there are too few observations to reveal one.
Different displays emphasize different information. A population-prediction overlay shows how the typical model prediction compares with the subject’s data, but may not follow a subject whose parameters differ from the population average. An individual-prediction overlay describes the model fit conditional on an estimated subject effect, but can overstate how much the subject’s own data establish. A points-only display emphasizes what was observed and avoids implying a continuous measured trajectory, but does not show model predictions. These views are complementary; none alone resolves weak individual information.
Individual-fit displays also do not replace uncertainty summaries. A single empirical Bayes curve is a point estimate, not an interval for the subject’s trajectory or parameters. Conditional uncertainty, uncertainty in estimated population parameters, and uncertainty about model adequacy are distinct considerations. Sparse data can make all individual-level conclusions fragile even when the population model is useful for describing or predicting the group.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Why can one concentration measurement fail to identify both volume and clearance in a one-compartment model?
- In the random-intercept model, what happens to the empirical Bayes shrinkage multiplier as the number of observations increases, with other parameters fixed?
- A subject’s individual prediction curve passes close to all three observed points. Does that establish that the curve is accurate between those times? Explain.
- What is one benefit and one limitation of showing population predictions alongside individual predictions?
Solutions. (1) A single measurement gives one equation involving two unknown parameters. As the worked example shows, distinct positive values of volume and clearance can produce the same concentration at the observed time. (2) The multiplier \(n_i\tau^2/(\sigma^2+n_i\tau^2)\) increases toward \(1\) as \(n_i\) increases, so the data receive more weight relative to the population mean under this model. (3) No. The curve between measurements is generated by the structural model; three close in-sample points do not independently verify the unobserved trajectory. (4) The population curve provides a reference for the typical model prediction, while the individual curve shows the fit conditional on the subject’s estimated effects. The latter is partly informed by the same observations being plotted, so a close fit does not by itself demonstrate strong individual-level evidence.