Two Different Questions About a Covariate
A covariate can help explain why typical clearance differs between people, why clearance varies more among some people than others, or both. These are distinct questions. A covariate effect on the typical value shifts the center of the individual-parameter distribution across covariate values. A covariate effect on between-subject variability (IIV) changes the spread of that distribution. Treating the two as interchangeable can lead to a model that answers the wrong scientific question.
For example, body weight might be associated with higher typical clearance, while the amount of unexplained clearance variability might be similar at all weights. Alternatively, typical clearance might not change with weight, while clearance is more variable in one weight range. Data can also support both patterns or neither. Each possibility requires a model comparison that isolates the effect being assessed.
Representing the Two Effects
Suppose \(CL_i\) is clearance for individual \(i\), and \(WT_i\) is that individual’s body weight in kilograms. One common individual-parameter model is
Here \(TVCL_i\) is the typical clearance for an individual with covariate value \(WT_i\). The random effect \(\eta_{CL,i}\) describes that individual’s deviation from the typical value, and \(\omega_{CL,i}^{2}\) is its variance. The usual lognormal parameterization ensures positive clearance. Conditional on weight, \(TVCL_i\) is the median of the individual clearance distribution; the arithmetic mean is \(TVCL_i\exp(\omega_{CL,i}^{2}/2)\).
A power model for the typical-value effect and an exponential model for the variance effect can be written as
The reference weight is 70 kg. The coefficient \(\beta_{WT}\) describes the weight effect on typical clearance: changing \(\beta_{WT}\) changes \(TVCL_i\). The coefficient \(\gamma_{WT}\) describes the weight effect on the variance of the random effect: changing \(\gamma_{WT}\) changes \(\omega_{CL,i}^{2}\). Setting \(\beta_{WT}=0\) removes the typical-value effect, while setting \(\gamma_{WT}=0\) makes the random-effect variance constant across weights. If the model is written for the standard deviation rather than the variance, its coefficient has a different scale; the model equation must therefore be checked before interpreting that coefficient.
These components can be distinguished in the model, but their estimates need not be statistically independent. In particular, when a covariate changes IIV, it can also change the arithmetic mean clearance through the factor \(\exp(\omega_{CL,i}^{2}/2)\), even if \(TVCL_i\) is unchanged. Thus “effect on typical value” and “effect on the population arithmetic mean” are not automatically equivalent statements.
Four Models Make the Distinction Explicit
Let the baseline model include the same structural and residual error models in every fit. For one covariate effect on one PK parameter, define four versions of that model:
| Model | Typical-value covariate effect | IIV covariate effect |
|---|---|---|
| \(M_{00}\) | Absent | Absent |
| \(M_{10}\) | Included | Absent |
| \(M_{01}\) | Absent | Included |
| \(M_{11}\) | Included | Included |
For the weight example, \(M_{10}\) estimates \(\beta_{WT}\) while fixing \(\gamma_{WT}=0\); \(M_{01}\) estimates \(\gamma_{WT}\) while fixing \(\beta_{WT}=0\); and \(M_{11}\) estimates both. The models should use the same observations, structural model, residual error model, and other included covariates. Otherwise, a change in fit cannot be attributed solely to the effect under consideration.
The comparison depends on the question. To assess the typical-value effect when no IIV effect is included, compare \(M_{00}\) with \(M_{10}\). To assess it while allowing an IIV effect, compare \(M_{01}\) with \(M_{11}\). Similarly, compare \(M_{00}\) with \(M_{01}\) to assess the IIV effect without a typical-value effect, or \(M_{10}\) with \(M_{11}\) to assess it conditional on the typical-value effect. Comparing \(M_{00}\) with \(M_{11}\) asks whether adding both effects improves fit; by itself, it does not identify which effect accounts for the improvement.
Why the Comparisons Are Nested
When a covariate coefficient is set to zero, the model containing that coefficient reduces to the model without it. This makes each pair above a nested comparison. For example, setting \(\beta_{WT}=0\) in \(M_{11}\) gives \(M_{01}\). If \(\ell\) denotes the maximized log-likelihood, the likelihood-ratio statistic for a larger model and its nested smaller model is
The statistic measures the improvement in maximized likelihood, while the degrees of freedom equal the difference in the number of freely estimated parameters. Under standard likelihood-ratio conditions—including nested models, identifiable parameters, and regularity of the null model—the reference distribution is approximately chi-square with those degrees of freedom. For a single added coefficient, the difference is one degree of freedom. A categorical covariate with \(K\) levels typically requires \(K-1\) contrasts, so its joint effect may add more than one parameter.
These conditions matter. Both models need to be fitted to the same data with comparable likelihoods, and the estimates must be sufficiently well identified for the likelihood approximation to be credible. This comparison concerns a covariate effect on an already modeled IIV distribution; it is not the same question as testing whether IIV itself is zero. If an estimate is on a boundary or the model is poorly identified, ordinary reference calculations may not be reliable.
Worked Example: Calculating Separate Likelihood Improvements
Suppose four models for a hypothetical PK data set produce the following maximized log-likelihoods: \(M_{00}=-1200\), \(M_{10}=-1188\), \(M_{01}=-1196\), and \(M_{11}=-1182\). Each covariate effect adds one coefficient. The structural and residual error models and the observations are identical across fits.
Typical-value effect without an IIV effect. Compare \(M_{10}\) with \(M_{00}\):
This is a one-parameter comparison of adding the typical-value coefficient when the IIV variance is kept constant across the covariate.
IIV effect without a typical-value effect. Compare \(M_{01}\) with \(M_{00}\):
This is a separate one-parameter comparison: it adds the variance covariate coefficient while leaving the typical-value model unchanged.
Conditional comparisons. With the IIV effect already included, the typical-value comparison is \(M_{11}\) against \(M_{01}\), giving \(2[-1182-(-1196)]=28\). With the typical-value effect already included, the IIV comparison is \(M_{11}\) against \(M_{10}\), giving \(2[-1182-(-1188)]=12\). These are still separate one-parameter comparisons, but they answer conditional questions.
Finally, \(M_{11}\) versus \(M_{00}\) gives \(2[-1182-(-1200)]=36\) on two added parameters. That joint comparison cannot be used to say that either the typical-value effect or the IIV effect alone produced the improvement. The numerical results are hypothetical and illustrate the comparisons, not a decision about whether either effect should be retained.
Interpreting the Parameters in Context
For a typical-value effect, interpretation follows the covariate function. In the power model, a one-unit increase in \(\log(WT/70)\) changes typical clearance by a multiplicative factor of \(\exp(\beta_{WT})\). For the variance model above, the same increase changes \(\omega_{CL}^{2}\) by a factor of \(\exp(\gamma_{WT})\). A positive \(\gamma_{WT}\) means the random-effect variance increases as weight increases; it does not mean typical clearance increases.
The reference value and scaling affect coefficient interpretation, not the underlying distinction. Centering a continuous covariate can make the typical parameter more interpretable at a chosen reference value. It does not turn a typical-value coefficient into an IIV coefficient. For a categorical covariate, the same separation applies: one set of contrasts can describe changes in typical values, while another set describes changes in the IIV distribution.
Worked Example: Interpreting Both Effects for Body Weight
Consider a hypothetical model with \(\theta_{CL}=5.00\) L/h, \(\beta_{WT}=0.75\), \(\omega_{CL,0}^{2}=0.0900\), and \(\gamma_{WT}=0.60\). Compare an individual weighing 50 kg with the 70 kg reference individual. Use \(\log(50/70)\approx-0.3365\).
Typical clearance at 50 kg. The typical value is
The reference typical clearance is 5.00 L/h, so this model’s typical clearance is lower at 50 kg. This is the contribution of \(\beta_{WT}\), not \(\gamma_{WT}\).
Variance of the clearance random effect at 50 kg.
At 70 kg, the variance is 0.0900. The corresponding standard deviations of the random effect are \(\sqrt{0.0735}\approx0.271\) and \(\sqrt{0.0900}=0.300\), respectively. These are spreads on the log scale, not clearance values in L/h. The example shows why the typical-value and IIV coefficients require separate interpretations and comparisons.
Checks, Alternatives, and Common Pitfalls
Before interpreting the comparisons, check that the intended model components actually differ. Confirm which equation contains the covariate, whether the coefficient acts on the variance or standard deviation, and whether the reference and covariate scaling are the same across fits. Inspect convergence, parameter uncertainty, and whether the IIV estimates are plausible. A failed or poorly identified fit can produce misleading likelihood differences.
Plots of individual empirical Bayes estimates against a covariate may be useful for exploration, but they do not replace the model comparisons. In particular, the estimated individual deviations can be imprecise, and an apparent pattern in them is not by itself evidence that the population variance changes with the covariate. Evaluate the fitted model’s overall adequacy as well as the covariate terms; a covariate effect can appear to compensate for misspecification elsewhere.
Likelihood-ratio comparisons are useful when the models are nested and the assumptions for their reference distribution are reasonable. AIC or BIC can compare candidate models using different complexity penalties, but those criteria address a broader model-selection objective; they do not, by themselves, label an improvement as a typical-value effect or an IIV effect. A joint comparison can be informative about the combined model, but separate nested comparisons are needed to isolate the contributions.
Common mistake: “The model with both weight effects fits better than the model with neither, so weight affects typical clearance.” This conclusion is not justified by that comparison alone: the improvement could arise from the IIV effect, the typical-value effect, or both. Compare models that differ only by the typical-value term to assess that contribution, and models that differ only by the IIV term to assess the other. When both are scientifically plausible, also report the conditional comparisons so readers can see how each contribution is evaluated when the other is included.
Check Your Understanding
Answer each question, then check the solutions below.
- In a model where \(\beta=0\) removes a covariate from \(TVCL\), what model comparison isolates the typical-value effect when an IIV covariate effect is already included?
- A comparison adds one typical-value coefficient and one IIV coefficient at once. What does that comparison assess, and what can it not establish on its own?
- If \(\gamma=0.40\) is a coefficient on log variance, what multiplicative change in variance corresponds to a one-unit increase in the covariate’s model scale?
- Two nested models have maximized log-likelihoods \(-500\) and \(-494\), with one parameter added in the larger model. Calculate the likelihood-ratio statistic.
Solutions. (1) Compare the model with both effects against the model with the IIV effect only; the typical-value coefficient is the sole addition. (2) It assesses the joint contribution of both effects, but cannot attribute the improvement to either one individually. (3) The variance is multiplied by \(\exp(0.40)\approx1.49\), an increase of about 49% per one-unit increase on that covariate scale. (4) \(LR=2[-494-(-500)]=2(6)=12\), with one added parameter.