Random-Effects Structures for TMDD Parameters

Why Random-Effects Placement Matters in TMDD

A TMDD model can describe substantial differences between individuals in binding, target abundance, or target turnover. Those differences should not be represented by adding random effects indiscriminately. A random effect changes the individual parameter values that drive the state equations, and its placement therefore determines which biological quantities vary across individuals and how those variations are related.

The basic choice is both statistical and mechanistic. For example, baseline target abundance can be represented directly, or it can be derived from target synthesis and degradation. These choices imply different patterns of individual variation. Likewise, allowing binding, target abundance, and turnover random effects to correlate may be reasonable, but a large covariance matrix can be difficult to estimate when the measurements contain limited information about those parameters.

Use the TMDD quantities established in the earlier material on mass-action state equations and parameterizing binding, complex turnover, and clearance. Here, $k_{\mathrm{on}}$ and $k_{\mathrm{off}}$ describe binding and dissociation, $R_0$ is baseline target abundance, and $k_{\mathrm{deg}}$ is the first-order degradation rate of unbound target. In the absence of drug, the steady-state relation is $R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}$.

Log-Normal Random Effects for Positive Parameters

Definition: For a positive TMDD parameter $P$, a log-normal random-effects model is $$ P_i=\theta_P\exp(\eta_{P,i}), $$ where $\theta_P>0$ is the population parameter and $\eta_{P,i}$ is subject $i$'s random effect on the log scale. For a vector of random effects, the usual assumption is $\boldsymbol{\eta}_i\sim N(\mathbf{0},\Omega)$, independently across subjects. The covariance matrix $\Omega$ describes between-subject variability and correlation on the log scale.

Exponentiation ensures that every individual parameter is positive, regardless of the value of its random effect. A zero random effect gives $P_i=\theta_P$; a positive random effect gives a value above $\theta_P$, and a negative one gives a value below it. The model is multiplicative: the same random effect represents a proportional change, rather than a fixed additive change, in the parameter.

When $\eta_{P,i}\sim N(0,\omega_P^2)$, $\theta_P$ is the median of the individual parameter distribution, not generally its arithmetic mean. The distinction follows directly from the normal moment-generating function:

\operatorname{median}(P_i)=\theta_P,\qquad E(P_i)=\theta_P E\{\exp(\eta_{P,i})\} =\theta_P\exp\left(\frac{\omega_P^2}{2}\right).

For this one-parameter model, the coefficient of variation is $\sqrt{\exp(\omega_P^2)-1}$. Thus, $\omega_P$ is a standard deviation on the log scale, not a percentage coefficient of variation. If a model reports $\theta_P$, the interpretation should specify whether it is being described as the typical value (the median under this model) or converted to an arithmetic mean.

The random-effects distribution is an assumption, not a guarantee that every individual is well described by a log-normal distribution. The model also assumes that the chosen parameters are meaningful and positive, that the random-effects distribution is appropriate for the population, and that its covariance can be supported by the data. The conditional observation model and residual errors remain separate from between-subject random effects.

Worked Example: Calculating Individual Binding and Target Parameters

Suppose a hypothetical model has population values $\theta_{k_{\mathrm{on}}}=0.0800\ \mathrm{nM}^{-1}\mathrm{h}^{-1}$, $\theta_{R_0}=12.0\ \mathrm{nM}$, and $\theta_{k_{\mathrm{deg}}}=0.100\ \mathrm{h}^{-1}$. For one hypothetical subject, let $\eta_{k_{\mathrm{on}}}=-0.2231$, $\eta_{R_0}=0.1823$, and $\eta_{k_{\mathrm{deg}}}=0.09531$. Calculate the individual values and the implied synthesis rate.

For binding association, $$ k_{\mathrm{on},i}=0.0800\exp(-0.2231) \approx 0.0800(0.8000)=0.0640\ \mathrm{nM}^{-1}\mathrm{h}^{-1}. $$

For baseline target abundance, $$ R_{0,i}=12.0\exp(0.1823) \approx 12.0(1.200)=14.4\ \mathrm{nM}. $$

For target degradation, $$ k_{\mathrm{deg},i}=0.100\exp(0.09531) \approx 0.100(1.100)=0.110\ \mathrm{h}^{-1}. $$

If synthesis is derived to maintain baseline target equilibrium, then $$ k_{\mathrm{syn},i}=k_{\mathrm{deg},i}R_{0,i} =(0.110\ \mathrm{h}^{-1})(14.4\ \mathrm{nM}) =1.584\ \mathrm{nM}\,\mathrm{h}^{-1}. $$ Using the displayed rounded values, $k_{\mathrm{syn},i}/k_{\mathrm{deg},i}=1.584/0.110=14.4\ \mathrm{nM}$, as required. The calculation shows why the definition of the random-effects model matters: random effects on $R_0$ and $k_{\mathrm{deg}}$ also determine the individual synthesis rate when synthesis is derived.

Choosing Which TMDD Parameters Vary

A model should put random effects on the quantities the data and scientific question can support. Plausible candidates include a binding parameter such as $k_{\mathrm{on}}$, a dissociation parameter such as $k_{\mathrm{off}}$, baseline target abundance $R_0$, and turnover parameters such as $k_{\mathrm{deg}}$ or complex internalization. This list is not a recommendation to vary all of them. Different measurements—free drug, total drug, complex, or target—can contribute different information about these quantities, as considered in the earlier material on likelihoods for free, total, and target measurements.

Two parameterizations can express related mechanisms but induce different individual variation. One option is to model $R_0$ and $k_{\mathrm{deg}}$ as random effects and calculate $k_{\mathrm{syn},i}=R_{0,i}k_{\mathrm{deg},i}$. Another is to model $k_{\mathrm{syn}}$ and $k_{\mathrm{deg}}$ and derive $R_{0,i}=k_{\mathrm{syn},i}/k_{\mathrm{deg},i}$. The choice should match the biological interpretation and the information in the data. In either case, avoid assigning independent random effects to all three quantities $R_0$, $k_{\mathrm{syn}}$, and $k_{\mathrm{deg}}$ while also enforcing their exact relationship: that would be an inconsistent or redundant specification.

Worked Example: Comparing Two Target-Turnover Parameterizations

Consider hypothetical population values $R_0=10.0\ \mathrm{nM}$ and $k_{\mathrm{deg}}=0.120\ \mathrm{h}^{-1}$. The corresponding synthesis rate at equilibrium is $$ k_{\mathrm{syn}}=R_0 k_{\mathrm{deg}} =(10.0\ \mathrm{nM})(0.120\ \mathrm{h}^{-1}) =1.20\ \mathrm{nM}\,\mathrm{h}^{-1}. $$ Suppose the individual random effects on $R_0$ and $k_{\mathrm{deg}}$ are $0.1823$ and $-0.2231$, respectively.

Under a model that varies $R_0$ and $k_{\mathrm{deg}}$, $$ R_{0,i}=10.0\exp(0.1823)\approx12.0\ \mathrm{nM},\qquad k_{\mathrm{deg},i}=0.120\exp(-0.2231)\approx0.0960\ \mathrm{h}^{-1}. $$ The derived synthesis rate is $k_{\mathrm{syn},i}=(12.0)(0.0960)=1.152\ \mathrm{nM}\,\mathrm{h}^{-1}$. The same result follows from $1.20\exp(0.1823-0.2231)$, because the log of the product is the sum of the two log-scale deviations.

Alternatively, if the model places random effects directly on $k_{\mathrm{syn}}$ and $k_{\mathrm{deg}}$, their ratio determines $R_{0,i}$. That specification will generally give a different distribution of $R_0$, depending on the variances and covariance of the two random effects. Therefore, the two parameterizations are not interchangeable descriptions of individual variation merely because both satisfy the equilibrium equation. Choose one, state the implied relationship, and do not count a derived parameter as an additional independent random-effect dimension.

Diagonal and Full Covariance Structures

For $q$ random effects, $\Omega$ is a $q\times q$ positive-semidefinite covariance matrix. A diagonal $\Omega$ allows different amounts of between-subject variability but assumes zero covariance among the included log-scale random effects. A full $\Omega$ estimates the variances and covariances, allowing the effects to vary together. For example, a positive correlation between random effects on binding and target abundance means that subjects with above-typical binding parameters tend also to have above-typical target abundance, on the model's log scale.

The number of distinct covariance parameters rises quickly. A diagonal matrix for $q$ random effects has $q$ variances. An unrestricted symmetric matrix has $q(q+1)/2$ distinct entries: $q$ variances and $q(q-1)/2$ covariances. A full structure may be scientifically plausible, but its extra flexibility is useful only if the data identify those additional quantities.

Worked Example: Recognizing a Weakly Determined Covariance

Suppose a hypothetical fit includes random effects on $k_{\mathrm{on}}$ and $R_0$, with estimated log-scale standard deviations $0.30$ and $0.40$, and estimated correlation $0.98$. The implied covariance is $$ \operatorname{Cov}(\eta_{k_{\mathrm{on}}},\eta_{R_0}) =(0.30)(0.40)(0.98)=0.1176. $$ Thus the estimated covariance matrix is $$ \Omega= \begin{pmatrix} 0.0900 & 0.1176\\ 0.1176 & 0.1600 \end{pmatrix}. $$ Its determinant is $(0.0900)(0.1600)-(0.1176)^2=0.0144-0.01382976=0.00057024$, which is positive but small. Equivalently, the determinant of the corresponding correlation matrix is $1-(0.98)^2=0.0396$. The covariance matrix is therefore close to singular: the two estimated random effects are nearly linearly dependent on the log scale.

This is not, by itself, proof that the biological correlation is impossible. It is a warning to examine uncertainty and stability: a near-boundary correlation may be weakly determined, particularly if the observations do not separately inform binding and target abundance. A diagonal structure would estimate two variances and impose zero correlation; the full two-effect structure estimates those same two variances plus one covariance. Whether the extra parameter is warranted is a modeling and evidence question, not a decision made from the correlation estimate alone.

Justifying a Parsimonious Structure

A useful starting point is a biologically interpretable set of random effects with a simple covariance structure. Add complexity only when the data provide evidence for it and the added terms have a defensible interpretation. The following sequence makes the reasoning explicit.

Choose the varying quantities.
Specify which positive parameters vary between subjects and whether any other parameter is derived from them. Preserve relationships such as $R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}$.

Begin with a supported covariance model.
A diagonal structure estimates variability without estimating correlations. A full structure should be considered only when the number and type of observations can inform its additional covariances.

Check the fit and parameter behavior.
Review uncertainty in variance and covariance estimates, extreme individual predictions, boundary estimates, and whether different random effects are difficult to distinguish. Reconsider a term if estimates are unstable or poorly supported.

Retain complexity for a reason.
Compare a simpler and a more complex specification using appropriate model assessment and scientific interpretation. A more flexible covariance matrix is not automatically a better-supported population model.

Diagnostics should be interpreted in combination. Very small estimated variances can indicate that the data do not support a proposed random effect, although they can also reflect a genuinely homogeneous population. Correlations near $+1$ or $-1$, large uncertainty in covariance estimates, implausibly extreme individual parameters, or substantial sensitivity to small modeling changes are reasons to investigate. A poor fit to measurements may also indicate that the structural model, observation model, or random-effects structure is inadequate; adding covariance parameters does not automatically repair a structural mismatch.

Common Mistakes and Limitations

Treating log-scale variability as a percentage. The standard deviation $\omega_P$ is on the log scale. For a log-normal parameter, convert it to a coefficient of variation with $\sqrt{\exp(\omega_P^2)-1}$ when that summary is needed.
Calling $\theta_P$ the arithmetic mean. Under a mean-zero normal random effect, $\theta_P$ is the median; the arithmetic mean is $\theta_P\exp(\omega_P^2/2)$.
Giving a random effect to a derived parameter as if it were independent. If synthesis is defined by $k_{\mathrm{syn},i}=R_{0,i}k_{\mathrm{deg},i}$, then all three cannot vary independently while that identity is also imposed.
Estimating every covariance because the software permits it. Each additional covariance requires information. A large or near-singular covariance estimate may be unstable rather than biologically informative.
Assuming random effects compensate for limited measurement information. If the available observations do not separate two mechanistic parameters, a richer $\Omega$ cannot create that missing information.

Log-normal random effects are convenient because they preserve positivity and give a clear multiplicative interpretation, but they are not uniquely correct for every parameter or population. The assumed distribution, parameterization, and covariance structure should be assessed against the design, measurements, and scientific purpose. In particular, the variability attributed to individuals is conditional on the chosen structural and observation models.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If $\theta_P=5.0$ units and an individual has $\eta_{P,i}=\log(1.5)$, what is $P_i$?
For $R_0=8.0\ \mathrm{nM}$ and $k_{\mathrm{deg}}=0.050\ \mathrm{h}^{-1}$, calculate the equilibrium $k_{\mathrm{syn}}$.
How many distinct covariance parameters are in a full covariance matrix for four random effects?
What does a positive correlation between random effects on $k_{\mathrm{on}}$ and $R_0$ mean?
Give one reason to prefer a diagonal covariance structure over a full structure.

Solutions. (1) $P_i=5.0\exp\{\log(1.5)\}=7.5$ units. (2) $k_{\mathrm{syn}}=R_0k_{\mathrm{deg}}=(8.0\ \mathrm{nM})(0.050\ \mathrm{h}^{-1})=0.400\ \mathrm{nM}\,\mathrm{h}^{-1}$. (3) A full matrix has $q(q+1)/2=4(5)/2=10$ distinct covariance parameters. (4) Subjects with above-typical $k_{\mathrm{on}}$ random effects tend, on the model's log scale, to have above-typical $R_0$ random effects; correlation alone does not establish a causal relationship. (5) A diagonal structure may be preferable when the data cannot reliably estimate the additional covariances, or when zero correlations are a defensible simplifying assumption.