Residual Error and Assay Precision in Design Information

Why residual error changes design information

A sampling time can be highly informative when the predicted concentration changes sharply with a parameter. But sensitivity alone is not enough: a measurement is less informative when its residual variability is large. In design calculations, residual error therefore determines how much each observation contributes, and concentration-dependent error can change the relative value of different sampling times.

The calculations here describe information from a normally distributed observation, conditional on the model quantities stated. In an NLME model, this may mean conditioning on an individual’s random effects. If those effects are integrated out, observations can be dependent and the marginal information must account for that dependence; the conditional contributions cannot simply be treated as the full marginal information.

Specify the residual-error model first

A common combined-error model assumes that, conditional on the model prediction, the observation is normal with variance \(v_i=\sigma_{\mathrm{add}}^2+(\sigma_{\mathrm{prop}}m_i)^2\). Here \(\sigma_{\mathrm{add}}\) is an additive residual standard deviation in concentration units, while \(\sigma_{\mathrm{prop}}\) is a dimensionless proportional-error coefficient. The corresponding residual standard deviation is the square root of this variance. This tutorial uses the sum of the two variance components; it is not the alternative model that adds standard deviations.

With additive error alone, \(v_i=\sigma_{\mathrm{add}}^2\), so the absolute residual spread is constant across concentrations. With proportional error alone, \(v_i=(\sigma_{\mathrm{prop}}m_i)^2\), so the residual standard deviation is a constant fraction of the predicted concentration. In the combined model, the two terms are equal when \(m_i=\sigma_{\mathrm{add}}/\sigma_{\mathrm{prop}}\), for positive predicted concentrations. Below this crossover, additive error tends to dominate; above it, proportional error tends to dominate.

These models assume a correctly specified conditional mean, a normal conditional residual distribution, and the stated variance structure. The calculations also require the variance to be positive. A proportional-only model needs positive predictions to describe positive concentrations sensibly; it is not a remedy for zeros or values near zero. Whether residuals are independent conditional on the model is another assumption to check, not something guaranteed by the variance formula.

Information from a normal observation

Suppose \(Y_i\mid\boldsymbol{\theta}\sim N(m_i(\boldsymbol{\theta}),v_i(\boldsymbol{\theta}))\), where the variance is positive and both functions are differentiable. Write \(m_{ia}=\partial m_i/\partial\theta_a\) and \(v_{ia}=\partial v_i/\partial\theta_a\). The observation’s Fisher information for components \(\theta_a\) and \(\theta_b\) is

The first term is information carried by changes in the mean relative to residual variance. The second term is information carried by changes in the variance. When \(v_i\) is fixed with respect to the parameters of interest, \(v_{ia}=0\), leaving only the first term. For independent observations, sum the observation matrices to obtain the conditional total information. In vector notation, the mean term is \(\mathbf{J}_i^\mathsf{T}\mathbf{J}_i/v_i\), where \(\mathbf{J}_i\) is the row of mean sensitivities.

Why the variance term appears

For one observation, let \(r=Y_i-m_i\). The log likelihood, apart from constants, is \(-\tfrac12\log(v_i)-r^2/(2v_i)\). Differentiating with respect to \(\theta_a\) gives the score

Under the assumed normal model, \(E(r)=0\), \(E(r^2)=v_i\), and \(E(r^4)=3v_i^2\). Thus the two score components have zero expected cross-product: their product contains only terms proportional to \(E(r)\) or \(E(r^3)\), both of which are zero for a centered normal variable. The mean-score product has expectation \(m_{ia}m_{ib}/v_i\). For the variance-score product, the centered quantity \(r^2/v_i^2-1/v_i\) has variance \(2/v_i^2\), giving \(v_{ia}v_{ib}/(2v_i^2)\). Since Fisher information is the expected product of score components, the displayed formula follows.

In the combined-error model, if \(\sigma_{\mathrm{add}}\) and \(\sigma_{\mathrm{prop}}\) are known and \(m_i\) depends on \(\theta_a\), then \(\partial v_i/\partial\theta_a=2\sigma_{\mathrm{prop}}^2m_i\,\partial m_i/\partial\theta_a\). Some design calculations evaluate the variance at nominal predictions and hold it fixed while differentiating the mean. That gives the mean-information term only. It is a deliberate approximation or convention, not the exact Fisher information for a likelihood whose variance changes with the parameter.

Concentration changes the value of an observation

Under the mean-information term, an observation is valuable when its parameter sensitivity is large relative to its residual standard deviation. Under pure proportional error, for example, that term for a scalar parameter is \((\partial m_i/\partial\theta)^2/(\sigma_{\mathrm{prop}}m_i)^2\). Equivalently, it is the squared relative sensitivity \((\partial m_i/\partial\theta)/m_i\), divided by the squared proportional-error coefficient. A large concentration by itself does not guarantee high information: what matters is sensitivity relative to noise.

With combined error, increasing concentration can move an observation from the additive-dominated to the proportional-dominated region. If the sensitivity does not grow enough to offset that increase in variance, its mean-information contribution falls. This is why a design that ranks times using sensitivities alone can favor noisy observations and give a misleading picture of their value.

Assay precision and technical replication

Assay precision enters the information calculation through the relevant variance component. Improving a known additive assay standard deviation reduces its variance contribution by the square of the improvement factor. For instance, halving an additive standard deviation quarters its variance. If proportional or other residual variability remains, however, the total variance does not generally fall by that same factor.

Technical replicates can provide repeated measurements of the same underlying concentration. If their assay errors are conditionally independent, have common known variance \(v\), and share the same mean sensitivity, their mean-information contributions add: \(r\) replicates contribute \(r(\partial m/\partial\theta)^2/v\), where \(r\) is the number of replicates. This calculation applies to the independently replicated error component. If the model also has shared biological or occasion-level variability, treating all replicate measurements as independent draws from the entire variance model can overstate information.

Checks, alternatives, and limitations

A useful design check is to calculate the predicted residual standard deviation and the mean sensitivity at each candidate time, then compare their ratio through the mean-information term. Inspect the additive and proportional variance components separately. If the high-concentration variance is unexpectedly dominant, verify the units and scale of the proportional coefficient; a coefficient intended as a fraction should not be entered as a percentage without conversion.

Residual diagnostics from suitable fitted models can reveal patterns inconsistent with the design assumption: residual spread increasing with prediction challenges an additive-only model, while a nearly constant absolute spread may challenge a proportional-only model. Curvature, outliers, censoring, and serial dependence can also make a normal independent-error calculation inadequate. Diagnostics cannot prove the model correct, and the assumed error structure should be justified for the assay and measurement process.

For a population design, conditional information given random effects is not automatically the information in the marginal likelihood. Marginalization can induce dependence among observations and changes the relevant mean and covariance structure. Likewise, if residual-error parameters are estimated rather than known, their uncertainty must be included as nuisance-parameter information when assessing precision for other parameters. The computations above isolate the effect of the specified residual model; they do not replace those broader calculations.