Choosing Sampling Times for Mechanism Discrimination

Why Sampling Time Is Part of Mechanism Discrimination

Two biological mechanisms can produce nearly identical measurements at one stage of a response and markedly different measurements later. For example, a delayed inhibitory feedback loop may only alter a biomarker after the initial response has developed. Similarly, mechanisms that both reduce a biomarker may predict different rates of recovery. A sampling schedule that misses those periods can leave the mechanisms difficult to distinguish, even if their full time courses differ.

The design question here is specifically when to measure. For a fixed biomarker, intervention, and set of competing mechanisms, compare the predicted measurements at candidate times while accounting for the variability that can obscure their differences. This is a design criterion, not a claim that a particular mechanism is true. The earlier material on designing experiments and expected information gain provides broader perspectives; here the emphasis is on translating time-course contrasts into practical sampling choices.

Build the Criterion From Predictions and Variability

The predictions should correspond to the actual design: the same biomarker, dose or intervention, time origin, and population being considered. The numerator is a contrast in the measurement’s units. The denominator is a standard deviation in those same units, so \(d(t)\) has no units. A large raw difference is not necessarily useful if the measurement is also highly variable.

For a simple setting, suppose individual \(i\)'s measurement under mechanism \(M_m\) at time \(t\) is

where \(b_i(t)\) represents between-subject variation and \(\epsilon_i(t)\) represents assay error. If both terms have mean zero, are independent of one another, and have variances \(\tau^2(t)\) and \(\sigma^2(t)\), respectively, then the total variance for an individual measurement is

This variance is for an individual observation. If the design compares sample means from \(n\) independent subjects and the two mechanisms have the same variance, the variance of the difference in sample means is \(2v(t)/n\). That comparison has a different denominator from the individual-level score. State which design is being scored before using a separation measure.

The simple criterion assumes that the predictive means and variability are meaningful for the intended population, and that observations have the assumed error structure. With repeated measurements, errors and subject effects can be correlated over time. In that case, use a covariance matrix for the full schedule rather than treating every time point as independent.

Why Standardized Separation Is a Useful Starting Point

At one time point, suppose measurements under each mechanism are normally distributed with the same variance \(v(t)\). The log likelihood ratio for an observed value \(y\), comparing \(M_1\) with \(M_2\), is

Taking the expectation under \(M_1\), where \(E_1[Y]=\mu_1(t)\), gives

Thus, under equal-variance normal observation models, \(d(t)^2/2\) is the expected log likelihood ratio in favor of \(M_1\) when \(M_1\) generates the observation. This derivation explains why a squared, noise-scaled contrast is informative. It does not make the score a posterior probability that either mechanism is correct, and the equal-variance normal assumptions matter.

For several times measured on the same subject, let \(\boldsymbol{\mu}_m\) be the vector of predicted means at the selected times, and let \(\Sigma\) be the common covariance matrix of an individual’s measurements, including assay and subject-level variation. The schedule’s squared separation is

This is the multivariate counterpart of \(d(t)^2\), assuming a common, positive-definite covariance matrix. It accounts for correlation: two measurements that tend to move together may contribute less independent information than two uncorrelated measurements with the same marginal variances. For \(n\) independent subjects with the same schedule and fixed model predictions, the squared separation for the sample mean vector scales to \(nD^2\).

Choose a Schedule, Not Just an Isolated Time

When more than one sample per subject is feasible, the goal is to choose a set of times jointly. A time with a modest individual contrast can still contribute to a schedule if it adds a distinct temporal pattern. Conversely, several individually promising times may provide redundant information when their measurements are strongly correlated.

Recovery Kinetics and a Practical Design Workflow

Recovery is another setting in which the best time may not be the earliest or latest available. For example, one mechanism might predict slow receptor recycling and another might predict delayed recovery downstream of the receptor. Both may predict a similar nadir after treatment, but different concentrations during recovery. The useful sampling window is where the predicted trajectories differ after scaling by the variability at that time.

A practical selection process makes the modeling assumptions visible before optimizing a schedule:

Diagnostics, Alternatives, and Limits

A separation score is a summary of specified predictions and variability, so inspect its ingredients rather than reporting only the winning time. Plot both predicted trajectories and the chosen times. Check whether the apparent contrast is driven by a narrow transient that could be missed because of timing error, or by a region where assay performance is poor. For repeated measurements, check whether the covariance matrix is based on a defensible correlation structure and is invertible. If it is nearly singular, the schedule may contain redundant observations or the covariance estimates may be unstable.

Subject variability deserves particular attention. If it is omitted, the denominator can be too small and the design can favor times whose population means differ but whose individual measurements overlap substantially. If variability differs between mechanisms, the equal-variance criterion is incomplete: variance differences can themselves be informative, but should be evaluated with a criterion based on the full predictive distributions. For non-normal measurements, heavy tails, censoring, or substantial parameter uncertainty, a Gaussian mean-separation score may not reflect actual discrimination. Simulation-based assessment of how often a design distinguishes the mechanisms is then a useful alternative.

The standardized criterion is especially useful for screening candidate times and explaining why a time is promising. It does not account automatically for every design feature, such as dropout, unequal group sizes, cost, parameter uncertainty, or a decision-specific penalty for choosing the wrong mechanism. Nor should a high score be interpreted as causal evidence from data not yet collected. It is a prediction-based design aid: the observed data and a suitable analysis are still needed to assess the mechanisms.

Common Pitfall