Posterior Probabilities Of Bioequivalence

From a Model-Derived Ratio to a Posterior Probability

A model-based analysis can estimate a formulation contrast while accounting for features of the concentration data, such as repeated measurements or within-subject variation. The resulting estimate alone does not answer a Bayesian bioequivalence question. The question is instead: given the data, model, and prior distribution, how much posterior probability lies inside the prespecified equivalence limits?

Let \(R\) be the model-derived exposure ratio for the chosen estimand, test formulation relative to reference formulation. For the familiar limits 0.80 and 1.25, the bioequivalence event is \(0.80<R<1.25\). Other limits can be used when justified and specified in advance. The probability is calculated from the posterior distribution of the quantity defining \(R\), not from a point estimate alone.

For a positive ratio, it is often convenient to define \(\theta=\log(R)\). The usual ratio limits become \(\log(0.80)=-0.2231436\) and \(\log(1.25)=0.2231436\), so the event can be evaluated on the log scale without changing its meaning. The numerical limits are reciprocal, making the interval symmetric about zero on the log scale.

A posterior probability is not the probability that the null hypothesis is true, nor is it a long-run error rate. It describes uncertainty about the estimand conditional on the data and the specified Bayesian analysis. A decision rule may require this probability to exceed a prespecified threshold, but there is no universal threshold implied by the calculation itself.

Calculating the Probability

Suppose the model gives a normal posterior for the log ratio, \(\theta\mid\text{data}\sim N(m,s_\theta^2)\), where \(m\) is the posterior mean and \(s_\theta\) is the posterior standard deviation. Define the standardized limits \(z_L\) and \(z_U\) by subtracting \(m\) from each log limit and dividing by \(s_\theta\). If \(\Phi\) is the standard normal cumulative distribution function, then the desired probability is the upper-limit cumulative probability minus the lower-limit cumulative probability.

This result follows by standardizing the posterior variable: \(Z=(\theta-m)/s_\theta\) has a standard normal posterior distribution under the stated normal-posterior assumption. The event \(\log(0.80)<\theta<\log(1.25)\) is therefore equivalent to \(z_L<Z<z_U\). Subtracting the cumulative probability below \(z_L\) from the probability below \(z_U\) gives \(\Phi(z_U)-\Phi(z_L)\).

The calculation assumes that the posterior distribution used is appropriate for the stated estimand and that the normal approximation is adequate over the range relevant to the limits. In practice, a model may provide posterior draws rather than a normal distribution. In that case, calculate \(R\) for each draw and estimate the probability by the fraction of draws inside the limits. This empirical fraction is reliable only when the draws adequately represent the posterior; poor mixing or too few effective draws can make it unstable.

Why the Prior Can Matter

Bayesian estimation combines the likelihood, which describes how the data vary with the parameter, and the prior, which represents uncertainty about the parameter before observing these data. The posterior probability of bioequivalence can therefore change when the prior changes, especially when the data are sparse or the likelihood is relatively diffuse. Reporting the probability without identifying the prior and model leaves out information needed to interpret it.

A simple conjugate calculation makes this dependence explicit. Suppose a hypothetical likelihood for \(\theta\) is normal, centered at 0.040 with standard deviation 0.080. If the prior is normal with mean \(\mu_0\) and standard deviation \(\tau\), the posterior variance is the reciprocal of the sum of the likelihood and prior precisions, and the posterior mean is the precision-weighted average of their means.

For example, a prior with mean 0 and standard deviation 0.080 gives posterior mean 0.020 and posterior standard deviation approximately 0.05657. A prior with mean 0.100 and standard deviation 0.040 gives posterior mean 0.088 and posterior standard deviation approximately 0.03578. The second prior places more prior weight on positive effects, so it pulls the posterior toward a larger log ratio. These calculations do not establish which prior is appropriate; they show why prior choice should be justified and its influence examined.

The Estimand Defines What Is Being Judged

The exposure ratio must be defined before calculating its posterior probability. For example, a ratio of population geometric means is not generally the same estimand as a ratio of population arithmetic means. If log exposure under each formulation is normally distributed, the arithmetic mean depends both on the mean log exposure and on the log-scale variance. Consequently, a formulation contrast for log means alone does not necessarily determine the arithmetic-mean ratio.

Similarly, a population-level exposure comparison and a probability statement about individual subjects answer different questions. A posterior probability for a population contrast should not be described as the posterior probability that every individual response lies within the limits. If a different model-derived metric is chosen, such as a population distributional measure or an individual-level quantity, its own posterior distribution and equivalence event must be evaluated.

Assumptions, Diagnostics, and Alternatives

The probability has a defensible interpretation only if the model targets the intended exposure estimand, its likelihood and dependence assumptions are reasonable, and the prior is appropriate for the scientific context. The model-derived ratio may depend on fitted concentration curves or other derived quantities; uncertainty in those quantities should be carried through to the posterior distribution used for the event calculation. Treating a fitted ratio as fixed when it is uncertain will generally understate uncertainty.

For a normal-posterior calculation, inspect whether the posterior is approximately symmetric and whether a normal approximation is credible near both limits. For posterior simulation, check that chains explore the same posterior region, that summaries are stable, and that there are enough effective draws to estimate a tail-sensitive probability. A probability close to a decision threshold deserves particular care: Monte Carlo error or modest prior sensitivity may change the decision. These checks diagnose computation and approximation; they do not validate the structural or residual assumptions of the exposure model.

The Bayesian probability is an alternative to frequentist approaches such as confidence-interval inclusion or two one-sided tests. Those procedures have different interpretations and operating characteristics. A posterior probability directly summarizes mass inside the specified limits conditional on the model and prior; it is not interchangeable with a frequentist confidence level or a p-value. A decision threshold for the posterior probability should be chosen and justified for the intended regulatory or scientific use, rather than selected after seeing the result.

Common pitfalls include using the wrong ratio direction, evaluating limits on the ratio scale while treating a log-scale parameter as if it were a ratio, and reporting a probability without naming the estimand. Another mistake is to interpret a high probability as proof that the model is correct. The correction is to state the event, formulation direction, estimand, model, prior, and probability together, and to acknowledge that the conclusion is conditional on those choices.