Posterior Risk Intervals in Bayesian Logistic Regression Models

Why Risk Intervals Matter in Dose Finding

A dose-toxicity model does not give a single known toxicity probability at each candidate dose. It gives a posterior distribution for that probability, reflecting uncertainty in the model parameters after the observed dose-limiting toxicity (DLT) data and prior information are combined. A posterior risk interval summarizes part of that distribution at a particular dose. It can show whether the estimated risk is fairly well determined or remains uncertain, including when the estimated risk is near the target toxicity rate.

The interval is not itself a dose recommendation. In particular, a risk interval that contains the target does not establish that the dose has an acceptable overdose probability, nor does a narrow interval establish that the model is appropriate. The interval answers a more focused question: under the specified model, prior, and observed data, what range of toxicity probabilities is supported by a chosen portion of the posterior distribution at this dose?

Model and Definition

For a binary DLT outcome \(Y_i\) observed within a defined assessment window, let \(Y_i=1\) indicate a DLT and \(Y_i=0\) otherwise. At dose \(d_i\), the logistic model specifies

Here \(d_{\mathrm{ref}}\) is a positive reference dose, \(\alpha\) is the log-odds of DLT at that reference dose, and \(\beta\) is the slope per unit of log dose. The inverse-logit function is \(\operatorname{logit}^{-1}(z)=1/\{1+\exp(-z)\}\). If the model imposes \(\beta>0\), its risk curve increases with dose. A prior distribution on \((\alpha,\beta)\), together with the likelihood, yields a posterior distribution for these parameters and therefore for \(p(d)\).

The dose must be held fixed when interpreting this interval. The interval is a marginal, dose-specific summary: it accounts for posterior uncertainty in the model parameters at that dose. A collection of such intervals across doses is not automatically a simultaneous 95% band for the entire risk curve. Nor is a Bayesian credible interval a frequentist confidence interval: its stated posterior probability applies to the parameter or risk under the posterior distribution, conditional on the model and data.

Transforming Posterior Uncertainty to Risk

The direct calculation uses posterior draws. For draw \(m\), transform its parameter values \((\alpha^{(m)},\beta^{(m)})\) to a risk at dose \(d\):

The empirical quantiles of these transformed draws estimate the posterior risk quantiles. This approach preserves the joint posterior dependence between \(\alpha\) and \(\beta\), which matters because the linear predictor at a dose is a combination of both parameters. With a large, well-mixed posterior sample, the quantiles can be estimated directly without assuming that the risk itself is normally distributed.

A useful justification follows from monotonicity. The inverse-logit function is strictly increasing, so for a scalar posterior linear predictor \(\eta(d)=\alpha+\beta x(d)\), its quantiles transform in order:

Thus, if the posterior quantiles of \(\eta(d)\) are available, applying the inverse-logit to the lower and upper quantiles gives the corresponding risk interval endpoints. This does not mean that the interval width can be found by transforming the width on the log-odds scale: the inverse-logit is nonlinear, so endpoint transformation is required.

Intervals From a Joint Normal Approximation

When posterior draws are unavailable, a local approximation can be made from a bivariate normal posterior for the parameter vector. Suppose \((\alpha,\beta)\) has posterior mean \((\mu_\alpha,\mu_\beta)\), variances \(V_\alpha,V_\beta\), and covariance \(C_{\alpha\beta}\). At a fixed dose, the linear predictor is a linear combination of the parameters. Its posterior mean and variance are therefore

If the joint normal approximation is adequate, \(\eta(d)\) is approximately normal. An approximate equal-tailed 95% interval for its value is \(\mu_\eta(d)\pm1.96\sqrt{V_\eta(d)}\); inverse-logit transformation gives an approximate risk interval. This calculation must use the covariance term, not just the separate parameter variances. It also assumes that the normal approximation is reasonable in the relevant posterior region.

Reading Interval Width Near the Target

The width is the upper risk endpoint minus the lower risk endpoint, measured in probability units or percentage points. It describes posterior spread on the risk scale, not the distance between a point estimate and the target. Width is affected both by uncertainty in the linear predictor and by the nonlinear inverse-logit transformation. In particular, equal uncertainty on the logit scale can produce different risk-scale widths at different locations: the inverse-logit changes most rapidly near zero and flattens toward risks of zero and one.

Diagnostics, Alternatives, and Limitations

Before interpreting the interval, check that the posterior computation and the model support it. For simulation-based output, inspect chain mixing, effective sample sizes, convergence diagnostics, and stability of the reported quantiles. A small effective sample size in the tails can make 2.5th and 97.5th percentiles unstable even when the posterior median appears stable. Recompute intervals from longer or better-mixed runs if relevant tail quantiles vary materially across chains or runs.

Check the model inputs as well: the dose units and reference dose must be consistent; the DLT assessment window must match the outcome being modeled; and the likelihood and prior must correspond to the intended analysis. For a monotone dose-toxicity model, inspect whether the posterior slope behaves as intended and whether the risk curve is plausible over the candidate-dose range. A credible interval computed precisely from an implausible or poorly calibrated model does not become reliable merely because its endpoints are numerically stable.

The posterior median or mean risk is an alternative point summary, but neither replaces an interval when uncertainty is important. The posterior mean risk at a dose is the average of the transformed posterior draws, not generally the inverse-logit of the posterior mean linear predictor. A posterior probability such as \(P\{p(d)>\phi\mid\text{data}\}\), where \(\phi\) is a specified risk threshold, answers a different question from a central interval: it is the posterior probability that risk exceeds that threshold. A dose-specific interval also differs from a simultaneous curve band, which must account for uncertainty across doses jointly.

A common mistake is to say, “The interval contains the 30% target, so the dose has a 95% probability of being acceptable.” The correction is that a central 95% interval describes the posterior distribution of risk at that dose; it does not assign 95% probability to the target being met, and inclusion of the target does not quantify the probability of exceeding a safety threshold. That probability requires evaluating the relevant posterior event directly. Likewise, a narrow interval does not guarantee good prediction if the logistic form, monotonicity assumption, prior, or data structure is inappropriate.