From Accrued DLT Counts to Posterior Curves
A dose-toxicity model is useful not only for predicting toxicity at one dose, but for describing uncertainty across the full set of candidate doses. After participants have accrued and their dose-limiting toxicity (DLT) outcomes are observed, the model combines those data with its prior distribution. The result is a posterior distribution over model parameters. Each parameter value implies a toxicity probability at every candidate dose, so the posterior induces a distribution of entire dose-toxicity curves.
The distinction between a curve and a single estimate matters. A posterior mean curve gives one summary, but it does not show how uncertain the estimates are, whether uncertainty is concentrated at particular doses, or how parameter uncertainty links predictions across doses. Those features are visible when posterior draws are evaluated over the full dose grid.
Likelihood and Posterior Distribution
For the two-parameter Bayesian logistic regression model (BLRM), write the toxicity probability at dose \(d\) as
where \(d_{\mathrm{ref}}\) is the reference dose, \(\alpha\) is the log-odds of toxicity at that reference dose, and \(\beta\) is the positive dose slope. Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The inverse-logit transformation gives \(p(d\mid\alpha,\beta)=\exp(\eta)/(1+\exp(\eta))\), where \(\eta=\alpha+\beta\log(d/d_{\mathrm{ref}})\). The earlier material on the BLRM establishes this parameterization and the role of a positive slope in enforcing increasing toxicity with dose.
The grouped-binomial likelihood assumes that, conditional on the model parameters, participants' DLT outcomes are independent Bernoulli observations and all participants treated at a given dose share the same probability \(p(d_j\mid\alpha,\beta)\). Outcomes are assumed observed over the relevant DLT assessment period, with no unaccounted-for differences in follow-up or DLT definition. Given these assumptions, \(Y_j\), the number of DLTs at dose \(d_j\), has a binomial distribution.
With prior density \(\pi(\alpha,\beta)\), Bayes' rule gives the posterior density up to a normalizing constant:
where \(\mathbf{y}=(y_1,\ldots,y_J)\). In general, the BLRM posterior is not available in a simple closed form. Numerical methods produce posterior draws \((\alpha^{(m)},\beta^{(m)})\), for \(m=1,\ldots,M\). For each draw, calculate \(p^{(m)}(d_j)=p(d_j\mid\alpha^{(m)},\beta^{(m)})\) at every candidate dose. These vectors, not isolated dose-specific samples shuffled independently, represent posterior uncertainty in the curves.
Worked Example: Updating Two Plausible Curves
Worked Example: Updating Two Plausible Curves
Suppose four participants receive the reference dose of 20 mg, with two DLTs among those four. To make the updating arithmetic transparent, consider a deliberately simplified prior with two equally weighted possible BLRM curves. Curve A has toxicity probabilities 0.1010, 0.2000, and 0.3576 at 10, 20, and 40 mg. Curve B has probabilities 0.2315, 0.5000, and 0.7685 at those doses. Both are increasing dose-toxicity curves. This two-point prior is an illustration of posterior weighting, not a substitute for fitting a continuous BLRM.
For the observed count of two DLTs among four participants, the likelihood at 20 mg is the binomial probability. Under curve A, \(p=0.2000\); under curve B, \(p=0.5000\).
The equal prior weights cancel when normalizing. Thus the posterior probabilities of A and B are
At each dose, the posterior mean probability is the probability-weighted average of the curve values. At 10 mg this is \(0.2905789(0.1010)+0.7094211(0.2315)\approx0.1936\). At 20 mg it is \(0.2905789(0.2000)+0.7094211(0.5000)\approx0.4128\). At 40 mg it is \(0.2905789(0.3576)+0.7094211(0.7685)\approx0.6491\). The data shift posterior weight toward curve B, which assigns a higher probability at the observed reference dose. The posterior means remain dose-specific summaries; the full posterior here also says that the possible curve is A or B with the weights just calculated.
Summarizing Posterior Uncertainty
For each candidate dose, common summaries include the posterior mean or median and a credible interval. For example, an equal-tail 95% credible interval is bounded by the 2.5th and 97.5th posterior quantiles of \(p(d_j\mid\alpha,\beta)\). Under the specified model and prior, the interval contains 95% of the posterior probability for the toxicity probability at that dose. It is not a statement that there is a 95% frequentist probability that a fixed parameter lies in the realized interval.
Posterior summaries can be estimated from \(M\) posterior draws. For instance, the Monte Carlo estimate of the posterior mean at dose \(d_j\) is the average of the \(M\) values \(p^{(m)}(d_j)\). Quantiles are calculated from those same dose-specific values. Intervals at separate doses are pointwise: each describes uncertainty at its own dose. They do not by themselves give a 95% probability that the entire unknown curve lies inside all plotted intervals simultaneously.
Worked Example: Summarizing Posterior Draws Across Doses
Suppose four equally weighted posterior draws from a fitted model produce the following probabilities. The small number of draws is chosen only to show the arithmetic; real analyses require enough well-behaved draws for reliable Monte Carlo summaries.
| Posterior draw | 10 mg | 20 mg | 40 mg |
|---|---|---|---|
| 1 | 0.1111 | 0.2000 | 0.3333 |
| 2 | 0.1976 | 0.3000 | 0.4273 |
| 3 | 0.3034 | 0.5000 | 0.6966 |
| 4 | 0.4974 | 0.6000 | 0.6946 |
The posterior mean at 10 mg is \((0.1111+0.1976+0.3034+0.4974)/4=0.2774\), to four significant figures. At 20 mg it is \((0.2000+0.3000+0.5000+0.6000)/4=0.4000\). At 40 mg it is \((0.3333+0.4273+0.6966+0.6946)/4=0.5380\). These are posterior means of the toxicity probabilities, not observed DLT rates.
For an empirical equal-tail 50% interval using the inverse empirical distribution quantiles, sort the values at each dose and take the 25th and 75th percentiles. The intervals are [0.1976, 0.3034] at 10 mg, [0.3000, 0.5000] at 20 mg, and [0.4273, 0.6946] at 40 mg. They illustrate how posterior uncertainty can vary with dose. With only four draws, these quantiles are extremely coarse and should not be treated as useful clinical estimates.
Why the Likelihood Uses Counts, Not a Smoothed Curve
Observed DLT proportions at the candidate doses need not increase with dose, even when the model constrains the underlying toxicity probabilities to increase. The likelihood evaluates how probable the observed counts are under each parameter value; it does not require the observed proportions themselves to be monotone. This is one reason a model-based posterior curve can be more stable than connecting observed proportions, especially with small cohorts.
Worked Example: A Nonmonotone Pattern of Observed DLT Rates
Suppose hypothetical accrued data are 1 DLT among 3 participants at 10 mg, 0 among 3 at 20 mg, and 2 among 3 at 40 mg. The observed rates are \(1/3\), \(0\), and \(2/3\), so they are nonmonotone. Consider a posterior parameter draw with \(d_{\mathrm{ref}}=20\) mg, \(\alpha=\operatorname{logit}(0.2)\), and \(\beta=1\). Its model probabilities are \(1/9\), \(1/5\), and \(1/3\) at 10, 20, and 40 mg, respectively: for example, at 10 mg the odds are \((0.2/0.8)\exp\{-\log 2\}=1/8\), giving probability \(1/9\).
The grouped-binomial likelihood of the observed counts under this draw is
This likelihood is the weight the data contribute at this parameter value, before combining it with the prior and normalizing over parameter values. It does not mean the model has to pass through the three observed rates. A different posterior draw has a different likelihood, and the posterior balances those likelihoods against the prior.
Computation, Diagnostics, and Alternatives
In practice, posterior computation may use Markov chain Monte Carlo or another validated sampling method. Each retained parameter draw must be transformed through the dose model at all candidate doses. A useful display overlays a manageable number of posterior curves with a posterior mean or median curve and pointwise credible intervals. A table of dose-specific summaries is useful for precise reporting, but it hides the fact that a single parameter draw jointly determines values at all doses.
Check that the posterior computation has converged and that the chains explore the posterior rather than remaining in a narrow region. Inspect trace plots and mixing, effective sample sizes, and Monte Carlo error for the parameters and dose-specific summaries of interest. Also check whether the fitted posterior predictions are plausible in light of the accrued data and clinical knowledge. Strong prior-to-posterior similarity despite informative data, extreme sensitivity to modest prior changes, or implausible curve shapes are prompts to investigate the parameterization, prior calibration, data coding, and model fit. A failure to explore posterior tails can make credible intervals look falsely narrow.
The main alternatives answer different questions or make different assumptions:
- Observed rates by dose are direct summaries of accrued data, but can be highly variable with small \(n_j\), can equal zero or one, and do not provide a coherent smooth curve.
- Separate binomial models by dose avoid imposing a logistic shape, but do not borrow information across doses and do not inherently enforce increasing toxicity.
- A frequentist logistic fit can summarize a dose-response relationship, but does not directly provide a posterior distribution over curves or posterior credible intervals.
These approaches are not interchangeable: the BLRM curve reflects the chosen dose-response form, prior, and monotonicity assumption. It can be misleading if the logistic shape is a poor representation of toxicity, the prior is poorly calibrated, participants differ in ways not captured by the model, or DLT ascertainment is incomplete. With sparse data, posterior summaries may be strongly influenced by the prior; with very little information at extreme doses, interval widths and tails deserve particular scrutiny. A posterior mean curve also need not itself be a curve associated with one representative parameter value, although it summarizes the marginal mean at each dose.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At a dose with 5 DLTs among 12 participants, write the likelihood contribution, omitting no term that depends on the probability \(p\).
- Why does a posterior draw at one dose need to be retained together with its values at the other candidate doses?
- Under a BLRM with positive slope, the observed DLT rates at three increasing doses are 0.25, 0.00, and 0.50. Does this pattern contradict the model's monotonicity assumption?
- What does a pointwise 95% posterior credible interval at one dose describe, and what does it not establish about the entire curve?
Solutions. (1) The contribution is \({12\choose5}p^5(1-p)^7\); the binomial coefficient is constant in \(p\) but is part of the likelihood. (2) The same parameter draw determines probabilities at every dose, preserving posterior dependence and the shape of each curve. (3) No. The observed proportions are noisy data, whereas monotonicity applies to the model probabilities for each positive-slope parameter draw. (4) It contains 95% of the marginal posterior probability for the toxicity probability at that dose, conditional on the model and prior. It does not assert a 95% probability that the whole curve lies within the pointwise intervals at all doses simultaneously.