Plotting Posterior Results Against the Discount

Why Plot Posterior Results Against the Discount?

A power-prior analysis uses a fixed discount \(a\) to control how much historical information contributes to the current analysis, with \(0\leq a\leq1\). A result reported at only one value of \(a\) can hide how sensitive that result is to the borrowing choice. A plot makes this dependence visible: it can show whether a posterior mean shifts gradually, whether uncertainty contracts, and whether a clinically relevant posterior probability crosses a decision threshold.

The horizontal axis represents the chosen discount, not time, a patient characteristic, or a quantity estimated from the trial. Each point is the result of a separate posterior calculation conditional on that fixed value of \(a\). The resulting curve is therefore a sensitivity display, not a posterior distribution for the discount.

A useful plot states the estimand or parameter, its units, the discount range, the prior and likelihood model, and whether each interval is a credible interval conditional on that fixed discount. It should also identify any clinical threshold used in a decision-probability curve. These details let the reader distinguish a change caused by the discount from a change caused by redefining the model or summary.

Constructing a Posterior Mean and Interval Curve

Consider a continuous control outcome with current and historical sample means \(\bar{x}_c\) and \(\bar{x}_h\). Suppose the two sample means have known sampling variances \(\sigma_c^2/n_c\) and \(\sigma_h^2/n_h\), respectively. Assume conditional normal sampling, a common parameter \(\theta\) for the current and historical controls, and a normal prior \(\theta\sim N(\mu_0,\tau_0^2)\). Under a power prior, the historical likelihood is raised to a fixed power \(a\). This model assumes that the historical and current data measure the same endpoint on the same scale and that the common-parameter assumption is reasonable; those assumptions are not established merely by making a plot.

Write the prior precision as \(P_0=1/\tau_0^2\), the current-data precision as \(P_c=n_c/\sigma_c^2\), and the historical-data precision as \(P_h=n_h/\sigma_h^2\). The current and historical precision-weighted means are \(B_0=P_0\mu_0\), \(B_c=P_c\bar{x}_c\), and \(B_h=P_h\bar{x}_h\). Completing the square in the normal prior times the current likelihood and the power-weighted historical likelihood gives posterior precision \(P(a)=P_0+P_c+aP_h\). The posterior mean is the precision-weighted sum of the prior mean and the two data means, divided by that precision:

Thus a central \(100(1-\gamma)\%\) credible interval at each fixed \(a\) is \(m(a)\pm z_{1-\gamma/2}/\sqrt{P(a)}\), where \(z_{1-\gamma/2}\) is the corresponding standard-normal quantile. Plotting the lower and upper endpoints around the mean displays both location and posterior uncertainty. The interval has its stated posterior coverage conditional on the model and fixed discount; it is not a frequentist confidence interval, nor does it assign a probability to \(a\).

What the Curves Can—and Cannot—Show

The mean curve has a simple form in the normal model. Let \(B=B_0+B_c\) and \(C=P_0+P_c\), so \(m(a)=(B+aB_h)/(C+aP_h)\). Differentiating gives:

Because the denominator is positive, the sign of the derivative is determined by the constant numerator \(B_hC-P_hB\). The mean is therefore monotone in \(a\) under this particular model, or constant if that numerator is zero. In the example, \(B=37.875\), \(C=0.2525\), \(B_h=58\), and \(P_h=0.4\), so \(B_hC-P_hB=58(0.2525)-0.4(37.875)=-0.505\). The posterior mean decreases throughout the discount range. This conclusion is specific to the stated conjugate model; do not assume every posterior summary in every model must be monotone.

A second useful curve is a posterior probability tied to a clinical question. For example, if lower systolic pressure is favorable, a decision-maker might examine \(P(\theta<148\text{ mmHg}\mid\text{data},a)\). Under the normal model, this is the standard-normal cumulative distribution function evaluated at \((148-m(a))/\sqrt{1/P(a)}\). The threshold must be justified clinically and expressed in the parameter’s units. A probability curve reports posterior support for that event, conditional on the model and the selected discount; it does not by itself determine the trial’s decision rule.

Binary Outcomes: The Same Plotting Principle

For a binary control outcome, suppose the current data contain \(y_c\) responses among \(n_c\) participants, and the historical data contain \(y_h\) responses among \(n_h\) participants. Assume conditionally independent Bernoulli observations with a common response probability \(p\), and use a \(\operatorname{Beta}(\alpha_0,\beta_0)\) prior. With a fixed power-prior discount, the posterior is \(\operatorname{Beta}(\alpha_0+y_c+ay_h,\ \beta_0+n_c-y_c+a(n_h-y_h))\). The model again requires that a common response probability be appropriate for the current and historical controls.

Building a Reliable Discount Plot

In practice, calculate posterior summaries over a sufficiently fine grid from 0 to 1, including both endpoints and any values that are scientifically prespecified. A grid can be dense enough to reveal curvature without implying more precision than the model or data warrant. For every grid value, keep the likelihood, prior, endpoint, analysis population, and summary definition unchanged; vary only the fixed discount. A multi-step workflow helps prevent accidental changes in the analysis:

A table of selected fixed-discount results is a useful alternative when only a few values are of interest, and it can be easier to read exact numbers from a table than from a graph. A plot is usually better for seeing the overall pattern and possible threshold crossings. Neither replaces model diagnostics or justification of the historical data. A fixed-discount plot also differs from an analysis that assigns a distribution to the discount and averages over it: the former displays conditional results separately, rather than combining them.

Diagnostics, Failure Modes, and Limitations

Check that the plotted means and interval endpoints reproduce direct calculations at selected discounts, especially \(a=0\), \(a=1\), and any clinically emphasized value. Confirm that \(a=0\) removes the historical likelihood contribution and that \(a=1\) corresponds to full power-prior weighting under the chosen model. For simulated or numerically integrated posteriors, check Monte Carlo error or numerical accuracy at every grid point; a jagged curve may be numerical noise rather than a real feature.

• Misleading axis limits. A narrow vertical range can exaggerate a small mean change. Show a scale that supports a clinically meaningful comparison.
• Unclear interval display. Label the interval level and state that it is conditional on the fixed discount. Do not call it a probability that the parameter lies within the realized interval.
• Mixing summaries. A posterior mean curve and a probability curve answer different questions. Label the vertical axis so readers do not mistake one for the other.
• Unjustified common model. Smooth curves do not demonstrate that historical and current data are comparable. Differences in endpoint, population, or data quality can undermine the likelihood assumptions.
• Overinterpreting threshold crossings. A curve crossing a decision threshold marks a change in the displayed posterior probability, not a universal decision rule. The threshold and action rule require clinical and design justification.

A discount plot is conditional on the model and the fixed values selected for display. It cannot repair bias in historical data, establish transportability, or show that borrowing is justified. It is also not a causal analysis. Interpret it alongside the rationale for selecting and discounting historical information, and report the fixed-discount results needed to make the display auditable.