Borrowing Functions Across Data Configurations

Why Compare Borrowing Across Data Configurations?

A borrowing rule is not defined by a single posterior estimate. Its behavior depends on how much historical information is available, how precise the current data are, and how compatible the observed effects appear. Two models can give similar estimates in one configuration yet respond differently when sample sizes or effects change. Examining these changes helps assess whether a model’s borrowing is appropriate for the trial at hand.

We compare a robust mixture prior with a normal commensurate prior using hypothetical treatment-effect data. Effects are measured in points, and the residual standard deviation is known to be 2 points. The historical estimate is \(y_H\), based on \(n_H\) participants, and the current estimate is \(y_C\), based on \(n_C\) participants. For this illustration, the sampling variances are \(4/n_H\) and \(4/n_C\), respectively. We assume independent historical and current sampling errors conditional on their effects.

Models and assumptions: The current likelihood is \(y_C\mid\theta_C\sim N(\theta_C,4/n_C)\). For the mixture model, the prior for \(\theta_C\) is \(wN(y_H,4/n_H)+(1-w)N(0,100)\), with \(w=0.8\). The second component is a protective prior: it is broad, but its center at zero is still a modeling choice. For the commensurate model, suppose \(\theta_H\mid y_H\sim N(y_H,4/n_H)\) and \(\theta_C\mid\theta_H\sim N(\theta_H,1/\tau)\), with \(\tau=1\). Thus its marginal prior for \(\theta_C\) is \(N(y_H,4/n_H+1)\). These calculations treat the stated sampling variances and model parameters as fixed.

Mixture Borrowing: Component Weight and Within-Component Shrinkage

In the mixture model, current data can update the probability that the historical component generated the current effect. Let \(f_H(y_C)\) and \(f_R(y_C)\) be the predictive densities of \(y_C\) under the historical and protective components. These are obtained by integrating the current likelihood over each component prior. The posterior historical-component probability is

q_H=\frac{w f_H(y_C)}{w f_H(y_C)+(1-w)f_R(y_C)}.

For normal prior and likelihood distributions, the predictive distributions are normal: \(y_C\) has mean \(y_H\) and variance \(4/n_H+4/n_C\) under the historical component, and mean zero and variance \(100+4/n_C\) under the protective component. Within either component, the posterior mean is a precision-weighted average of its prior mean and \(y_C\). The overall posterior mean averages the two component-specific posterior means using \(q_H\) and \(1-q_H\).

Interpretation: \(q_H\) is the posterior probability of the historical mixture component under the specified model; it is not itself the proportion of historical information in the posterior estimate. The historical component may still shrink substantially toward \(y_C\), and the protective component may also contribute to the final estimate.

Worked Example: A Baseline Configuration

Suppose \(n_H=4\), \(n_C=16\), \(y_H=2\) points, and \(y_C=1\) point. The historical estimate variance is \(4/4=1\) point² and the current sampling variance is \(4/16=0.25\) point². The historical predictive variance is \(1+0.25=1.25\) point², so

f_H(1)=\frac{\exp[-(1-2)^2/(2(1.25))]}{\sqrt{2\pi(1.25)}}\approx0.2392 \quad\text{per point}.

The protective predictive variance is \(100+0.25=100.25\) point², giving

f_R(1)=\frac{\exp[-1^2/(2(100.25))]}{\sqrt{2\pi(100.25)}}\approx0.03965 \quad\text{per point}.

Therefore,

q_H=\frac{0.8(0.2392)}{0.8(0.2392)+0.2(0.03965)} \approx0.9602.

Under the historical component, the posterior mean is

\frac{(0.25)(2)+(1)(1)}{1+0.25}=1.200\text{ points}.

Under the protective component, it is \(\{(0.25)(0)+(100)(1)\}/100.25\approx0.9975\) points. Thus the overall posterior mean is \(0.9602(1.200)+0.0398(0.9975)\approx1.192\) points. The estimate is near the historical-component result, but not equal to it.

How Sample Sizes Change Borrowing

Holding the observed effects fixed, increasing \(n_H\) reduces the uncertainty in the historical estimate. In the mixture model, this narrows the historical predictive distribution. Increasing \(n_C\) reduces the current sampling variance; this makes the current observation more influential within each component and changes how well each component predicts it. Neither change can be summarized by the sample size alone: the observed \(y_C\) matters because it determines the predictive densities.

Worked Example: More Historical Information

Keep \(y_H=2\), \(y_C=1\), and \(n_C=16\), but increase \(n_H\) from 4 to 16. Now the historical estimate variance is \(4/16=0.25\) point². The historical predictive variance is \(0.25+0.25=0.5\) point², and

f_H(1)=\frac{\exp[-(1-2)^2/(2(0.5))]}{\sqrt{2\pi(0.5)}}\approx0.2076 \quad\text{per point}.

The protective predictive density remains about \(0.03965\) per point. Hence

q_H=\frac{0.8(0.2076)}{0.8(0.2076)+0.2(0.03965)} \approx0.9544.

The posterior historical-component mean is now \(\{(0.25)(2)+(0.25)(1)\}/0.5=1.500\) points. The protective-component mean remains approximately \(0.9975\) points, so the mixture posterior mean is \(0.9544(1.500)+0.0456(0.9975)\approx1.477\) points. The posterior historical-component probability decreases slightly from the baseline even though historical data have increased. This happens because the observation one point below \(y_H\) is less probable under the narrower historical predictive distribution. Conditional on that component, however, the more precise historical estimate pulls the component posterior mean less toward the current estimate.

Worked Example: More Precise Current Data

Return to \(n_H=4\), \(y_H=2\), and \(y_C=1\), and increase \(n_C\) to 64. The current sampling variance is \(4/64=0.0625\) point². The historical predictive variance is \(1.0625\) point², giving \(f_H(1)\approx0.2418\) per point; the protective predictive variance is \(100.0625\) point², giving \(f_R(1)\approx0.03968\) per point. Substitution yields

q_H\approx \frac{0.8(0.2417)}{0.8(0.2417)+0.2(0.03968)} \approx0.9606.

Within the historical component, the posterior mean is \(\{(0.0625)(2)+(1)(1)\}/1.0625\approx1.0588\) points. Within the protective component it is \(100/(100.0625)\approx0.9994\) points. The mixture posterior mean is approximately \(0.9606(1.0588)+0.0394(0.9994)=1.0565\) points. The current estimate is now much more precise, so the historical-component posterior mean is pulled close to \(y_C\), despite a high posterior probability of that component.

Commensurate Shrinkage Across the Same Configurations

The commensurate model does not assign a posterior probability to a historical mixture component. Instead, under the stated normal model, it gives a single normal prior for \(\theta_C\), centered at \(y_H\), with variance \(V_H=4/n_H+1/\tau\). Combining that prior with the current likelihood gives a posterior mean whose coefficient on \(y_H\) is \(s_C^2/(V_H+s_C^2)\), where \(s_C^2=4/n_C\). This coefficient describes the historical estimate’s weight in the posterior mean for this normal setup. It is not interchangeable with \(q_H\).

E(\theta_C\mid y_H,y_C) = \frac{s_C^2}{V_H+s_C^2}y_H+ \frac{V_H}{V_H+s_C^2}y_C, \qquad \operatorname{Var}(\theta_C\mid y_H,y_C) = \frac{V_Hs_C^2}{V_H+s_C^2}.

For the baseline configuration, \(V_H=1+1=2\) point² and \(s_C^2=0.25\) point². The historical coefficient is \(0.25/(2.25)=0.1111\), giving a posterior mean of \(0.1111(2)+0.8889(1)=1.111\) points. With \(n_H=16\), \(V_H=0.25+1=1.25\) point², so the historical coefficient becomes \(0.25/1.5=0.1667\), and the posterior mean is \(1.167\) points. With \(n_C=64\) and \(n_H=4\), the coefficient is \(0.0625/2.0625=0.03030\), and the posterior mean is \(1.030\) points. In this fixed-\(\tau\) example, more precise current data reduce the historical estimate’s coefficient.

Why the commensurate expression holds: Multiplying the normal prior density with mean \(y_H\) and variance \(V_H\) by the normal likelihood with mean \(y_C\) and variance \(s_C^2\) gives a normal posterior. Its precision is \(1/V_H+1/s_C^2\); its mean is that precision’s reciprocal times \(y_H/V_H+y_C/s_C^2\). Rearranging gives the displayed weighted average and variance.

Effect Discrepancy and Model Response

A direct way to examine conflict is to vary \(y_C\) while holding sample sizes and \(y_H\) fixed. In a mixture model, this changes the relative predictive densities and therefore \(q_H\). In a commensurate model with fixed \(\tau\), it changes the posterior mean through the current-data term but does not change the historical coefficient. A commensurate model with an estimated or uncertain precision can behave differently; the fixed-\(\tau\) calculation here illustrates only the specified model.

Worked Example: A Larger Current–Historical Difference

Use \(n_H=1\), \(n_C=16\), \(y_H=2\), and change the current estimate to \(y_C=5\). The historical and current sampling variances are 4 and 0.25 point². The mixture historical predictive variance is \(4.25\) point², and the protective predictive variance is \(100.25\) point². Their densities at 5 are

f_H(5)=\frac{\exp[-(5-2)^2/(2(4.25))]}{\sqrt{2\pi(4.25)}}\approx0.0671, \qquad f_R(5)=\frac{\exp[-5^2/(2(100.25))]}{\sqrt{2\pi(100.25)}}\approx0.03517 \quad\text{per point}.

Thus \(q_H=0.8(0.067123)/\{0.8(0.067123)+0.2(0.03517)\}\approx0.8842\). The historical-component posterior mean is \(\{(0.25)(2)+(4)(5)\}/4.25\approx4.824\) points; the protective-component mean is \(100(5)/100.25\approx4.988\) points. The mixture posterior mean is approximately \(0.8842(4.824)+0.1158(4.988)=4.843\) points.

For the commensurate model with \(\tau=1\), \(V_H=4+1=5\) point², so the historical coefficient is \(0.25/(5.25)=0.04762\). Its posterior mean is \(0.04762(2)+0.95238(5)=4.857\) points. Both posterior means are close to the current estimate, but for different reasons: the commensurate model has a small fixed historical coefficient, whereas the mixture model combines component-specific means using a data-updated component probability.

Diagnostics, Comparisons, and Limitations

A useful comparison is to tabulate posterior means, uncertainty, and borrowing summaries over a planned set of configurations: vary \(n_H\), \(n_C\), and \(y_C-y_H\) separately, then vary them jointly. Check that the calculations use variances on the same scale, that predictive densities are evaluated at the observed current estimate, and that reported density units are inverse points. For the mixture, also inspect each component-specific posterior and its posterior probability. A high \(q_H\) with a component posterior pulled strongly toward \(y_C\) is not evidence that the posterior estimate is mostly historical.

The mixture and commensurate models offer distinct summaries, not a universally ordered amount of borrowing. The mixture permits current data to shift posterior support between a historical and a protective component. The fixed-precision commensurate model expresses borrowing through a continuous normal shrinkage weight; in the formulation above, that weight changes with the specified variances but not with observed effect discrepancy. A robust mixture can be sensitive to the protective component’s center, spread, and prior weight. A commensurate analysis can be sensitive to the precision specification. Neither model automatically resolves differences caused by changes in population, trial conduct, or endpoint measurement.

A common mistake is to read a large posterior historical-component probability as the fraction of the posterior estimate borrowed from historical data. That is incorrect: \(q_H\) weights a component-specific posterior mean, and that mean itself combines the current and historical estimates. Conversely, the commensurate historical coefficient is a weight in a posterior mean under a particular normal model, not a posterior probability that the historical data are commensurate. Keep each summary tied to its model and state which aspect of borrowing it measures.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In the mixture model, what two predictive densities determine \(q_H\), and what does \(q_H\) represent?
With \(n_H=4\), \(n_C=16\), \(y_H=2\), and \(y_C=1\), calculate the historical-component posterior mean.
For the fixed-precision commensurate model, explain how increasing \(n_C\) affects the coefficient on \(y_H\), holding \(n_H\) and \(\tau\) fixed.
Why can \(q_H\) and the commensurate historical coefficient not be interpreted as the same borrowing measure?

Solutions. (1) They are the current-data predictive densities under the historical and protective components; \(q_H\) is the posterior probability of the historical component under the mixture model. (2) The mean is \(\{(0.25)(2)+(1)(1)\}/1.25=1.200\) points. (3) Increasing \(n_C\) reduces \(s_C^2=4/n_C\), so \(s_C^2/(V_H+s_C^2)\) decreases; the current estimate receives greater weight in this posterior mean. (4) \(q_H\) is a probability weighting two component-specific posterior distributions, whereas the commensurate coefficient is a continuous weight in one normal posterior mean. They summarize different model features.