A Worked Continuous-Control Borrowing Analysis

Why Borrowing for a Continuous Control Outcome Needs a Model

A historical control group can contribute information about the control outcome in a current trial, but only to the extent that the two groups can reasonably be treated as informative about a shared parameter. For a continuous outcome, the contribution depends on the observed means, sample sizes, outcome variance, prior information, and the fixed power-prior discount. This tutorial works through that calculation for normally distributed outcomes.

The calculation is simple once its assumptions are explicit. In particular, the model below assumes that historical and current control outcomes have a common mean and a common, known variance. These assumptions make the normal likelihoods and normal prior conjugate: the posterior distribution is also normal. If the groups differ in ways that make a common mean or variance implausible, the calculation can give misleadingly precise results.

Model and assumptions: Let \(Y_{hj}\) be outcome \(j\) in the historical control group and \(Y_{cj}\) outcome \(j\) in the current control group. Conditional on a common mean \(\mu\), assume independent observations with \(Y_{hj}\sim N(\mu,\sigma^2)\) and \(Y_{cj}\sim N(\mu,\sigma^2)\), where the common variance \(\sigma^2\) is known. Assign \(\mu\sim N(\mu_0,\tau_0^2)\), with \(\tau_0^2>0\). Use the current likelihood at full weight and raise the historical likelihood to a fixed power-prior discount \(\delta\), where \(0\leq\delta\leq1\).

Here \(n_h\) and \(n_c\) denote the historical and current sample sizes, and \(\bar{y}_h\) and \(\bar{y}_c\) their sample means. The outcome and all mean parameters use the same measurement units; \(\sigma^2\) and \(\tau_0^2\) have squared units. The discount is dimensionless. A value of \(\delta=0\) excludes the historical likelihood, while \(\delta=1\) gives it full likelihood weight under this model. Intermediate values discount its contribution.

The fixed discount does not make the historical observations disappear or change their observed mean. It multiplies their likelihood contribution to the posterior. With a common known variance, the historical contribution to posterior precision is equivalent algebraically to \(\delta n_h\) observations. This is an effective-information interpretation, not a claim that the historical sample literally contained that many patients.

Posterior Distribution and Derivation

For normal observations with known variance, the parts of the current and historical likelihoods that depend on \(\mu\) are proportional to \(\exp\{-n_c(\bar{y}_c-\mu)^2/(2\sigma^2)\}\) and \(\exp\{-n_h(\bar{y}_h-\mu)^2/(2\sigma^2)\}\), respectively. Raising the historical likelihood to \(\delta\) multiplies its quadratic contribution by \(\delta\). Multiplying these likelihood contributions by the normal prior gives the posterior kernel.

\begin{aligned} \pi(\mu\mid \text{data},\delta) &\propto \exp\left\{-\frac{1}{2}\left[ \frac{(\mu-\mu_0)^2}{\tau_0^2} +\frac{n_c(\mu-\bar{y}_c)^2}{\sigma^2} +\frac{\delta n_h(\mu-\bar{y}_h)^2}{\sigma^2} \right]\right\}. \end{aligned}

Define the posterior precision as the coefficient of \(\mu^2\) inside the brackets, and define \(V_\delta\) as its reciprocal. Collecting the quadratic terms gives the posterior variance \(V_\delta\) and posterior mean \(m_\delta\) below.

V_\delta= \left( \frac{1}{\tau_0^2}+\frac{n_c+\delta n_h}{\sigma^2} \right)^{-1}, \qquad m_\delta= V_\delta\left( \frac{\mu_0}{\tau_0^2} +\frac{n_c\bar{y}_c+\delta n_h\bar{y}_h}{\sigma^2} \right).

Posterior result: Under the stated model, \(\mu\mid\text{data},\delta\sim N(m_\delta,V_\delta)\). Its mean is a precision-weighted combination of the prior mean and the two sample means; its variance is the reciprocal of the combined prior and likelihood precision. The posterior standard deviation is \(\sqrt{V_\delta}\), in the original outcome units.

The sign in the mean follows directly from completing the square. Write \(A=1/\tau_0^2+n_c/\sigma^2+\delta n_h/\sigma^2\) for the coefficient of \(\mu^2\) and \(B=\mu_0/\tau_0^2+n_c\bar{y}_c/\sigma^2+\delta n_h\bar{y}_h/\sigma^2\) for the weighted sum. Expanding each squared term gives the linear term \(-2B\mu\) inside the brackets, not \(+2B\mu\). Thus the bracketed expression has the form \(A\mu^2-2B\mu+C\), which can be written as \(A(\mu-B/A)^2+C-B^2/A\). Therefore the posterior center is \(B/A\), the formula for \(m_\delta\), and the variance is \(1/A\).

A\mu^2-2B\mu+C = A\left(\mu-\frac{B}{A}\right)^2+C-\frac{B^2}{A}.

For a central 95% posterior credible interval in this normal model, use \(m_\delta\pm1.96\sqrt{V_\delta}\). This interval describes posterior uncertainty about \(\mu\), conditional on the model, data, prior, and chosen discount. It is not a statement that the fixed parameter has a 95% probability of lying in this particular interval under a frequentist interpretation.

Worked Calculations

Worked Example: Discounted Historical Blood-Pressure Outcomes

Suppose a hypothetical trial measures change in systolic blood pressure in mmHg. Assume the common known outcome variance is \(\sigma^2=100\;(\text{mmHg})^2\). The historical controls have \(n_h=40\) and mean \(\bar{y}_h=-8\) mmHg; the current controls have \(n_c=20\) and mean \(\bar{y}_c=-4\) mmHg. Use the prior \(\mu\sim N(0,20^2)\) and fixed discount \(\delta=0.25\). Assume the common-mean and common-variance model is scientifically defensible for this illustration.

The prior precision is \(1/20^2=0.0025\) per squared mmHg. The discounted historical precision is \(0.25(40)/100=0.10\), and the current precision is \(20/100=0.20\). Hence

V_\delta= \left(0.0025+0.10+0.20\right)^{-1} =3.3058\;(\text{mmHg})^2.

The weighted numerator is \(0/20^2+0.10(-8)+0.20(-4)=-0.8-0.8=-1.6\) mmHg\(^{-1}\). Therefore

m_\delta=\frac{-1.6}{0.3025}=-5.289\text{ mmHg}, \qquad \sqrt{V_\delta}=\sqrt{3.3058}=1.818\text{ mmHg}.

The 95% posterior credible interval is \(-5.289\pm1.96(1.818)\), or approximately \((-8.853,-1.726)\) mmHg. Conditional on the specified model and discount, the posterior mean change is \(-5.289\) mmHg, and the interval expresses uncertainty about the common control mean. As a contrast, with \(\delta=0\), the posterior mean would be \(-0.8/(0.2025)=-3.951\) mmHg and the posterior standard deviation would be \(\sqrt{1/0.2025}=2.222\) mmHg. Here borrowing shifts the posterior mean toward the more negative historical mean and reduces posterior uncertainty.

Worked Example: Historical and Current Means Disagree

Consider a hypothetical continuous outcome measured in units. Let \(\sigma^2=144\), \(n_h=30\), \(\bar{y}_h=5\), \(n_c=15\), and \(\bar{y}_c=1\). Use \(\mu\sim N(0,30^2)\) and \(\delta=0.40\). Assume the same common-mean, common-known-variance model for the calculation, while recognizing that the four-unit difference between sample means deserves scrutiny.

The prior precision is \(1/900=0.001111\), the discounted historical precision is \(0.40(30)/144=0.083333\), and the current precision is \(15/144=0.104167\). Their sum is \(0.188611\), so

V_\delta=\frac{1}{0.188611111\ldots}=5.3019, \qquad m_\delta= \frac{0.083333(5)+0.104167(1)}{0.188611} = \frac{0.520833}{0.188611} =2.761.

The posterior standard deviation is \(\sqrt{3600/679}=2.303\) outcome units, giving the 95% credible interval \(\frac{1875}{679}\pm1.96\sqrt{\frac{3600}{679}}\approx(-1.752,7.274)\) units. Without historical borrowing, the posterior mean would be \(0.104167/(0.001111+0.104167)=0.990\) units, close to the current mean. With borrowing, the posterior mean moves toward the historical mean of 5. This is exactly what the formula does; it does not establish that the historical and current populations truly share a mean. The wide interval also makes clear that borrowing does not guarantee a precise estimate.

Worked Example: Prior Information and the Current Data

Suppose a hypothetical outcome is measured in points, with known common variance \(\sigma^2=64\). Historical controls have \(n_h=25\) and mean \(0\); current controls have \(n_c=25\) and mean \(4\). Set \(\mu\sim N(2,10^2)\) and \(\delta=0.20\). For calculation, suppose the model’s shared mean and variance assumptions are appropriate.

The prior precision is \(0.01\), the historical precision contribution is \(0.20(25)/64=0.078125\), and the current contribution is \(25/64=0.390625\). Thus total precision is \(0.47875\), while the weighted numerator is \(2(0.01)+0(0.078125)+4(0.390625)=1.5825\). It follows that

m_\delta=\frac{1.5825}{0.47875}=3.3055, \qquad V_\delta=\frac{1}{0.47875}=2.0888, \qquad \sqrt{V_\delta}=1.445.

The 95% credible interval is \(3.3055\pm1.96(1.445)=(0.473,6.138)\) points. The current group contributes more precision than either the historical data or the prior in this example. The posterior is therefore pulled toward the current mean of 4, while still reflecting the prior mean and the discounted historical mean.

Assumptions, Diagnostics, and Alternatives

The formula is appropriate only when its assumptions support the intended inference. Conditional independence is needed for the likelihood to factor across observations. A common mean says that historical and current controls target the same control-outcome parameter. A common known variance says that each observation has the same specified outcome variance. The prior must be a proper normal distribution for the stated prior interpretation, and \(\delta\) must be fixed rather than estimated from these data in this calculation.

Before interpreting the posterior, inspect the historical and current data in their clinical and trial context. Check whether outcome definitions, measurement procedures, eligibility criteria, follow-up, and background care are sufficiently comparable to make a shared mean meaningful. Look for unusual observations, skewness or marked departures from normality, and evidence that the groups’ variability differs. In practice \(\sigma^2\) is rarely known exactly; treating it as known is an idealized modeling assumption, not a consequence of observing sample variances. If these checks raise concerns, the displayed normal-normal calculation may understate uncertainty or borrow information that does not apply to the current controls.

A difference between \(\bar{y}_h\) and \(\bar{y}_c\) is a useful warning signal, not a definitive test of exchangeability. Conversely, similar observed means do not prove that the groups are comparable: small samples may conceal a difference, and measured means cannot diagnose every change in study conduct or population. The fixed discount is a modeling choice that reduces historical influence; it does not repair a biased outcome definition or make dissimilar populations exchangeable.

Approach	Historical contribution	What it implies
No borrowing, \(\delta=0\)	None	The posterior uses the prior and current controls only.
Fixed discount, \(0<\delta<1\)	Reduced by the chosen power	Historical information influences both posterior mean and precision, conditional on the model.
Full borrowing, \(\delta=1\)	Full historical likelihood weight	Historical and current data contribute as observations from the same modeled distribution.

No borrowing avoids dependence on historical outcomes but can leave less information about the control mean. Full borrowing is more efficient when the shared model is credible, but is most exposed to incompatibility. A fixed intermediate discount makes the compromise transparent, though its value is not learned from the data in this analysis. Results should therefore be understood conditional on the selected discount; examining more than one defensible fixed value can show how sensitive the posterior is to that choice.

Common Pitfall and Key Interpretation

Wrong reasoning: “A discount of 0.25 means the historical mean is moved 25% of the way toward the current mean.” Correction: the discount multiplies the historical likelihood’s contribution to precision and to the weighted numerator. The resulting posterior mean depends on the prior, both sample sizes, the variance, and both observed means; it is not generally a simple 25% interpolation.
Wrong reasoning: “The posterior interval is narrow, so historical and current controls must be comparable.” Correction: the interval measures uncertainty conditional on the assumed model. It cannot verify the common-mean assumption that made borrowing possible.
Wrong reasoning: “The historical data count as \(\delta n_h\) real patients.” Correction: \(\delta n_h\) is an algebraic effective-information interpretation under this normal, common-variance model. It is not a literal sample size.

Key takeaway: Under a common known-variance normal model and a normal prior, fixed power-prior borrowing adds \(\delta n_h/\sigma^2\) to posterior precision and \(\delta n_h\bar{y}_h/\sigma^2\) to the precision-weighted numerator. Both the mean and uncertainty depend on the chosen discount and the validity of the shared model.

Check Your Understanding

Work through each question, then compare with the solutions below.

For \(n_h=20\), \(\bar{y}_h=3\), \(n_c=10\), \(\bar{y}_c=1\), \(\sigma^2=25\), prior \(N(0,10^2)\), and \(\delta=0.50\), calculate the posterior mean and standard deviation.
What happens to the historical likelihood contribution when \(\delta=0\)? What happens when \(\delta=1\)?
Does a posterior 95% credible interval have a 95% frequentist probability of containing the fixed \(\mu\)? Explain.
Name two reasons why a small difference between the historical and current sample means does not establish that borrowing is appropriate.

Solutions. (1) The prior precision is \(0.01\); the historical contribution is \(0.50(20)/25=0.40\); and the current contribution is \(10/25=0.40\). Total precision is \(0.81\). The numerator is \(0.40(3)+0.40(1)=1.60\), so \(m_\delta=1.60/0.81=1.975\). The posterior standard deviation is \(\sqrt{1/0.81}=1.111\) outcome units. (2) At \(\delta=0\) the historical likelihood contributes nothing; at \(\delta=1\) it has full likelihood weight under the model. (3) No. The 95% posterior probability statement is conditional on the model, prior, data, and discount; it is not a repeated-sampling probability statement about a fixed parameter. (4) Small samples can conceal differences, and similar means do not rule out changes in eligibility, outcome measurement, follow-up, or other aspects of the study context.