What is already known?

• • Shrinkage estimation may be used to effectively and robustly borrow information between related data sources.

• • Shrinkage estimation may alternatively be motivated via a meta-analytic-predictive (MAP) approach.

What is new?

• • A MAP approach remains sensible down to the extreme case of only a single study.

• • The MAP prior’s usual features are retained, in addition, there are connections to power prior and bias allowance approaches.

Potential impact for RSM readers

• • MAP priors are useful for constructing empirically motivated priors based on external/historical data.

• • MAP priors may serve as an additional motivation for related approaches (bias allowance models and power priors).

• • Practical application is straightforward using existing software packages.

2.1 The normal–normal hierarchical model

The most common model for random-effects meta-analysis is given by the NNHM. It implements sampling error as well as between-study heterogeneity using normal distributions. The data are given in terms of k estimates $y_i$ and associated standard errors $s_i$ ( $i=1,\ldots ,k$ ). Each individual study aims to quantify a parameter $\theta _i$ , so that

(2.1) $$ \begin{align} y_i|\theta_i,s_i \;\sim\;{\mathrm{Normal}}(\theta_i, s_i^2){.} \end{align} $$

The underlying parameters $\theta _i$ are not necessarily identical for all studies, instead some amount of (between-study) heterogeneity is allowed for, expressed as

(2.2) $$ \begin{align} \theta_i|\mu,\tau \;\sim\; {\mathrm{Normal}}(\mu,\tau^2) {.} \end{align} $$

Often the overall mean $\mu $ is the aim of the analysis, while sometimes the study-specific parameters $\theta _i$ are also of interest.Reference Röver and Friede ^{9 ,} Reference Wandel, Neuenschwander, Röver and Friede ¹⁶ The heterogeneity, while important, usually remains a nuisance parameter. In the context of “shrinkage estimation” of the $\theta _i$ , an interesting aspect is that the problem may be motivated in two ways; classically, one may think of shrinkage estimation as a joint analysis of all (k) estimates, which also returns estimates of any $\theta _i$ parameter along the way; this is also denoted as the meta-analytic-combined (MAC) approach. The problem, however, may also be factored into the evidence stemming from the ith study alone, as well as the information provided by the remaining ( $k-1$ ) estimates. Shrinkage estimation then may be interpreted as the analysis of the ith study, based on a prior distribution that results as the predictive distribution derived from a meta-analysis of the other ( $k-1$ ) studies; this prior is denoted as the meta-analytic-predictive (MAP) prior. Both MAC and MAP approaches are equivalent and yield identical shrinkage estimates.Reference Schmidli, Gsteiger, Roychoudhury, O’Hagan, Spiegelhalter and Neuenschwander ⁸

In the following, we will focus on the special case of only two studies ( $k=2$ ). For the shrinkage estimate ( $\theta _2$ ), this implies a MAP prior that is based on a single study (i.e., the data provided through $y_1$ and $s_1$ ). While this may appear odd at first, the idea readily applies also in this special case, as will be demonstrated in the following. Analysis may generally be performed based on informative or uninformative priors for $\mu $ , while a proper, informative prior is required for $\tau $ .Reference Röver ^{7 ,} Reference Röver, Bender and Dias ¹² The case of only $k=2$ studies is also closely connected to the related concepts of power prior Reference Röver and Friede ⁹ or bias allowance models.Reference Welton, Sutton, Cooper, Abrams and Ades ^{13 ,} Reference Pocock ¹⁷

2.2 Uniform prior for the overall mean effect ( $\mu $ )

Priors for the overall mean parameter ( $\mu $ ) in the NNHM may be specified as informative or as uninformative. For (more or less informative) priors, normal distributions are an obvious choice, also since these lead to analytically simple inference. Quite commonly, effect priors however are chosen as uninformative and (improper) uniform, not least due to certain analogies to frequentist meta-analysis procedures.Reference Röver ⁷ In case of an (improper, non-informative) uniform prior for $\mu $ , certain expressions turn out particularly simple, which is also why we will focus on this particular, yet insightful, common and practically relevant case in the following. As the (improper) uniform prior constitutes the limiting case of an increasingly uninformative effect prior, the following considerations may also be viewed as relating to the limiting behavior for increasingly uninformative priors (e.g., for normal priors when their variance approaches infinity). The uniform effect prior leads to a normal conditional posterior for the overall mean effect $\mu $ , with moments given by

(2.3) $$ \begin{align} {\mathrm{E}}[\mu|y_1, s_1, \tau] \;=\; y_1 \quad \text{and} \quad {\mathrm{Var}}(\mu|y_1, s_1, \tau) \;=\; s_1^2 + \tau^2 {,} \end{align} $$

and a marginal heterogeneity likelihood that is constant (independent of $\tau $ ), so that the heterogeneity’s posterior equals its prior.Reference Röver ⁷ This seems reasonable, since (as long as the overall mean effect prior is uniform) a single observation $y_1$ does not provide information on the heterogeneity $\tau $ .

2.3 The MAP prior for the effect in a new study

The MAP prior results as the posterior predictive distribution for a “new,” second study’s (study-specific) effect $\theta _2$ given the data from the first study ( $y_1$ , $s_1$ ). In the NNHM framework, the conditional predictive distribution again is normal with mean

(2.4) $$ \begin{align} {\mathrm{E}}[\theta_2|y_1, s_1, \tau] \;=\; {\mathrm{E}}[\mu|y_1, s_1, \tau] \;=\; y_1 \end{align} $$

and variance

(2.5) $$ \begin{align} {\mathrm{Var}}(\theta_2|y_1, s_1, \tau) \;=\; {\mathrm{Var}}(\mu|y_1, s_1, \tau) + \tau^2 \;=\; s_1^2 + 2\tau^2 \end{align} $$

(see also (2.2) and (2.3), and the more detailed derivation in Appendix A.1). These expressions make sense in the present context: We know $\theta _1$ with accuracy given by the standard error $s_1$ , and we know that the difference between $\theta _1$ and $\theta _2$ is normally distributed with variance $2\tau ^2$ , so that the (conditional) variance expression results as a corresponding sum.Reference Röver and Friede ⁹ As pointed out above, information on heterogeneity ( $\tau $ ) so far is based on the prior only; the heterogeneity in this context generally requires a proper, informative prior (since $k<3$ ).Reference Röver ^{7 ,} Reference Röver, Bender and Dias ¹²

The eventual (marginal) predictive distribution (marginalized over the distribution of $\tau $ ) hence results as a normal scale mixture Reference Lee, McLachlan, Balakrishnan, Colton, Everitt, Piegorsch, Ruggeri and Teugels ^{18 ,} Reference Lindsay ¹⁹ with fixed mean (2.4) and with variance as given in (2.5), where $\tau $ is distributed according to the specified prior. In particular, one may think of the MAP prior as the first study’s point estimate $p(\theta _1|y_1,s_1)$ convolved with the prior predictive distribution $p(\theta _2|\theta _1,\tau )$ and then marginalized over $\tau $ . The MAP prior is symmetric around $y_1$ , and its (marginal) variance results from (2.5) as

(2.6) $$ \begin{align} {\mathrm{Var}}(\theta_2|y_1, s_1) \;=\; {\mathrm{E}}[s_1^2+2\tau^2] \;=\; s_1^2 + 2\,{\mathrm{E}}[\tau^2] {,} \end{align} $$

where the expectation ${\mathrm {E}}[\tau ^2]$ depends on the assigned heterogeneity prior. For a range of common prior specifications, this expectation may be derived analytically; Table A1 in Appendix A.4 lists some popular cases. From this expression, one can see that the relative magnitudes of $s_1^2$ and ${\mathrm {E}}[\tau ^2]$ determine whether the resulting MAP prior’s variance is dominated by estimation uncertainty (regarding $\theta _1$ ) or anticipated heterogeneity ( $\tau $ ). These two variance components may in fact also be considered to reflect so-called “type A” and “type B” uncertainties relating to measurement uncertainty and background knowledge, respectively, which together sum up to form the combined standard uncertainty $u_c$ (the square root of (2.6)).Reference Kirkup ^{20 ,} Reference van der Bles, van der Linden and Freeman ²¹

Since the MAP prior results as a normal scale mixture, it is generally heavier-tailed than a normal distribution, which has implications for the resulting operating characteristics. A heavy-tailed (MAP-) prior means that in combination with a (“shorter-tailed”) normal likelihood, the likelihood will dominate in case of a prior-data conflict.Reference O’Hagan and Pericchi ²² Such robustness properties have in fact been noted and demonstrated in the meta-analysis context, as these lead to a dynamic borrowing behavior.Reference Röver and Friede ⁹ The MAP priors’ tail behavior will also be illustrated for some examples below (Figures 4 and 5).

Another way to quantify the precision of a prior distribution is by relating it to a number of observations that would in a certain sense convey an equivalent amount of information. In the following, we will use two approaches to this effect. Firstly, a prior may be assessed in terms of a corresponding absolute effective sample size (ESS) (here: number of patients), this will be done by quoting ESSs based on the expected local-information-ratio ( ${\mathrm {ESS}_{\mathrm {ELIR}}}$ ). This measure is based on the prior density’s curvature and it ensures predictive consistency, that is, on expectation, the posterior’s ESS will be the sum of the prior’s ${\mathrm {ESS}_{\mathrm {ELIR}}}$ plus the actual sample size.Reference Neuenschwander, Weber, Schmidli and O’Hagan ²³ Secondly, the added information from including the prior in a specific analysis may be expressed in terms of the (relative) gain in ESS.Reference Röver and Friede ⁹ This is based on comparing the relative width of a confidence interval with and without considering the informative prior, and then determining by what factor the sample size would have needed to be increased to yield the same precision gain (see also Appendix A.2).

2.4 The bias allowance model connection

In the 2-study case, there is a one-to-one correspondence between the NNHM and a simple bias allowance model Reference Welton, Sutton, Cooper, Abrams and Ades ¹³ ; instead of the NNHM assumption (2.2) in combination with a uniform prior for the overall mean effect $\mu $ and heterogeneity prior $p_\star (\tau )$ as in Section 2.1, one may specify

(2.7) $$ \begin{align} \theta_2|\alpha,\beta &= \alpha\qquad\ \ \qquad \end{align} $$

(2.8) $$ \begin{align}\kern9pt \theta_1|\alpha,\beta &\sim {\mathrm{Normal}}(\alpha, \, \beta^2) \end{align} $$

with prior $p(\beta )=\frac {1}{\sqrt {2}}\,p_{\star }\bigl (\frac {\beta }{\sqrt {2}}\bigr )$ for the standard deviation $\beta $ .Reference Röver and Friede ⁹ This “reference model” is different in that one estimate (the reference, or “target” $y_2$ ) directly relates to $\alpha $ , while the other one (the “source” $y_1$ ) is associated with an additional offset to account for potential bias. The shrinkage estimates of $\theta _i$ , however, can be shown to be identical in both models as long as a uniform prior for the overall mean effect ( $\mu $ ) is used.Reference Röver and Friede ⁹ The reference model may be considered a variation of Pocock’s bias model or, more generally, a bias allowance model.Reference Welton, Sutton, Cooper, Abrams and Ades ^{13 ,} Reference Pocock ^{17 ,} Reference Neuenschwander, Schmidli, Lesaffre, Baio and Boulanger ²⁴ The $\beta $ parameter, which only differs from $\tau $ by a scaling factor of $\sqrt {2}$ , may also help motivating a heterogeneity prior, as it directly relates to the expected difference between $\theta _2$ and $\theta _1$ , without reference to a common overall mean $\mu $ .

2.5 The power prior connection

There is also a connection to a so-called power prior, which has been proposed as an approach for deliberate down-weighting of prior information. It is intended for a prior distribution that itself results as a posterior, and the power prior results from applying an exponent $a_0$ (with $0 \leq a_0 \leq 1$ ) to its likelihood contribution.Reference Neuenschwander, Schmidli, Lesaffre, Baio and Boulanger ^{24 ,} Reference Ibrahim and Chen ²⁵ When conditioning on a fixed $\tau $ value, the (conditional) MAP prior is normal with moments given in (2.4) and (2.5); in particular, note that $\tau ^2$ acts additively on the “plain” variance ( $s_1^2$ ). In this context, a power prior with fixed exponent $a_0$ on the other hand would correspond to a ${\mathrm {Normal}}\Bigl (y_1,\,\frac {s_1^2}{a_0}\Bigr )$ distribution, where the (inverse) exponent acts multiplicatively on the variance. Both MAP and power prior then are identical if $a_0=\bigl (2\frac {\tau ^2}{s_1^2}+1\bigr )^{-1}$ .Reference Röver and Friede ^{9 ,} Reference Chen and Ibrahim ^{26 ,} Reference Pawel, Aust, Held and Wagenmakers ²⁷ It is interesting to note that the relationship between $a_0$ and $\tau $ here depends on the ratio $\frac {\tau }{s_1}$ ; Pawel et al. (2024)Reference Pawel, Aust, Held and Wagenmakers ²⁷ point out that the exponent $a_0$ directly relates to the “relative” heterogeneity as expressed though the popular $I^2$ statistic,Reference Higgins and Thompson ²⁸ which in this case simply equals $I^2=\frac {\tau ^2}{\tau ^2+s_1^2}$ . The exponent may then directly be expressed as a function of the corresponding $I^2$ as $a_0=\frac {1-I^2}{1+I^2}$ . While a prior probability distribution for $\tau $ is readily motivated with reference to the effect scale of $y_i$ and $\theta _i$ ,Reference Röver, Bender and Dias ¹² specification of a fixed $\alpha _0$ value remains tricky, and a prior distribution may be even harder to motivate, as it would implicitly relate to the $I^2$ scale. On the other hand, through the above functional correspondence, any prior for the heterogeneity $\tau $ immediately implies a corresponding distribution for the exponent $a_0$ ; an example is illustrated in Appendix A.5.

3.1 Pediatric Alport example

3.1.1 Application

Gross et al. (2020)Reference Gross, Tönshoff and Weber ²⁹ performed a randomized controlled trial (RCT) in Alport syndrome to investigate the effects of ramipril, an angiotensin-converting enzyme inhibitor (ACEi). Recruitment of participants to the RCT was hampered by the rare and pediatric nature of the disease, and so the analysis of the RCT had been planned with the inclusion of observational data from an open-label arm and a natural disease cohort.Reference Gross, Friede and Hilgers ³⁰ Time to disease progression was a co-primary endpoint, and from the observational data, a hazard ratio (HR) of 0.53 [0.22, 1.29] was estimated based on 70 patients. Only 20 patients entered into the RCT, and an HR of 0.51 [0.12, 2.20] was estimated. The data are also summarized in Table 1.

Table 1 Alport example data from Gross et al. (2020).Reference Gross, Tönshoff and Weber ²⁹

The analysis was then performed by jointly considering both log-HR estimates in an NNHM, anticipating a reasonable amount of heterogeneity between them (expressed through a $\text {half-Normal}(0.5)$ prior for $\tau $ ), and deriving a shrinkage estimate for the RCT effect ( $\theta _2$ ). The resulting estimate then was substantially more precise than if the RCT data were considered in isolation; the HR was estimated at 0.52 [0.19, 1.39].Reference Gross, Tönshoff and Weber ²⁹

To make the flow of information transparent, we may now derive the corresponding MAP prior reflecting the information contributed by the observational data. Figure 1 illustrates MAP-prior, likelihood, and posterior (shrinkage estimate) for the Alport example. The MAP prior here has a mean of $y_i=-0.63$ and a variance of $s_1^2 + 2\,{\mathrm {E}}[\tau ^2] = 0.45^2 + 2\times 0.5^2 = 0.84^2$ . While the observational sample size was 70 patients, the MAP prior’s effective sample size ( ${\mathrm {ESS}_{\mathrm {ELIR}}}$ ) is at only 26 patients (i.e., 37% of originally 70 actual patients).Reference Neuenschwander, Weber, Schmidli and O’Hagan ²³ The eventual shrinkage interval is only 67% as wide as the original, implying a substantial “effective gain in sample size”Reference Röver and Friede ⁹ ; such a precision increase would otherwise have required more than doubling the sample size (by the addition of 24 extra patients). So in this case, the absolute ( ${\mathrm {ESS}_{\mathrm {ELIR}}}$ ) estimate matches well the observed precision gain.

Figure 1 Illustration of MAP-prior, likelihood, and (shrinkage-) posterior for the Alport example discussed in Section 3.1.Reference Gross, Tönshoff and Weber ²⁹ The horizontal lines at the bottom indicate point estimates and corresponding 95% intervals.

3.1.2 Variations of the MAP prior

The heterogeneity ( $\tau $ ) prior was specified as a ${\text {half-Normal}}(0.5)$ distribution, which is a reasonably conservative choice for endpoints such as HRs, as it covers “reasonable” and up to “fairly high” levels of heterogeneity ( $\tau \leq 1$ ) and leaves a small prior probability for “fairly extreme” amounts ( $\tau>1$ ).Reference Röver, Bender and Dias ^{12 ,} Reference Friede, Röver, Wandel and Neuenschwander ³¹ Since conclusions heavily depend on the heterogeneity prior settings, it may however be interesting to investigate the effects of a range of reasonable alternative specifications; in particular, we will consider different prior scales and different distribution families. Among the various assumptions implemented in the analysis, a “too optimistic” heterogeneity prior (favoring small heterogenity) might yield results inappropriately close to a common-effect analysis, while overly “pessimistic” or “conservative” assumptions may on the other hand eventually lead to very little borrowing of information.

Assuming that $s_1=0.451$ (as in the present example, see Table 1), we can illustrate the resulting MAP prior when varying the heterogeneity prior scale. Figure 2 shows the likelihood of the observational estimate along with the corresponding MAP priors for half-normal heterogeneity priors with scales 0.25, 0.50, and 1.00. Increasing the heterogeneity prior scale yields a MAP prior that becomes increasingly wider than the plain likelihood alone. In the present case, the effect scale was a logarithmic HR, and on the logarithmic scale, the original MAP prior (based on a ${\text {half-Normal}}(0.5)$ heterogeneity prior) covers a range of $y_1 \pm 1.72$ with 95% probability. On the exponentiated scale (see also the top axis in Figure 2), a difference of 1.72 in log-HR would correspond to a 5.6-fold larger HR. If one switched to a ${\text {half-Normal}}(0.25)$ or ${\text {half-Normal}}(1.0)$ prior instead, the range would change to $y_1\pm 1.13$ or $y_1\pm 3.18$ instead, corresponding to multiplicative factors of 3.1 or 24.0, respectively.

Figure 2 Illustration of the resulting MAP-prior for varying heterogeneity prior scales. The dashed line indicates the likelihood of the observational data alone for comparison.

Half-normal distributions are a common and obvious choice as heterogeneity priors, possible reasons may be familiarity and availability, as well as a “flat” shape near the origin and a rather short tail.Reference Röver, Bender and Dias ¹² Variations of the distribution family commonly do not alter conclusions dramatically as long as they cover a similar range, as manifested, for example, in a common prior median.

Figure 3 Illustration of MAP-prior’s dependence on the heterogeneity prior distribution family. The different heterogeneity priors shown here all share the same prior median.

Figure 4 The MAP-priors’ cumulative distribution functions corresponding to the densities shown in Figure 3.

Figure 5 The MAP-priors’ densities on a logarithmic scale (see also Figure 3). Note that the likelihood for the observational data alone follows a parabola shape here, while the corresponding MAP priors are clearly much heavier-tailed.

MAP priors corresponding to alternative specifications to a ${\text {half-Normal}}(0.5)$ for the heterogeneity are illustrated in Figure 3. A range of distribution families is used, with their scale parameters specified such that all correspond to a common prior median for $\tau $ (of $0.34$ ). These different heterogeneity prior families are also shown in Appendix A.3. The resulting MAP prior densities themselves are hard to distinguish. Differences are more noticeable when focusing on the tail behavior, for example, considering cumulative distribution functions (as shown in Figure 4) or logarithmic densities (shown in Figure 5). One can see that heavier-tailed heterogeneity priors also yield correspondingly heavier-tailed MAP-priors. The heterogeneity and the corresponding MAP priors’ properties are also summarized and compared in Table 2 in terms of prior quantiles and effective sample sizes ( ${\mathrm {ESS}_{\mathrm {ELIR}}}$ ).

Table 2 Summaries of MAP priors resulting from several settings for the heterogeneity ( $\tau $ ) prior. The half-normal(0.5) prior is contrasted with half-normal priors of differing scale, as well as with priors of differing distributional families, but with matching prior medians. Note that in the context of the present example, the MAP prior’s domain corresponds to logarithmic hazard ratios (log-HRs). Quantiles are centered at $y_1$

It is sometimes also instructive to observe the effects of variations of the prior on the resulting estimates; for example, varying the heterogeneity prior scale allows for a sensitivity (or tipping point) analysis. Such an analysis is shown in Appendix A.6; the amount of borrowing is reflected in the shrinkage interval’s width, but in the present example, inference would not change qualitatively, and a log-HR of zero always remains included.

3.2 Heart failure example

The Spirit-HF trial has been designed in order to test the efficacy of spironolactone in patients with heart failure (HF).Reference Pieske ³² Spironolactone is expected to reduce cardiovascular mortality as well as hospitalizations due to HF, and had previously been investigated in the Topcat trial.Reference Pitt, Pfeiffer and Assmann ³³ Both studies refer to the composite of (recurrent) HF hospitalization and cardiovascular death as the primary endpoint to evaluate treatment efficacy. Despite a sizeable sample size of 3,445 patients and a mean follow-up duration of more than three years, the Topcat trial failed to demonstrate statistical significance; the estimated HR was at 0.89 (0.77, 1.04) ( $p=0.14$ ).Reference Pitt, Pfeiffer and Assmann ³³

The analysis of the new Spirit-HF trial meanwhile is being planned, and may take into consideration the evidence already generated in the Topcat trial. One idea may be to derive a shrinkage estimate, anticipating some between-study heterogeneity, and dynamically borrowing information from the earlier study based on the corresponding MAP prior.Reference Röver and Friede ^{9 ,} Reference Röver and Friede ¹⁰ For the between-study heterogeneity $\tau $ , use of a ${\text {half-Normal}}(0.25)$ prior may be appropriate. The heterogeneity prior may be motivated referring to anticipated levels of heterogeneity based on general considerationsReference Röver, Bender and Dias ¹² or using empirical evidence, in particular in view of the similar study designs and the effect measure being a log-HR.Reference Lilienthal, Sturtz and Schürmann ³⁴

In the present case, analysis is based on the logarithmic HR; the HR estimated in the Topcat trial corresponds to a log-HR of $-0.117$ with a standard error of $0.077$ . The corresponding Topcat likelihood along with the resulting MAP prior is illustrated in Figure 6. The variance (squared standard error) of the Topcat study’s estimate was $s_1^2=0.077^2$ while for the assumed heterogeneity prior the expected heterogeneity variance is ${\mathrm {E}}[\tau ^2]=0.25^2$ (see also Table A1), so that the resulting MAP prior’s variance (2.6) is $s_1^2 + 2\,{\mathrm {E}}[\tau ^2] = 0.362^2$ , and the majority of the variance is due to epistemic uncertainty relating to the anticipated similarity of the Topcat and Spirit-HF parameters. The 95% prediction interval for the MAP prior is centered at the Topcat log-HR estimate and ranges from $-0.899$ to $+0.665$ , corresponding to HRs in the range [ $0.407$ , $1.945$ ]. According to the MAP prior, the probability of a beneficial treatment effect (a log-HR below zero) is 71%. The MAP prior has an ${\mathrm {ESS}_{\mathrm {ELIR}}}$ of 399, that is, only 12% of the 3,445 actual Topcat patients, and 31% of the estimated enrolment of 1,300 Spirit-HF patients.Reference Pieske ³² This means that the prior derived from the Topcat data will not enter the eventual analysis as an additional 3,445 patients (as would be the case if both study populations were pooled naïvely), but instead we expect an accuracy corresponding to a total of some $1,300+399$ patients. The Spirit-HF study’s contribution to its own shrinkage estimate may also be assessedReference Röver and Friede ¹⁰ ; assuming that both studies show the same dependence of standard error and sample size, the Spirit-HF study will account for a minimum of $61\%$ in weight to the eventual effect estimate.

Figure 6 Illustration of likelihood and corresponding MAP-prior for the heart failure example, using a ${\text {half-Normal}}(0.25)$ prior for $\tau $ . The horizontal line at the bottom indicates the 95% prediction interval.