Within the Bayesian regression modeling in Stan framework brms priors are required to perform inference. This introductory material is intended to provide some pragmatic considerations on how to setup priors and point readers to material to follow-up on. The case studies themselves provide details on setting up priors for the respective problem. By default these priors have been chosen to have minimal impact on the posterior whenever appropriate and also ensure stable numerical inference. As a consequence, the results from the case studies will resemble respective Frequentist model results in most cases. Nonetheless, priors are defined explicitly for all models, since these are an integral part of any Bayesian analysis. This can be easily seen by considering Bayes rule to obtain the posterior:

The marginal likelihood term \(p(y)\) can be dropped, since this is merely a normalization constant given that we condition the inference on the observed data \(y\). What is left is hence the product of the likelihood \(p(y|\theta)\) multiplied by the prior \(p(\theta)\). The posterior information on \(\theta\) is hence provided equally by likelihood and prior.

From a practical perspective one may ask under which circumstances it actually matters which prior we set. In many cases the posterior is dominated by the data, which means that the likelihood term \(p(y|\theta)\) is much larger than the prior term \(p(\theta)\). This is the case for most case studies. It is tempting to drop the prior in these cases and not worry about it. However, this is not recommended as it is in many instances of interest to study a model with small sample sizes eventually (by applying the model to subsets, increasing model complexity, etc.). Another practical aspect is the numerical stability and quality of the Markov Chain Monte Carlo (MCMC) sample we obtain from Stan. Without a prior the inference problem becomes much harder to solve. That is, the Markov Chain Monte Carlo (MCMC) sampler Stan has to consider the full sampling space of the prior. Dropping the prior entirely implies an improper prior on the sampling space which becomes un-countably infinitely large. Rather than dropping the prior entirely we strongly recommend to consider so-called weakly informative priors. While no formal definition is given in the literature, these priors aim to identify the scale of parameters. This requires a basic understanding of the parameters for which priors are being defined. Thus, a prior can only be understood in the overall context of the likelihood, parametrization and problem at hand (see also Gelman, Simpson, and Betancourt 2017). For most statistical analyses this means to consider the endpoint and the applied transformations.

To just get started with brms one may choose to not specify priors when calling brm. Doing so will let brms provide in most cases reasonable default priors. These default priors are intended to avoid any influence on the calculated posterior. Hence, the results are fully data driven and will be very close to the respective Frequentist maximum likelihood inference result. However, the default prior is not guaranteed to stay stable between releases and can thus change whenever the brms version changes.

Given that any Bayesian analysis requires a prior, we recommend to always explicitly define these - even if these just repeat the default prior from brms, which one can easily obtain. Here we use as example binomially distributed data (responder in a control arm) and we use the simplest possible model, which will pool the information across the different studies (in practice one would allow for between-trial heterogeneity):

model <-bf(r |trials(n) ~1, family=binomial)## with brms >= 2.21.0 we can directly write... ## default_prior(model, data=RBesT::AS)## a workaround is to create an empty brms model and obtain the## defined prior likeprior_summary(brm(model, data=RBesT::AS, empty=TRUE)) |>kable()

prior

class

coef

group

resp

dpar

nlpar

lb

ub

source

student_t(3, 0, 2.5)

Intercept

default

Note that we have defined first the model via the bf call. This defines the linear predictor and the likelihood using the family argument of bf. We recommend to always explicitly do this step first as it defines the likelihood and the model parametrization in one step. Choosing these is a critical first step in defining the model and, most importantly, the chosen priors can only be understood in the context of the model (likelihood and parametrization) (see Gelman, Simpson, and Betancourt 2017).

Rather than running the model with no explicit prior definition we encourage users to explicitly define the priors used for the analysis - even if these are simply the default priors:

## standard error estimatesqrt(vcov(fit_freq)[1,1])

[1] 0.1022971

In this case the data with a total of 513 subjects dominates the posterior. Nonetheless, it is illustrative to consider in more detail the prior here as an example. Importantly, the intercept of the model is defined on the logit scale of the response probability as it is common for a standard logistic regression. The transformation changes the scale from the interval 0 to 1 to a new scale which covers the entire space of reals. While formally very large or very small logits are admissible, the range from 5% to 95% response rate corresponds to -3 to 3 on the logit space approximately. Thus, logit values smaller or greater than these values would be considered extreme effects.

When defining a prior for a model we would typically want to compare different choices of the prior to one another. As example prior for the intercept only model we may compare here three different priors: (i) a very wide prior \(\mbox{Normal}(0,10^2)\), (2) the default brms prior and (3) the density \(\mbox{Normal}(0,2^2)\) as used as default prior in RBesT for this problem. In a first step we sample these priors and assess them graphically. We could use brms to sample these priors directly with the argument sample_prior="only", but we instead use basic R functions for simplicity:

As one can easily see, the wide prior has density at very extreme logit values of -20 and 20. In comparison, the brms and the RBesT default prior place most probability mass in the vicinity of zero. While the wide prior may suggest to be very non-informative it’s meaning becomes clearer when back transforming to the original response scale from 0 to 1. It is clear that the wide prior implies that we expect extreme response rates of either 0 or 1. In contrast, the RBesT default prior has an almost uniform distribution on the 0 to 1 range while the default brms prior has some “U” shape. To now further discriminate these priors a helpful concept is to consider interval probabilities for pre-defined categories. The definition of cut-points of the categories is problem specific, but can be defined with usual common sense for most endpoints. The critical step here is in defining these categories and documenting these along with the analysis. Here we categorize the response as extreme 0-1% & 99-100%, very small 1-5% & 95-99%, small 5-10% & 90-95% and moderate for the remainder. Comparing then these priors gives:

It is apparent that the seemingly non-informative prior “wide” is in fact an informative prior implying that the expected response rates are extreme. This is the consequence of transforming to the logit scale.

In practice one would most certainly not fit the intercept only model, which pools the information. The RBesT::AS data set is in fact the standard example for the use of historical control data.

kable(RBesT::AS)

study

n

r

Study 1

107

23

Study 2

44

12

Study 3

51

19

Study 4

39

9

Study 5

139

39

Study 6

20

6

Study 7

78

9

Study 8

35

10

To use this data as historical control data one uses instead of the intercept only model a random intercept model, which casts this into a meta-analtyic model:

Running MCMC with 4 sequential chains...
Chain 1 finished in 0.1 seconds.
Chain 2 finished in 0.1 seconds.
Chain 3 finished in 0.1 seconds.
Chain 4 finished in 0.1 seconds.
All 4 chains finished successfully.
Mean chain execution time: 0.1 seconds.
Total execution time: 0.9 seconds.

summary(fit_meta)

Family: binomial
Links: mu = logit
Formula: r | trials(n) ~ 1 + (1 | study)
Data: RBesT::AS (Number of observations: 8)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Multilevel Hyperparameters:
~study (Number of levels: 8)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.34 0.17 0.04 0.73 1.01 1186 1066
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -1.11 0.17 -1.47 -0.75 1.00 1622 1731
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

The meta-analytic model is generative in the sense of allowing to predict the mean response rate for future studies by way of the hierarchical model structure. Now the importance of the prior on the overall intercept becomes more relevance as the information from each study is discounted through the random effects model leading to less data to estimate the parameter in comparison to the full pooling model. However, the key parameter in this model is the between-trial heterogeneity parameter. In the example a half-normal prior with scale 1 is used (brms knows that the parameter must be positive and hence truncates the prior at zero automatically). This distribution of a half-normal density has been studied extensively in the literature and found to be a robust choice in a wide range of problems. For this reason, it is a very rational choice to use this prior in the current (and future) analyses. The choice of the scale of 0.5 is often referred to as a conservative choice in this problem while 1 is a very conservative choice and 0.25 a less conservative choice. A complication with the hierarchical model is that \(\tau\) itself implies a distribution. To simplify this, we first consider the case of a known between-trial heterogeneity parameter and study what this implies. For a known \(\tau\) the range of implied log odds can be characterized by the respective 95% probability mass range given by \(2
\cdot 1.96 \, \tau\), which is interpret-able as the largest log odds ratio. Recalling that the between-study heterogeneity parameter controls the (random) differences in the logits of response rates between studies, it is helpful to consider the distribution of implied log odds ratios between random pairs of two studies. These differences (log odds ratios) have a distribution of \(\mbox{Normal}(0, (\sqrt{2} \,
\tau)^2)\). Considering the absolute value of these differences, the distributions becomes a half-normal with scale \(\sqrt{2} \, \tau\), which has it’s median value at \(1.09 \, \tau\). This results for a range of values of \(\tau\) on the respective odds scale to:

tau

largest_odds_ratio

median_odds_ratio

0.000

1.000

1.000

0.125

1.632

1.146

0.250

2.664

1.313

0.500

7.099

1.725

1.000

50.400

2.974

1.500

357.809

5.129

2.000

2540.205

8.846

In (Neuenschwander and Schmidli 2020) suggest the categorization of the between-study heterogeneity \(\tau\) parameter the values of \(1\) for being large, \(0.5\) substantial, \(0.25\) moderate, and \(0.125\) small. Considering these with the table above qualifies these categories as plausible. An alternative approach to the categorization is to consider what is an extreme value for \(\tau\) (like unity) and then choose the prior on \(\tau\) such that a large quantile (like the 95% quantile) corresponds to this extreme value.

With the categorization of specific values for \(\tau\) we can now proceed and consider the interval probabilities for these categories for the different choices of \(\tau \sim \mbox{Normal}^+(s^2)\) for \(s=1\) and \(s=1/2\).

We can see that the “conservative” choice has an about 5% tail probability exceeding the value of large. This implies that some degree of homogeneity between the studies is given. This is usually the case whenever meta-analyses are conducted, since the inclusion & exclusion criteria for studies are aligned to a certain degree in order to ensure that a similar patient population is included in the analysis. In contrast, the “very conservative” choice admits with 30% probability mass the possibility of values in the domain “very large” for which practically no borrowing of historical information occurs. The “less conservative” choice on the hand has 5% tail probability above substantial. This way a large heterogeneity is considered to be unlikely as it can be the case for twin Phase III studies, for example.

Additional literature for consideration:

Empirical priors study for HTA treatment effect evaluation by the German IQWIG (Lilienthal et al. 2023)

Empirical priors for meta-analyses organized in disease specific manner (Turner et al. 2015)

Endpoint specific considerations for between-trial heterogeneity parameter priors in random effect meta-analyses (Röver et al. 2021)

Comprehensive introductory book to applied Bayesian data analysis with detailed discussion on many examples (Gelman et al. 2014)

Live wiki document maintained by Stan user community (heavily influenced by Andrew Gelman & Aki Vehtari) (Stan 2024)

Prior strategy based on nested modeling considerations (penalization of more complex models), (Simpson et al. 2014)

Gelman, Andrew, John B Carlin, Hal S Stern, David B Dunson, Aki Vehtari, and Donald B Rubin. 2014. Bayesian Data Analysis. Chapman Texts in Statistical Science Series. Third edit. https://doi.org/10.1007/s13398-014-0173-7.2.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.”Entropy 19 (October). https://doi.org/10.3390/e19100555.

Lilienthal, Jona, Sibylle Sturtz, Christoph Schürmann, Matthias Maiworm, Christian Röver, Tim Friede, and Ralf Bender. 2023. “Bayesian Random‐effects Meta‐analysis with Empirical Heterogeneity Priors for Application in Health Technology Assessment with Very Few Studies.”Research Synthesis Methods, December. https://doi.org/10.1002/jrsm.1685.

Piironen, Juho, and Aki Vehtari. 2017. “Sparsity Information and Regularization in the Horseshoe and Other Shrinkage Priors.”Electronic Journal of Statistics 11 (January): 5018–51. https://doi.org/10.1214/17-EJS1337SI.

Röver, Christian, Ralf Bender, Sofia Dias, Christopher H. Schmid, Heinz Schmidli, Sibylle Sturtz, Sebastian Weber, and Tim Friede. 2021. “On Weakly Informative Prior Distributions for the Heterogeneity Parameter in Bayesian Random-Effects Meta-Analysis.”Research Synthesis Methods 12 (January): 448–74. https://doi.org/10.1002/jrsm.1475.

Simpson, Daniel P., Håvard Rue, Thiago G. Martins, Andrea Riebler, and Sigrunn H. Sørbye. 2014. “Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors.”Statistical Science 32 (February): 1–28. https://doi.org/10.1214/16-STS576.

Turner, Rebecca M, Dan Jackson, Yinghui Wei, Simon G Thompson, and Julian P T Higgins. 2015. “Predictive Distributions for Between-Study Heterogeneity and Simple Methods for Their Application in Bayesian Meta-Analysis.”Statistics in Medicine 34: 984–98. https://doi.org/10.1002/sim.6381.

Zhang, Yan Dora, Brian P. Naughton, Howard D. Bondell, and Brian J. Reich. 2022. “Bayesian Regression Using a Prior on the Model Fit: The R2-D2 Shrinkage Prior.”Journal of the American Statistical Association 117: 862–74. https://doi.org/10.1080/01621459.2020.1825449.

---author: - Sebastian Weber - <sebastian.weber@novartis.com>bibliography: references.bib---{{< include _macros.qmd >}}# Model setup & priors {#sec-model-priors}```{r, include=FALSE}source(here::here("src", "setup.R"))##knitr::opts_chunk$set(cache=FALSE) library(ggplot2)library(dplyr)library(knitr)library(brms)library(posterior)library(ggdist)library(bayesplot)library(tidybayes)library(forcats)library(RBesT)library(here)# instruct brms to use cmdstanr as backend and cache all Stan binariesoptions(brms.backend="cmdstanr", cmdstanr_write_stan_file_dir=here("_brms-cache"))# create cache directory if not yet availabledir.create(here("_brms-cache"), FALSE) set.seed(593467)theme_set(theme_bw(12))```Within the Bayesian regression modeling in Stan framework `brms`priors are required to perform inference. This introductory materialis intended to provide some pragmatic considerations on how to setuppriors and point readers to material to follow-up on. The case studiesthemselves provide details on setting up priors for the respectiveproblem. By default these priors have been chosen to have minimalimpact on the posterior whenever appropriate and also ensure stablenumerical inference. As a consequence, the results from the casestudies will resemble respective Frequentist model results in mostcases. Nonetheless, priors are defined explicitly for all models,since these are an integral part of any Bayesian analysis. This can beeasily seen by considering Bayes rule to obtain the posterior:$$ p(\theta | y) = \frac{p(y|\theta) \, p(\theta)}{p(y)} \propto p(y|\theta) \, p(\theta)$$The marginal likelihood term $p(y)$ can be dropped, since this ismerely a normalization constant given that we condition the inferenceon the observed data $y$. What is left is hence the product of thelikelihood $p(y|\theta)$ multiplied by the prior $p(\theta)$. Theposterior information on $\theta$ is hence provided equally bylikelihood and prior.From a practical perspective one may ask under which circumstances itactually matters which prior we set. In many cases the posterior isdominated by the data, which means that the likelihood term$p(y|\theta)$ is much larger than the prior term $p(\theta)$. This isthe case for most case studies. It is tempting to drop the prior inthese cases and not worry about it. However, this is not recommendedas it is in many instances of interest to study a model with smallsample sizes eventually (by applying the model to subsets, increasingmodel complexity, etc.). Another practical aspect is the numericalstability and quality of the Markov Chain Monte Carlo (MCMC) sample weobtain from Stan. Without a prior the inference problem becomes muchharder to solve. That is, the Markov Chain Monte Carlo (MCMC) samplerStan has to consider the full sampling space of the prior. Droppingthe prior entirely implies an improper prior on the sampling spacewhich becomes un-countably infinitely large. Rather than dropping theprior entirely we strongly recommend to consider so-called weaklyinformative priors. While no formal definition is given in theliterature, these priors aim to identify the *scale* ofparameters. This requires a basic understanding of the parameters forwhich priors are being defined. Thus, a prior can only be understoodin the overall context of the likelihood, parametrization and problemat hand [see also @Gelman2017]. For most statistical analyses thismeans to consider the endpoint and the applied transformations.<!--A notable exception to this are the case studies on the metaanalytic predictive (MAP) priors. These case studies infer betweenstudy heterogeneity from very few studies and therefore the prior usedfor the heterogeneity parameter will not be dominated by the data. TheMAP priors themselves represent *data* derived priors and they havebecome very popular nowadays.-->To just get started with `brms` one may choose to not specify priorswhen calling `brm`. Doing so will let `brms` provide in most casesreasonable default priors. These default priors are intended to avoidany influence on the calculated posterior. Hence, the results arefully data driven and will be very close to the respective Frequentistmaximum likelihood inference result. However, the default prior is notguaranteed to stay stable between releases and can thus change wheneverthe `brms` version changes.Given that any Bayesian analysis requires a prior, we recommend toalways explicitly define these - even if these just repeat the defaultprior from `brms`, which one can easily obtain. Here we use as examplebinomially distributed data (responder in a control arm) and we usethe simplest possible model, which will pool the information across thedifferent studies (in practice one would allow for between-trialheterogeneity):```{r}model <-bf(r |trials(n) ~1, family=binomial)## with brms >= 2.21.0 we can directly write... ## default_prior(model, data=RBesT::AS)## a workaround is to create an empty brms model and obtain the## defined prior likeprior_summary(brm(model, data=RBesT::AS, empty=TRUE)) |>kable()```Note that we have defined first the model via the `bf` call. Thisdefines the linear predictor and the likelihood using the `family`argument of `bf`. We recommend to always explicitly do this step firstas it defines the likelihood and the model parametrization in onestep. Choosing these is a critical first step in defining the modeland, most importantly, the chosen priors can only be understood in thecontext of the model (likelihood and parametrization) [see @Gelman2017].Rather than running the model with no explicit prior definition weencourage users to explicitly define the priors used for theanalysis - even if these are simply the default priors:```{r warning=FALSE, message=FALSE}model_def <- bf(r | trials(n) ~ 1, family=binomial)model_prior <- prior(student_t(3, 0, 2.5), class=Intercept)fit <- brm(model_def, data=RBesT::AS, prior=model_prior, seed=467657, refresh=0)fixef(fit)```The provided default prior is indeed not informative, as can be seenby comparing the posterior estimate to the respective Frequentist fitfrom `glm`:```{r warning=FALSE, message=FALSE}fit_freq <- glm(cbind(r, n-r) ~ 1, data=RBesT::AS, family=binomial)## intercept estimatecoef(fit_freq)## standard error estimatesqrt(vcov(fit_freq)[1,1])```In this case the data with a total of `r sum(RBesT::AS$n)` subjectsdominates the posterior. Nonetheless, it is illustrative to consider inmore detail the prior here as an example. Importantly, the interceptof the model is defined on the *logit* scale of the responseprobability as it is common for a standard logistic regression. Thetransformation changes the scale from the interval 0 to 1 to a newscale which covers the entire space of reals. While formally verylarge or very small logits are admissible, the range from 5% to 95%response rate corresponds to -3 to 3 on the logit spaceapproximately. Thus, logit values smaller or greater than these valueswould be considered extreme effects.When defining a prior for a model we would typically want to comparedifferent choices of the prior to one another. As example prior for the intercept only model we may compare here three different priors:(i) a very wide prior $\N(0,10^2)$, (2) the default `brms` prior and(3) the density $\N(0,2^2)$ as used as default prior in `RBesT` forthis problem. In a first step we sample these priors and assess themgraphically. We could use `brms` to sample these priors directly withthe argument `sample_prior="only"`, but we instead use basic Rfunctions for simplicity:```{r warning=FALSE, message=FALSE}#| code-fold: true#| code-summary: "Show the code"num_draws <- 1E4priors_cmp <- tibble(case=c("wide", "brms", "RBesT"), density=c("N(0,10^2)", "S(3, 0, 2.5^2)", "N(0, 2^2)"), prior=rvar(cbind(rnorm(num_draws, 0, 10), rstudent_t(num_draws, 3, 0, 2.5), rnorm(num_draws, 0, 2))))kable(priors_cmp)priors_logit <- priors_cmp |> ggplot(aes(y=case, xdist=prior)) + stat_slab() + xlab("Intercept\nlinear predictor") + ylab(NULL) + labs(title="Prior logit scale") + coord_cartesian(xlim=c(-30, 30))priors_response <- priors_cmp |> ggplot(aes(y=case, xdist=rdo(inv_logit(prior)))) + stat_slab() + xlab("Intercept\nresponse scale") + ylab(NULL) + labs(title="Prior response scale") + coord_cartesian(xlim=c(0, 1))bayesplot_grid(priors_logit, priors_response, grid_args=list(nrow=1))```As one can easily see, the wide prior has density at very extremelogit values of -20 and 20. In comparison, the `brms` and the `RBesT`default prior place most probability mass in the vicinity ofzero. While the wide prior may suggest to be very non-informative it'smeaning becomes clearer when back transforming to the originalresponse scale from 0 to 1. It is clear that the wide prior impliesthat we expect extreme response rates of either 0 or 1. In contrast,the `RBesT` default prior has an almost uniform distribution on the 0to 1 range while the default `brms` prior has some "U" shape. To nowfurther discriminate these priors a helpful concept is to consider**interval probabilities** for pre-defined categories. The definitionof cut-points of the categories is problem specific, but can bedefined with usual common sense for most endpoints. The critical stephere is in defining these categories and documenting these along withthe analysis. Here we categorize the response as extreme 0-1% &99-100%, very small 1-5% & 95-99%, small 5-10% & 90-95% andmoderate for the remainder. Comparing then these priors gives:```{r warning=FALSE, message=FALSE}#| code-fold: true#| code-summary: "Show the code"response_category <- function(r) { fct_rev(cut(pmin(r, 1-r), c(0, 0.01, 0.05, 0.1, 0.5), labels=c("extreme", "very_small", "small", "moderate")))}priors_cmp |> unnest_rvars()|> ggplot(aes(x=case, fill=response_category(inv_logit(prior)))) + geom_bar(position="fill") + xlab("Prior") + ylab("Probability") + labs(title="Prior interval probabilities", fill="Category") + scale_y_continuous(breaks=seq(0,1,by=0.2)) + theme(legend.position="right")mutate(priors_cmp, prior_response=rfun(inv_logit)(prior), summarise_draws(prior_response, extreme=~100*E(.x < 0.01 | .x > 0.99), very_small=~100*E( (0.01 <= .x & .x < 0.05) | (0.95 < .x & .x <= 0.99) ), small=~100*E( (0.05 <= .x & .x < 0.1) | (0.9 < .x & .x <= 0.95) ), moderate=~100*E( (0.1 <= .x & .x < 0.9) ) )) |> select(case, density, extreme, very_small, small, moderate) |> kable(digits=1, caption="Interval probabilities per prior given in percent")```It is apparent that the seemingly non-informative prior "wide" is infact an informative prior implying that the expected response ratesare extreme. This is the consequence of transforming to the logitscale.In practice one would most certainly not fit the intercept only model,which pools the information. The `RBesT::AS` data set is in fact thestandard example for the use of historical control data. ```{r warning=FALSE, message=FALSE}kable(RBesT::AS)```To use this data as historical control data one uses instead of theintercept only model a random intercept model, which casts thisinto a meta-analtyic model:```{r warning=FALSE, message=FALSE}model_meta_def <- bf(r | trials(n) ~ 1 + (1 | study), family=binomial)model_meta_prior <- prior(normal(0, 2), class=Intercept) + prior(normal(0, 0.5), class=sd, coef=Intercept, group=study)fit_meta <- brm(model_meta_def, data=RBesT::AS, prior=model_meta_prior, seed=982345, refresh=0, control=list(adapt_delta=0.95))summary(fit_meta)```The meta-analytic model is generative in the sense of allowing topredict the mean response rate for future studies by way of thehierarchical model structure. Now the importance of the prior on theoverall intercept becomes more relevance as the information from eachstudy is discounted through the random effects model leading to lessdata to estimate the parameter in comparison to the full poolingmodel. However, the key parameter in this model is the between-trialheterogeneity parameter. In the example a half-normal prior with scale1 is used (`brms` knows that the parameter must be positive and hencetruncates the prior at zero automatically). This distribution of ahalf-normal density has been studied extensively in the literature andfound to be a robust choice in a wide range of problems. For thisreason, it is a very rational choice to use this prior in the current(and future) analyses. The choice of the scale of 0.5 is oftenreferred to as a conservative choice in this problem while 1 is a veryconservative choice and 0.25 a less conservative choice. Acomplication with the hierarchical model is that $\tau$ itself impliesa distribution. To simplify this, we first consider the case of aknown between-trial heterogeneity parameter and study what thisimplies. For a known $\tau$ the range of implied log odds can becharacterized by the respective 95% probability mass range given by $2\cdot 1.96 \, \tau$, which is interpret-able as the largest log oddsratio. Recalling that the between-study heterogeneity parametercontrols the (random) differences in the logits of response ratesbetween studies, it is helpful to consider the distribution of impliedlog odds ratios between random pairs of two studies. These differences(log odds ratios) have a distribution of $\N(0, (\sqrt{2} \,\tau)^2)$. Considering the absolute value of these differences, thedistributions becomes a half-normal with scale $\sqrt{2} \, \tau$,which has it's median value at $1.09 \, \tau$. This results for arange of values of $\tau$ on the respective odds scale to:```{r echo=FALSE}tau_known <- tibble(tau=c(0, 0.125, 0.25, 0.5, 1.0, 1.5, 2)) |> mutate(largest_odds_ratio=exp(3.92 * tau), median_odds_ratio=exp(1.09 * tau))tau_known |> kable(digits=3)```In [@Neuenschwander2020] suggest the categorization of thebetween-study heterogeneity $\tau$ parameter the values of $1$ forbeing large, $0.5$ substantial, $0.25$ moderate, and $0.125$small. Considering these with the table above qualifies thesecategories as plausible. An alternative approach to the categorizationis to consider what is an extreme value for $\tau$ (like unity) andthen choose the prior on $\tau$ such that a large quantile (like the95% quantile) corresponds to this extreme value. With the categorization of specific values for $\tau$ we can nowproceed and consider the interval probabilities for these categoriesfor the different choices of $\tau \sim \HN(s^2)$ for $s=1$ and$s=1/2$.```{r}#| code-fold: true#| code-summary: "Show the code"priors_cmp_tau <-tibble(case=c("less_conservative", "conservative", "very_conservative"),density=c("HN(0, (1/4)^2)", "HN(0, (1/2)^2)", "HN(0, 1^2)"),prior=rvar(cbind(abs(rnorm(num_draws, 0, 0.25)),abs(rnorm(num_draws, 0, 0.5)),abs(rnorm(num_draws, 0, 1)))) )kable(priors_cmp_tau)priors_hetero <- priors_cmp_tau |>ggplot(aes(y=case, xdist=prior)) +stat_slab() +xlab("Between-study heterogeneity") +ylab(NULL) +scale_x_continuous(breaks=0:3) +labs(title="Between-study\nheterogeneity") +coord_cartesian(xlim=c(0, 3))priors_lor <- priors_cmp_tau |>mutate(lor1=rvar_rng(rnorm, n=n(), mean=0, prior),lor2=rvar_rng(rnorm, n=n(), mean=0, prior)) |>ggplot(aes(y=case, xdist=lor1-lor2)) +stat_slab() +xlab("Log-odds ratio") +ylab(NULL) +scale_x_continuous(breaks=-5:5) +labs(title="Log-odds ratio\nof random pair") +coord_cartesian(xlim=c(-3, 3))bayesplot_grid(priors_hetero, priors_lor, grid_args=list(nrow=1))``````{r warning=FALSE, message=FALSE}#| code-fold: true#| code-summary: "Show the code"tau_category <- function(tau) { fct_rev(cut(tau, c(0, 0.125, 0.25, 0.5, 1, Inf), labels=c("small", "moderate", "substantial", "large", "very_large")))}priors_cmp_tau |> unnest_rvars()|> ggplot(aes(x=case, fill=tau_category(prior))) + geom_bar(position="fill") + xlab("Prior") + ylab("Probability") + labs(title="Prior interval probabilities", fill="Category") + ##scale_y_continuous(breaks=seq(0,1,by=0.2)) + theme(legend.position="right")mutate(priors_cmp_tau, summarise_draws(prior, small=~100*E(.x < 0.125), moderate=~100*E( 0.125 <= .x & .x < 0.25 ), substantial=~100*E( 0.25 <= .x & .x < 0.5 ), large=~100*E( 0.5 <= .x & .x < 1 ), very_large=~100*E( .x >= 1 ) )) |> select(case, density, small, moderate, substantial, large, very_large) |> kable(digits=1, caption="Interval probabilities per prior given in percent")```We can see that the "conservative" choice has an about 5% tailprobability exceeding the value of large. This implies that somedegree of homogeneity between the studies is given. This is usuallythe case whenever meta-analyses are conducted, since the inclusion &exclusion criteria for studies are aligned to a certain degree inorder to ensure that a similar patient population is included in theanalysis. In contrast, the "very conservative" choice admits with 30%probability mass the possibility of values in the domain "very large"for which practically no borrowing of historical informationoccurs. The "less conservative" choice on the hand has 5% tailprobability above substantial. This way a large heterogeneity isconsidered to be unlikely as it can be the case for twin Phase IIIstudies, for example.Additional literature for consideration:- Empirical priors study for HTA treatment effect evaluation by the German IQWIG [@Lilienthal2023]- Empirical priors for meta-analyses organized in disease specific manner [@Turner2015]- Endpoint specific considerations for between-trial heterogeneity parameter priors in random effect meta-analyses [@Rover2021]- Comprehensive introductory book to applied Bayesian data analysis with detailed discussion on many examples [@Gelman2014]- Live wiki document maintained by Stan user community (heavily influenced by Andrew Gelman & Aki Vehtari) [@stanwikiPrior]- Prior strategy based on nested modeling considerations (penalization of more complex models), [@Simpson2014]- Global model shrinkage regularized horseshoe prior [@Piironen2017] or R2D2 prior (overall $R^2$) [@Zhang2022]