The R Bayesian evidence synthesis Tools (RBesT) have been created to facilitate the use of historical information in clinical trials. Once relevant historical information has been identified, RBesT supports the derivation of informative priors via the Meta-Analytic-Predictive (MAP) approach [1], the evaluation of the trial's operating characteristics, and the data analysis with the actual trial data. RBesT has been developed for endpoints from a number of well known distributions. Here we consider an example for a normally distributed response variable.
Let's look at the Crohn's disease example data [2] (data-set crohn
in RBesT). The primary endpoint is the change from baseline in Crohn's
Disease Activity Index (CDAI), which is assumed to be normally
distributed. Note that for CDAI, an improved outcome corresponds to a
negative change from baseline.
First from historical studies we get the estimated standard deviation of the response variable as $\sigma$ = 88, which is used to obtain the standard errors of the effect estimates.
dat <- crohn crohn_sigma <- 88 dat$y.se <- crohn_sigma/sqrt(dat$n)
kable(dat, digits=2)
The MAP prior can be derived with the function gMAP
. The
between-trial heterogeneity parameter $\tau$ governs how much
information will be shared from the historical trials into the design
and analysis of the future trials. In the normal case with a known
sampling standard deviation $\sigma$, the amount of borrowing from
historical data depends on the ratio $\tau/\sigma$. A conservative
choice for the prior on $\tau$ is a HalfNormal(0,
$\sigma/2$)
distribution. For the prior on the intercept we
recommend a "unit-information" prior [3] which is set to a precision
corresponding to a single observation and centered here at no change
from baseline. Please refer to the help page ?gMAP
for detailed
information. The set.seed
function is used to make the results
exactly reproducible.
library(RBesT) set.seed(689654) map_mcmc <- gMAP(cbind(y, y.se) ~ 1 | study, weights=n,data=dat, family=gaussian, beta.prior=cbind(0, crohn_sigma), tau.dist="HalfNormal",tau.prior=cbind(0,crohn_sigma/2)) print(map_mcmc) ## a graphical representation is also available pl <- plot(map_mcmc) ## a number of plots are immediately defined names(pl) ## forest plot with model estimates print(pl$forest_model)
Next, the MCMC MAP prior from the previous section is converted to a
parametric representation with the automixfit
function. This
function fits a parametric mixture representation using
expectation-maximization (EM). The number of mixture components is
chosen automatically using AIC. One can also specify the number of
components for the mixture via mixfit
function and compare with
the automixfit
outcome.
map <- automixfit(map_mcmc) print(map) ## check accuracy of mixture fit plot(map)$mix
The main advantage of using historical information is the possibility
to reduce the number of control patients, as the informative prior is
effectively equivalent to a certain number of control patients. This
is called the effective sample size (ESS) and can be calculated in
RBesT with the ess
function.
In the study protocol of this Crohn's disease data example, the very conservative moment-based ESS of 20 was used to reduce the planned sample size of the control group.
round(ess(map)) ## default elir method round(ess(map, method="morita")) round(ess(map, method="moment"))
We recommend robustifying [5] the prior with the robustify
function, which protects against type-I error inflation in presence of
prior-data conflict. For the normal case we strongly recommend
explicitly choosing the mean of the robust component. We use $-50$
consistent with the mean of the MAP prior. Furthermore, 20%
probability is used for the additional robust (unit-information)
mixture component. The choice of such probability reflects the
confidence about the validitiy of the model assumptions, i.e. the
possibility of a non-exchangable control group to be enrolled per
inclusion/exclusion criteria in the current trial as compared to the
historical control group population. Note that robustification
decreases the ESS.
## add a 20% non-informative mixture component map_robust <- robustify(map, weight=0.2, mean=-50) print(map_robust) round(ess(map_robust))
Typically, operating characteristics are required to evaluate the
proposed design or compare a few design options. RBesT requires the
input of decision rules via decision2S
function and then
calculates the operating characteristics with the oc2S
function. The calculation is expedited based on analytic
expressions. In the following we compare a few design options, which
differ in the choice of the control priors and the sample size of the
control group for the planned trial. There may be other design factors
(such as the outcome standard deviation), which should be considered
when comparing design options. Such factors are not considered here
for simplicity purpose.
Consider this 2-arm design of placebo (with an informative prior) against an experimental treatment. The dual-criterion for success is defined as follows: $$ \begin{align} \textrm{Criterion 1:} & \Pr(\theta_{act} - \theta_{pbo} \lt 0) &> 0.95 \ \textrm{Criterion 2:} & \Pr(\theta_{act} - \theta_{pbo} \lt -50) &> 0.50. \end{align} $$
Equivalently, the second criterion requires that the posterior median difference exceeds -50. The dual-criteria account for statistical significance as well as clinical relevance. Note that a negative change from baseline in CDAI corresponds to improvement.
## dual decision criteria ## pay attention to "lower.tail" argument and the order of active and pbo poc <- decision2S(pc=c(0.95,0.5), qc=c(0,-50), lower.tail=TRUE) print(poc)
For the active group we use the same weakly informative (unit-information) prior as used in the robustification step of the MAP prior. Also, we set up the few design options with different choices of control prior and different sizes of control group.
## set up prior for active group weak_prior <- mixnorm(c(1,-50,1), sigma=crohn_sigma, param = 'mn') n_act <- 40 n_pbo <- 20 ## four designs ## "b" means a balanced design, 1:1 ## "ub" means 40 in active and 20 in placebo design_noprior_b <- oc2S(weak_prior, weak_prior, n_act, n_act, poc, sigma1=crohn_sigma, sigma2=crohn_sigma) design_noprior_ub <- oc2S(weak_prior, weak_prior, n_act, n_pbo, poc, sigma1=crohn_sigma, sigma2=crohn_sigma) design_nonrob_ub <- oc2S(weak_prior, map, n_act, n_pbo, poc, sigma1=crohn_sigma, sigma2=crohn_sigma) design_rob_ub <- oc2S(weak_prior, map_robust, n_act, n_pbo, poc, sigma1=crohn_sigma, sigma2=crohn_sigma)
The type I can be increased compared to the nominal $\alpha$ level in case of a conflict between the trial data and the prior. The robustified MAP prior can reduce the type I error inflation in this case to a lower level.
# the range for true values cfb_truth <- seq(-120, -40, by=1) typeI1 <- design_noprior_b(cfb_truth, cfb_truth) typeI2 <- design_noprior_ub(cfb_truth, cfb_truth) typeI3 <- design_nonrob_ub(cfb_truth, cfb_truth) typeI4 <- design_rob_ub(cfb_truth, cfb_truth) ocI <- rbind(data.frame(cfb_truth=cfb_truth, typeI=typeI1, design="40:40 with non-informative priors"), data.frame(cfb_truth=cfb_truth, typeI=typeI2, design="40:20 with non-informative priors"), data.frame(cfb_truth=cfb_truth, typeI=typeI3, design="40:20 with non-robust prior for placebo"), data.frame(cfb_truth=cfb_truth, typeI=typeI4, design="40:20 with robust prior for placebo") ) qplot(cfb_truth, typeI, data=ocI, colour=design, geom="line", main="Type I Error") + xlab(expression(paste('True value of change from baseline ', mu[act] == mu[pbo]))) + ylab('Type I error') + coord_cartesian(ylim=c(0,0.2)) + theme(legend.justification=c(1,1),legend.position=c(0.95,0.85))
The power shows the gain of using an informative prior for the control arm; i.e. 90% power is reached for smaller $\delta$ values compared to the design with weakly informative priors for both arms or the balanced design.
delta <- seq(-80,0,by=1) m <- summary(map)["mean"] cfb_truth1 <- m + delta # active for 1 cfb_truth2 <- m + 0*delta # pbo for 2 power1 <- design_noprior_b(cfb_truth1, cfb_truth2) power2 <- design_noprior_ub(cfb_truth1, cfb_truth2) power3 <- design_nonrob_ub(cfb_truth1, cfb_truth2) power4 <- design_rob_ub(cfb_truth1, cfb_truth2) ocP <- rbind(data.frame(cfb_truth1=cfb_truth1, cfb_truth2=cfb_truth2, delta=delta, power=power1, design="40:40 with non-informative priors"), data.frame(cfb_truth1=cfb_truth1, cfb_truth2=cfb_truth2, delta=delta, power=power2, design="40:20 with non-informative priors"), data.frame(cfb_truth1=cfb_truth1, cfb_truth2=cfb_truth2, delta=delta, power=power3, design="40:20 with non-robust prior for placebo"), data.frame(cfb_truth1=cfb_truth1, cfb_truth2=cfb_truth2, delta=delta, power=power4, design="40:20 with robust prior for placebo") ) qplot(delta, power, data=ocP, colour=design, geom="line", main="Power") + xlab('True value of difference (act - pbo)')+ ylab('Power') + scale_y_continuous(breaks=c(seq(0,1,0.2),0.9)) + scale_x_continuous(breaks=c(seq(-80,0,20),-70)) + geom_hline(yintercept=0.9,linetype=2) + geom_vline(xintercept=-70,linetype=2) + theme(legend.justification=c(1,1),legend.position=c(0.95,0.85))
When the actual trial data are available, the final analysis can be
run with RBesT via the postmix
function. The real data are used
for this example, where the trial data led to a negative
conclusion. However, note that postmix
assumes that the sampling
standard deviation is known and fixed. Therefore, no uncertainty in
its estimate is taken into account.
## one can either use summary data or individual data. See ?postmix. y.act <- -29.2 y.act.se <- 14.0 n.act <- 39 y.pbo <- -63.1 y.pbo.se <- 13.9 n.pbo <- 20 ## first obtain posterior distributions post_act <- postmix(weak_prior, m=y.act, se=y.act.se) post_pbo <- postmix(map_robust, m=y.pbo, se=y.pbo.se) ## then calculate probability for the dual criteria ## and compare to the predefined threshold values p1 <- pmixdiff(post_act, post_pbo, 0); print(p1) p2 <- pmixdiff(post_act, post_pbo, -50); print(p2) print(p1>0.95 & p2>0.5) ## or we can use the decision function poc(post_act, post_pbo)
[1] Neuenschwander B et. al, Clin Trials. 2010; 7(1):5-18
[2] Hueber W. et. al, Gut, 2012, 61(12):1693-1700
[3] Kass RE, Wasserman L, J Amer Statist Assoc; 1995, 90(431):928-934.
[4] Morita S. et. al, Biometrics 2008;64(2):595-602
[5] Schmidli H. et. al, Biometrics 2014;70(4):1023-1032
sessionInfo()
options(.user_mc_options)
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