---
title: "RBesT for a Binary Endpoint"
author: "Sebastian Weber"
date: "`r Sys.Date()`"
output: html_vignette
---
<!--
%\VignetteEngine{knitr::rmarkdown}
%\VignetteIndexEntry{Getting started with RBesT (binary)}
-->
  
```{r, include=FALSE}
library(RBesT)
library(knitr)
knitr::opts_chunk$set(
    fig.width = 1.62*4,
    fig.height = 4
    )
## setup up fast sampling when run on CRAN
is_CRAN <- !identical(Sys.getenv("NOT_CRAN"), "true")
## NOTE: for running this vignette locally, please uncomment the
## following line:
## is_CRAN <- FALSE
if (is_CRAN) {
    options(RBesT.MC.warmup=250, RBesT.MC.iter=500, RBesT.MC.chains=2)
}
```

# Introduction

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] and the evaluation of the trial's operating
characteristics. The MAP approach performs a standard meta-analysis
followed by a prediction for the control group parameter of a future
study while accounting for the uncertainty in the population mean (the
standard result from a meta-analysis) and the between-trial
heterogeneity. Therefore, RBesT can also be used as a meta-analysis
tool if one simply neglects the prediction part.

Let's consider a Novartis Phase II study in ankylosing spondylitis
comparing the Novartis test treatment secukinumab with placebo
[2]. The primary efficacy endpoint was percentage of patients with a
20% response according to the Assessment of SpondyloArthritis
international Society criteria for improvement (ASAS20) at week 6. For
the control group, the following historical data were used to derive
the MAP prior:

```{r,results="asis",echo=FALSE}
kable(AS)
```

This dataset is part of RBesT and available after loading
the package in the data frame `AS`.

RBesT supports all required steps to design a clinical trial with
historical information using the MAP approach.

# Prior Derivation

## Meta-Analytic-Predictive Analysis

The **`gMAP`** function performs the meta-analysis and the prediction,
which yields the MAP prior. The analysis is run using stochastic
Markov-Chain-Monte-Carlo with Stan. In order to make results exactly
reproducible, the `set.seed` function must be called prior to calling
**`gMAP`** .

A key parameter in a meta-analysis is the between-trial heterogeneity
parameter $\tau$ which controls the amount of borrowing from
historical information for the estimation of the population mean will
occur. As we often have only few historical trials, the prior is
important.  For binary endpoints with an expected response rate of
20%-80% we recommend a conservative `HalfNormal(0,1)` prior as a
default. Please refer to the help-page of **`gMAP`** for more
information.

The **`gMAP`** function returns an analysis object from which we can
extract information using the functions from RBesT. We do recommend to
look at the graphical model checks provided by RBesT as demonstrated
below. The most important one is the forest plot, with solid lines for
the MAP model predictions and dashed lines for the stratified
estimates. For a standard forest plot without the shrinkage estimates
please refer to the `forest_plot` function in RBesT.

```{r}
library(RBesT)
set.seed(34563)
map_mcmc <- gMAP(cbind(r, n-r) ~ 1 | study,
                 data=AS,
                 tau.dist="HalfNormal",
                 tau.prior=1,
                 beta.prior=2,
                 family=binomial)
print(map_mcmc)

## a graphical representation of model checks is available
pl <- plot(map_mcmc)

## a number of plots are immediately defined
names(pl)

## forest plot with model estimates
print(pl$forest_model)
```

An often raised concern with a Bayesian analysis is the choice of the
prior. Hence sensitivity analyses may sometimes be necessary. They can
be quickly performed with the **`update`** function. Suppose we want
to evaluate a more optimistic scenario (with less between-trial
heterogeneity), expressed by a `HalfNormal(0,1/2)` prior on
$\tau$. Then we can rerun the original analysis, but with modified
arguments of **`gMAP`**:

```{r}
set.seed(36546)
map_mcmc_sens <- update(map_mcmc, tau.prior=1/2)
print(map_mcmc_sens)
```

## Parametric Approximation

As a next step, the MAP prior, represented numerically using a large
MCMC simulation sample, 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 to best describe the MAP is chosen
automatically. Again, the `plot` function produces a graphical
diagnostic which allows the user to assess whether the marginal
mixture density (shown in black) matches well with the histogram of
the MAP MCMC sample.
```{r}
map <- automixfit(map_mcmc)
print(map)
plot(map)$mix
```

## Effective Sample Size

The (usual) intended use of a (MAP) prior is to reduce the number of
control patients in the trial. The prior can be considered equivalent
to a number of experimental observations, which is called the
effective sample size (ESS) of the prior. It can be calculated in
RBesT with the **`ess`** function. It should be noted, however, that the
concept of ESS is somewhat elusive. In particular, the definition of
the ESS is not unique and multiple methods have therefore been
implemented in RBesT. The default moment-based approach yields
conservative (small) ESS in comparison to the Morita [3] approach,
which yields liberal (large) ESS:

```{r}
ess(map)
ess(map, method="morita")
```
The Morita approach uses the curvature of the prior at the mode and
has been found to be sensitive to a large number of mixture
components. From experience, a realistic ESS estimate can be obtained
by using few mixture components for the MAP MCMC sample:

```{r}
map_2 <- mixfit(map_mcmc, Nc=2)
ess(map_2, method="morita")
```

## Robustification of the MAP Prior

Finally, we recommend to **`robustify`** [4] the prior which protects
against type-I error inflation in presence of prior-data conflict,
i.e. if the future trial data strongly deviate from the historical
control information.
```{r}
## add a 20% non-informative mixture component
map_robust <- robustify(map, weight=0.2, mean=1/2)
print(map_robust)
```
Note that robustification does reduce the ESS (numbers not shown).


# Design Evaluation

Now we have a prior which can be specified in the protocol. The
advantage of using historical information is the possible reduction of
the placebo patient group. The sample size of the control group
 is supplemented by the historical information. The reduction
in placebo patients can be about as large as the ESS of the MAP prior.

In the following, we compare designs with different sample sizes and
priors for the control group. The comparisons are carried out by
evaluating standard Frequentist operating characteristics (type-I
error, power). The scenarios are not exhaustive, but rather specific
ones to demonstrate the use of RBesT for design evaluation.


## Operating Characteristics

We consider the 2-arm design of the actual Novartis
trial in ankylosing spondylitis [2]. This trial tested 6 patients on
placebo against 24 patients on experimental treatment. Success was
declared whenever the condition

$$\Pr(p_{placebo} - p_{treat} \leq 0) > 0.95$$

was met. A MAP prior was used for the placebo response rate
parameter. Here we evaluate a few design options as an example.

The operating characteristics are setup in RBesT in a stepwise manner:

1. Definition of priors for each arm.
2. Definition of the decision criterion using the **`decision2S`** function.
3. Specification of design options with the **`oc2S`** function. This
   includes the overall decision function and per arm the prior and
   the sample size to use.
4. The object from step 3 is then used to calculate the operating
   characteristics.

Note that for a 1-sample situation the respective `decision1S` and
`oc1S` function are used instead.

### Type I Error

The type I can be increased compared to the nominal $\alpha$ level in
case of a conflict between the trial data and the prior. Note, that in
this example the MAP prior has a 95% interval of about 0.1 to 0.5.

```{r}
p_truth       <- seq(0.1,0.95,by=0.01)
uniform_prior <- mixbeta(c(1,1,1))
treat_prior   <- mixbeta(c(1,0.5,1)) # prior for treatment used in trial
lancet_prior  <- mixbeta(c(1,11,32)) # prior for control   used in trial
decision      <- decision2S(0.95, 0, lower.tail=TRUE)

design_uniform   <- oc2S(uniform_prior, uniform_prior, 6, 24, decision)
design_nonrobust <- oc2S(map          , treat_prior,   6, 24, decision)
design_robust    <- oc2S(map_robust   , treat_prior,   6, 24, decision)

typeI_uniform   <- design_uniform(  p_truth, p_truth)
typeI_nonrobust <- design_nonrobust(p_truth, p_truth)
typeI_robust    <- design_robust(   p_truth, p_truth)

ocI <- rbind(data.frame(p_truth=p_truth, typeI=typeI_robust,    prior="robust"),
             data.frame(p_truth=p_truth, typeI=typeI_nonrobust, prior="non-robust"),
             data.frame(p_truth=p_truth, typeI=typeI_uniform,   prior="uniform")
             )

library(ggplot2)
theme_set(theme_bw()) # nice plotting theme
qplot(p_truth, typeI, data=ocI, colour=prior, geom="line", main="Type I Error")
```

### Power

The power demonstrates the gain of using an informative prior;
i.e. 80% power is reached for smaller $\delta$ values in comparison to
a design with non-informative priors for both arms.

```{r}
delta <- seq(0,0.7,by=0.01)
m <- summary(map)["mean"]
p_truth1 <- m + 0*delta
p_truth2 <- m +   delta

power_uniform   <- design_uniform(p_truth1, p_truth2)
power_nonrobust <- design_nonrobust(p_truth1, p_truth2)
power_robust    <- design_robust(p_truth1, p_truth2)

ocP <- rbind(data.frame(p_truth1=p_truth1, p_truth2=p_truth2, delta=delta, power=power_robust,    prior="robust"),
             data.frame(p_truth1=p_truth1, p_truth2=p_truth2, delta=delta, power=power_nonrobust, prior="non-robust"),
             data.frame(p_truth1=p_truth1, p_truth2=p_truth2, delta=delta, power=power_uniform,   prior="uniform")
             )

qplot(delta, power, data=ocP, colour=prior, geom="line", main="Power")

```

### Data Scenarios

An alternative approach to visualize the study design to
non-statisticians is by considering data scenarios. They show the
decisions based on potential trial outcomes. The information needed
are the critical values at which the decision criterion flips. In the
2-sample case this means to calculate the decision boundary, see the
**`decision2S_boundary`** help for more information.

```{r}
## Critical values at which the decision are given conditional on the
## outcome of the second read-out
crit_uniform   <- decision2S_boundary(uniform_prior, uniform_prior, 6, 24, decision)
crit_nonrobust <- decision2S_boundary(map          , treat_prior,   6, 24, decision)
crit_robust    <- decision2S_boundary(map_robust   , treat_prior,   6, 24, decision)
treat_y2 <- 0:24
## Note that -1 is returned to indicated that the decision is never 1
ocC <- rbind(data.frame(y2=treat_y2, y1_crit=crit_robust(treat_y2),    prior="robust"),
             data.frame(y2=treat_y2, y1_crit=crit_nonrobust(treat_y2), prior="non-robust"),
             data.frame(y2=treat_y2, y1_crit=crit_uniform(treat_y2),   prior="uniform")
             )

qplot(y2, y1_crit, data=ocC, colour=prior, geom="step", main="Critical values y1(y2)")
```

The graph shows that the decision will always be negative if there are
less than 10 events in the treatment group. On the other hand, under a
non-robust prior and assuming 15 events in the treatment group, three
(or less) placebo events would be needed for success. To check this
result, we can directly evaluate the decision function:

```{r}
## just positive
decision(postmix(map, n=6, r=3), postmix(treat_prior, n=24, r=15))
## negative
decision(postmix(map, n=6, r=4), postmix(treat_prior, n=24, r=14))
```


# Trial Analysis

Once the trial has completed and data is collected, the final analysis
can be run with RBesT using the **`postmix`** function. Calculations
are performed analytically as we are in the conjugate mixture
setting.

```{r}
r_placebo <- 1
r_treat   <- 14

## first obtain posterior distributions...
post_placebo <- postmix(map_robust,  r=r_placebo, n=6)
post_treat   <- postmix(treat_prior, r=r_treat  , n=24)

## ...then calculate probability that the difference is smaller than
## zero
prob_smaller <- pmixdiff(post_placebo, post_treat, 0)

prob_smaller

prob_smaller > 0.95

## alternativley we can use the decision object
decision(post_placebo, post_treat)
```

### References

[1] Neuenschwander B et. al, _Clin Trials_. 2010; 7(1):5-18  
[2] Baeten D. et. al, _The Lancet_, 2013, (382), 9906, p 1705  
[3] Morita S. et. al, _Biometrics_ 2008;64(2):595-602  
[4] Schmidli H. et. al, _Biometrics_ 2014;70(4):1023-1032  

### R Session Info

```{r}
sessionInfo()
```

