---
title: "SAMprior for Continuous Endpoints"
author: "Peng Yang and Ying Yuan"
date: "`r Sys.Date()`"
output:
rmarkdown::html_vignette:
toc: true
html_vignette:
toc: true
html_document:
toc: true
number_sections: true
toc_float:
collapsed: false
smooth_scroll: false
pdf_document:
toc: true
word_document:
toc: true
vignette: >
%\VignetteIndexEntry{Getting started with SAMprior (continuous)}
%\VignetteEncoding{UTF-8}
%\VignetteEngine{knitr::rmarkdown}
---
```{r, include=FALSE}
library(SAMprior)
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
.user_mc_options <- list()
if (is_CRAN) {
.user_mc_options <- options(RBesT.MC.warmup=250, RBesT.MC.iter=500, RBesT.MC.chains=2, RBesT.MC.thin=1, RBesT.MC.control=list(adapt_delta=0.9))
}
```
# Introduction
The self-adapting mixture prior (SAMprior) package is designed to enhance
the effectiveness and practicality of clinical trials by leveraging historical
information or real-world data [1]. The package incorporate historical data
into a new trial using an informative prior constructed based on historical
data while mixing a non-informative prior to enhance the robustness of
information borrowing. It utilizes a data-driven way to determine a self-adapting
mixture weight that dynamically favors the informative (non-informative) prior
component when there is little (substantial) evidence of prior-data conflict.
Operating characteristics are evaluated and compared to the robust
Meta-Analytic-Predictive (rMAP) prior [2], which assigns a fixed weight of 0.5.
# SAM Prior Derivation
SAM prior is constructed by mixing an informative prior $\pi_1(\theta)$,
constructed based on historical data, with a non-informative prior
$\pi_0(\theta)$ using the mixture weight $w$ determined by **`SAM_weight`**
function to achieve the degree of prior-data conflict [1]. The following
sections describe how to construct SAM prior in details.
## Informative Prior Construction based on Historical Data
We assume three historical data as follows:
```{r,results="D_h",echo=FALSE}
set.seed(123)
std <- function(x) sd(x)/sqrt(length(x))
df_1 <- rnorm(40, 0, 3);
df_2 <- rnorm(50, 0, 3);
df_3 <- rnorm(60, 0, 3);
dat <- data.frame(study = c(1,2,3),
n = c(40, 50, 60),
mean = round(c(mean(df_1), mean(df_2), mean(df_3)), 3),
se = round(c(std(df_1), std(df_2), std(df_3)), 3))
kable(dat)
```
To construct informative priors based on the aforementioned three historical
data, we apply **`gMAP`** function from RBesT to perform meta-analysis. This
informative prior results in a representative form from a large MCMC samples,
and it can be converted to a parametric representation with the
**`automixfit`** function using expectation-maximization (EM) algorithm [3].
This informative prior is also called MAP prior.
```{r, message=FALSE}
sigma = 3
# load R packages
library(ggplot2)
theme_set(theme_bw()) # sets up plotting theme
set.seed(22)
map_mcmc <- gMAP(cbind(mean, se) ~ 1 | study,
weights=n,data=dat,
family=gaussian,
beta.prior=cbind(0, sigma),
tau.dist="HalfNormal",tau.prior=cbind(0,sigma/2))
map_automix <- automixfit(map_mcmc)
map_automix
plot(map_automix)$mix
```
The resulting MAP prior is approximated by a mixture of conjugate priors.
## SAM Weight Determination
Let $\theta$ and $\theta_h$ denote the treatment effects associated with the
current arm data $D$ and historical $D_h$, respectively. Let $\delta$ denote
the clinically significant difference such that is $|\theta_h - \theta| \ge \delta$,
then $\theta_h$ is regarded as clinically distinct from $\theta$, and it is
therefore inappropriate to borrow any information from $D_h$. Consider two
hypotheses:
$$
H_0: \theta = \theta_h, ~~ H_1: \theta = \theta_h + \delta ~ \text{or} ~ \theta = \theta_h - \delta.
$$
$H_0$ represents that $D_h$ and $D$ are consistent (i.e., no prior-data
conflict) and thus information borrowing is desirable, whereas $H_1$ represents
that the treatment effect of $D$ differs from $D_h$ to such a degree that no
information should be borrowed.
The SAM prior uses the likelihood ratio test (LRT) statistics $R$ to quantify
the degree of prior-data conflict and determine the extent of information
borrowing.
$$
R = \frac{P(D | H_0, \theta_h)}{P(D | H_1, \theta_h)} = \frac{P(D | \theta = \theta_h)}{\max \{ P(D | \theta = \theta_h + \delta), P(D | \theta = \theta_h - \delta) \}} ,
$$
where $P(D | \cdot)$ denotes the likelihood function. An alternative Bayesian
choice is the posterior probability ratio (PPR):
$$
R = \frac{P(D | H_0, \theta_h)}{P(D | H_1, \theta_h)} = \frac{P(H_0)}{P(H_1)} \times BF ,
$$
where $P(H_0)$ and $P(H_1)$ is the prior probabilities of $H_0$ and $H_1$
being true. $BF$ is the Bayes Factor that in this case is the same as LRT.
The SAM prior, denoted as $\pi_{sam}(\theta)$, is then defined as a mixture
of an informative prior $\pi_1(\theta)$, constructed based on $D_h$, with a
non-informative prior $\pi_0(\theta)$:
$$\pi_{sam}(\theta) = w \pi_1(\theta) + (1 - w) \pi_0(\theta)$$
where the mixture weight $w$ is calculated as:
$$w = \frac{R}{1 + R}.$$
As the level of prior-data conflict increases, the likelihood ratio $R$
decreases, resulting in a decrease in the weight $w$ assigned to the
informative prior and a decrease in information borrowing. As a result,
$\pi_{sam}(\theta)$ is data-driven and has the ability to self-adapt the
information borrowing based on the degree of prior-data conflict.
To calculate the SAM weight $w$, we first assume the sample size enrolled in
the control arm is $n = 30$, with $\theta = 0.4$ and $\sigma = 3$. Additionally,
we assume the effective size is $d = \frac{\theta - \theta_h}{\sigma} = 0.5$,
then we can apply
function **`SAM_weight`** in SAMprior as follows:
```{r, message=FALSE}
set.seed(234)
sigma <- 3 ## Standard deviation in the current trial
data.crt <- rnorm(30, mean = 0.4, sd = sigma)
wSAM <- SAM_weight(if.prior = map_automix,
delta = 0.5 * sigma,
data = data.crt)
cat('SAM weight: ', wSAM)
```
The default method to calculate $w$ is using LRT, which is fully data-driven.
However, if investigators want to incorporate prior information on prior-data
conflict to determine the mixture weight $w$, this can be achieved by using
PPR method as follows:
```{r, message=FALSE}
wSAM <- SAM_weight(if.prior = map_automix,
delta = 0.5 * sigma,
method.w = 'PPR',
prior.odds = 3/7,
data = data.crt)
cat('SAM weight: ', wSAM)
```
The **`prior.odds`** indicates the prior probability of $H_0$ over the prior
probability of $H_1$. In this case (e.g., **`prior.odds = 3/7`**), the prior
information favors the presence prior-data conflict and it results in a
decreased mixture weight.
When historical information is congruent with the current control arm, SAM
weight reaches to the highest peak. As the level of prior-data conflict
increases, SAM weight decreases. This demonstrates that SAM prior is data-driven
and self-adapting, favoring the informative (non-informative) prior component
when there is little (substantial) evidence of prior-data conflict.
```{r, echo=FALSE, message=FALSE, warning=FALSE}
weight_grid <- seq(-3, 3, by = 0.3)
weight_res <- lapply(weight_grid, function(x){
res <- c()
for(s in 1:300){
data.control <- rnorm(n = 35, mean = x, sd = sigma)
res <- c(res, SAM_weight(if.prior = map_automix,
delta = 0.5 * sigma,
data = data.control))
}
mean(res)
})
df_weight <- data.frame(grid = weight_grid,
weight = unlist(weight_res))
qplot(grid, weight, data = df_weight, geom = "line", main= "SAM Weight") +
xlab('Sample mean from control trial')+ ylab('Weight') +
geom_vline(xintercept = summary(map_automix)['mean'], linetype = 2, col = 'blue')
```
## SAM Prior Construction
To construct the SAM prior, we mix the derived informative prior $\pi_1(\theta)$
with a vague prior $\pi_0(\theta)$ using pre-determined mixture weight by
**`SAM_prior`** function in SAMprior as follows:
```{r, message=FALSE}
unit_prior <- mixnorm(nf.prior = c(1, summary(map_automix)['mean'], sigma))
SAM.prior <- SAM_prior(if.prior = map_automix,
nf.prior = unit_prior,
weight = wSAM, sigma = sigma)
SAM.prior
```
where the non-informative prior $\pi_0(\theta)$ follows an unit-information prior.
## Operating Characteristics
In this section, we aim to investigate the operating characteristics of
the SAM prior, constructed based on the historical data, via simulation.
The incorporation of historical information is expected to be beneficial
in reducing the required sample size for the current arms. To achieve this,
we assume a 1:2 ratio between the control and treatment arms.
We compare the operating characteristics of the SAM prior and rMAP prior
with pre-specified fixed weight under various scenarios. Specifically,
we will evaluate the relative bias and relative mean squared error (MSE)
of these methods. The relative bias and relative MSE are defined
as the differences between the bias/MSE of a given method and the bias/MSE
obtained when using a non-informative prior.
Additionally, we investigate the type I error and power of the methods
under different degrees of prior-data conflicts. The decision regarding
whether a treatment is superior or inferior to a standard control will be
based on the condition:
$$\Pr(\theta_t - \theta > 0) > 0.95.$$
In SAMprior, the operating characteristics can be considered in following steps:
1. Specify priors: This step involves constructing informative prior based
on historical data and non-informative prior.
2. Specify the decision criterion: The **`decision2S`** function
is used to initialize the decision criterion. This criterion determines
whether a treatment is considered superior or inferior to a standard
control.
3. Specify design parameters for the **`get_OC`** function: This step
involves defining the design parameters for evaluating the operating
characteristics. These parameters include the clinically significant
difference (CSD) used in SAM prior calculation, the method used to
determine the mixture weight for the SAM prior, the sample sizes for
the control and treatment arms, the number of trials used for simulation,
the choice of weight for the robust MAP prior used as a benchmark, and
the vector of mean for both the control and treatment arms.
### Type I Error
To compute the type I error, we consider four scenarios, with the first and
last two scenarios representing minimal and substantial prior-data conflicts,
respectively. In general, the results show that both methods effectively
control the type I error.
```{r, message=FALSE}
set.seed(123)
# weak_prior <- mixnorm(c(1, summary(map_automix)[1], 1e4))
TypeI <- get_OC(if.prior = map_automix, ## MAP prior from historical data
nf.prior = unit_prior, ## Weak-informative prior for treatment arm
delta = 0.5*sigma, ## CSD for SAM prior
n = 35, n.t = 70, ## Sample size for control and treatment arms
## Decisions
decision = decision2S(0.95, 0, lower.tail=FALSE),
ntrial = 1000, ## Number of trials simulated
if.MAP = TRUE, ## Output robust MAP prior for comparison
weight = 0.5, ## Weight for robust MAP prior
## Mean for control and treatment arms
theta = c(0, 0, -2, 4),
theta.t = c(0, -0.1, -2, 4),
sigma = sigma
)
kable(TypeI)
```
### Power
For power evaluation, we also consider four scenarios, with the first and
last two scenarios representing minimal and substantial prior-data conflicts,
respectively. In general, it is observed that the SAM prior achieves comparable
or better power compared to rMAP.
```{r, message=FALSE}
set.seed(123)
Power <- get_OC(if.prior = map_automix, ## MAP prior based on historical data
nf.prior = unit_prior, ## Non-informative prior for treatment arm
delta = 0.5*sigma, ## CSD for SAM prior
n = 35, n.t = 70, ## Sample size for control and treatment arms
## Decisions
decision = decision2S(0.95, 0, lower.tail=FALSE),
ntrial = 1000, ## Number of trials simulated
if.MAP = TRUE, ## Output robust MAP prior for comparison
weight = 0.5, ## Weight for robust MAP prior
## Mean for control and treatment arms
theta = c(0, 0.1, 0.5, -3),
theta.t = c(1, 1.1, 2.0, -1.5),
sigma = sigma
)
kable(Power)
```
# Decision Making
Finally, we present an example of how to make a final decision on whether the
treatment is superior or inferior to a standard control once the trial has been
completed and data has been collected. This step can be accomplished using the
**`postmix`** function from RBesT, as shown below:
```{r, message=FALSE}
## Simulate data for treatment arm
data.trt <- rnorm(60, mean = 3, sd = sigma)
## first obtain posterior distributions...
post_SAM <- postmix(priormix = SAM.prior, ## SAM Prior
data = data.crt)
post_trt <- postmix(priormix = unit_prior, ## Non-informative prior
data = data.trt)
## Define the decision function
decision = decision2S(0.95, 0, lower.tail=FALSE)
## Decision-making
decision(post_trt, post_SAM)
```
### References
[1] Yang P. et al., _Biometrics_, 2023; 00, 1–12. https://doi.org/10.1111/biom.13927 \
[2] Schmidli H. et al., _Biometrics_, 2014; 70(4):1023-1032. \
[3] Neuenschwander B. et al., _Clin Trials_, 2010; 7(1):5-18.
### R Session Info
```{r}
sessionInfo()
```
```{r,include=FALSE}
options(.user_mc_options)
```