Self-adapting Mixture (SAM) Priors

Description

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.

Installation

To install the package:

install.packages('devtools')
devtools::install_github("pengyang0411/SAMprior")

Usage

Consider a randomized clinical trial to compare a treatment with a control in patients with ankylosing spondylitis. The primary efficacy endpoint is binary, indicating whether a patient achieves 20% improvement at week six according to the Assessment of SpondyloArthritis International Society criteria [3]. Nine historical data available to the control were used to construct the MAP prior:

study	n	r
Baeten (2013)	6	1
Deodhar (2016)	122	35
Deodhar (2019)	104	31
Erdes (2019)	23	10
Huang (2019)	153	56
Kivitz (2018)	117	55
Pavelka (2017)	76	28
Sieper (2017)	74	21
Van der Heijde (2018)	87	35

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

To construct informative priors based on the aforementioned nine 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 [4]. This informative prior is also called MAP prior.

# load R packages
library(ggplot2)
theme_set(theme_bw()) # sets up plotting theme
set.seed(22)
map_ASAS20 <- gMAP(cbind(r, n-r) ~ 1 | study,
                   family = binomial,
                   data = ASAS20, 
                   tau.dist = "HalfNormal", 
                   tau.prior = 1,
                   beta.prior = 2)

## Assuming default prior location   for beta: 0

map_automix <- automixfit(map_ASAS20)
map_automix

## EM for Beta Mixture Model
## Log-Likelihood = 5005.173
## 
## Univariate beta mixture
## Mixture Components:
##   comp1      comp2     
## w  0.6347375  0.3652625
## a 42.5097011  7.1944569
## b 77.2077293 12.3741342

plot(map_automix)$mix

The resulting MAP prior is approximated by a mixture of conjugate priors, given by $\pi_1(\theta) = 0.63 Beta(42.5, 77.2) + 0.37 Beta(7.2, 12.4)$, with $\hat{\theta}_h \approx 0.36$.

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|⋅) 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 mixture weight $w$ of the SAM prior, we assume the sample size enrolled in the control arm is $n = 35$, with $r = 10$ responses, then we can apply function SAM_weight in SAMprior as follows:

n <- 35; r = 10 
wSAM <- SAM_weight(if.prior = map_automix, 
                   delta = 0.2,
                   n = n, r = r)
cat('SAM weight: ', wSAM)

## SAM weight:  0.7900602

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:

wSAM <- SAM_weight(if.prior = map_automix, 
                   delta = 0.2,
                   method.w = 'PPR',
                   prior.odds = 3/7,
                   n = n, r = r)
cat('SAM weight: ', wSAM)

## SAM weight:  0.6172732

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.

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:

SAM.prior <- SAM_prior(if.prior = map_automix, 
                       nf.prior = mixbeta(nf.prior = c(1,1,1)),
                       weight = wSAM)
SAM.prior

## Univariate beta mixture
## Mixture Components:
##   comp1      comp2      nf.prior  
## w  0.3918065  0.2254667  0.3827268
## a 42.5097011  7.1944569  1.0000000
## b 77.2077293 12.3741342  1.0000000

where the non-informative prior $\pi_0(\theta)$ follows a uniform distribution.

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:

## Sample size and number of responses for treatment arm
n_t <- 70; x_t <- 22 

## first obtain posterior distributions...
post_SAM <- postmix(priormix = SAM.prior,         ## SAM Prior
                    r = r,   n = n)
post_trt <- postmix(priormix = mixbeta(c(1,1,1)), ## Non-informative prior
                    r = x_t, n = n_t)

## Define the decision function
decision = decision2S(0.95, 0, lower.tail=FALSE)

## Decision-making
decision(post_trt, post_SAM)

## [1] 0

Maintainer information

Peng Yang (py11@rice.edu)

Citation

[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] Baeten D. et al., The Lancet, 2013; (382), 9906, p 1705.
[4] Neuenschwander B. et al., Clin Trials, 2010; 7(1):5-18.