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.
To install the package:
install.packages('devtools')
devtools::install_github("pengyang0411/SAMprior")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 is constructed by mixing an informative prior
π1(θ), constructed based on historical data, with a
non-informative prior π0(θ) 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.
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 0.6347378 0.3652622
## a 42.5096289 7.1944564
## b 77.2075968 12.3741335
plot(map_automix)$mix
The resulting MAP prior is approximated by a mixture of conjugate priors, given by π1(θ) = 0.63Bet**a(42.5,77.2) + 0.37Bet**a(7.2,12.4), with θ̂h ≈ 0.36.
Let θ and θh denote the treatment effects associated with the current arm data D and historical Dh, respectively. Let δ denote the clinically significant difference such that is |θh−θ| ≥ δ, then θh is regarded as clinically distinct from θ, and it is therefore inappropriate to borrow any information from Dh. Consider two hypotheses:
H0 : θ = θh, H1 : θ = θh + δ or θ = θh − δ.
H0 represents that Dh and D are consistent (i.e., no prior-data conflict) and thus information borrowing is desirable, whereas H1 represents that the treatment effect of D differs from Dh 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.
where P(D|⋅) denotes the likelihood function. An alternative Bayesian choice is the posterior probability ratio (PPR):
where P(H0) and P(H1) is the prior probabilities of H0 and H1 being true. B**F is the Bayes Factor that in this case is the same as LRT.
The SAM prior, denoted as πsam(θ), is then defined as a mixture of an informative prior π1(θ), constructed based on Dh, with a non-informative prior π0(θ):
πsam(θ) = w**π1(θ) + (1−w)π0(θ),
where the mixture weight w is calculated as:
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, πsam(θ) 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 H0
over the prior probability of H1. 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.
To construct the SAM prior, we mix the derived informative prior
π1(θ) with a vague prior π0(θ) 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 0.3918066 0.2254666 0.3827268
## a 42.5096289 7.1944564 1.0000000
## b 77.2075968 12.3741335 1.0000000
where the non-informative prior π0(θ) follows a uniform distribution.
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
Peng Yang ([email protected])
[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.