- Functions, loops, and vectorized programming
- Writing and simulation a function from a statistical paper
- Debugging, profiling, and improving efficiency
We did not have enough time for most to finish lab 2, and today’s lab time is dedicating to finishing the lab.
If you complete the lab 2, profiling your functions and see if there are rooms to improve efficiency.
Devote any extra time to working on homework 1 which is a continuation of the lab.
Today’s lab is the first steps toward designing a response-adaptive randomization (RAR) trial. RAR designs are used in precision medicine trials, such as the BATTLE trial, to gather early evidence of treatment arms that work best for a given biomarker. Throughout RAR, the treatment allocation adjusts depending on which treatment arm looks most promising. We will focus on the initial steps of coding this design.
The lab is motivated by the paper by Kurt Viele: Comparison of methods for control allocation in multiple arm studies using response adaptive randomization. It practices:
- Pre-allocating vectors
- Using a loop
- Writing a function
Notation and criteria for study to successfully declare a treatment as efficacious:
-
$i = 1, \dots, N$ participants -
$t = 0, 1, 2, 3$ study arms (t = 0 is control) -
$Y_i \mid t \sim Bern(p_t)$ and$y_t$ is a vector of$n_t$ observed outcomes on arm$t$ - The prior on
$p_t \sim Beta(\alpha_t, \beta_t)$
Posterior Distribution
Quoting from the paper: The trial is considered successful at the final analysis if there is a high posterior probability that at least one arm has a higher rate than control.
where
- Equal allocation to four arms throughout design.
- RAR where the allocation probability is updated at an interim analysis as follows:
-
$V_t = P_t$ (Max) $V_0 = min{\sum V_t \frac{(n_t + 1)}{(n_0 + 1)}, max(V_1, V_2, V_3) }$
Note: A way to estimate cbind K = [1000] draws from
the posterior distribution of each arm and to see how frequently (across
the K draws from each arm) each arm is drawn to be the largest.
Write a function for each study design to simulate one trial.
- N = 228 with interim analyses after every 40th participant starting at 40.
- Use equal allocation for first 40 patients for both designs.
- Assume a setting where treatment effect is 0.35 for each study arm (the null scenario). (But allow flexibility in function for other treatment effects).
-
$\alpha_t = 0.35$ for all$t$ and$\beta_t = 0.65$ for all arms. - Use the following
$\delta$ thresholds to determine a successful trial:- Design 1,
$\delta = 0.9912$ - Design 2,
$\delta = 0.9892$
- Design 1,
For simplicity, have your function return a list of at least the following output:
- The probability that the best treatment arm is better than control.
- The number of patients assigned to each treatment arm.
mymodel <- function(threshold, N = N, K = 1000, design,
alpha_t =.35, beta_t = .65, seed = 1500){
set.seed(seed)
interim_n <- c(40, 80, 120, 160, 200)
mydatt <- NULL
norm_v <- c(0.25, 0.25, .25, .25)
names(norm_v) <- c("0", "1", "2", "3")
dat <- data.frame(id = 1:10000, y = rbinom(10000, 1, .35))
if(design ==1){
df <- lapply(split(sample(nrow(dat), N), c(0, 1, 2, 3)), function(i){dat[i, ]})
mydatt <- cbind(do.call(rbind, df),
trt_arm = rep(names(df),
sapply(df, function(y){dim(y)[1]})))
}else{
norm_v <- c(0.25, 0.25, .25, .25)
names(norm_v) <- c("0", "1", "2", "3")
trt <- apply(t(rmultinom(40, size = 1, prob = as.vector(norm_v))), 1, function(x) which(x==1)-1)
for(i in 1:(length(interim_n)+1)){
if(i<=(length(interim_n))){
n0 <- 40
df <- lapply(split(sample(nrow(dat), n0), trt), function(j){dat[j, ]})
}else{
n0 <- N-(interim_n[i-1])
df <- lapply(split(sample(nrow(dat), n0), trt), function(j){dat[j, ]})
}
mydat <- cbind(do.call(rbind, df),
trt_arm = rep(names(df),
sapply(df, function(y){dim(y)[1]})))
post_probs <- sapply(df,
function(m){rbeta(K,
(alpha_t + sum(m[, "y"])),
(beta_t + nrow(m) - sum(m[, "y"])))})
n_t <- as.vector(sample_per_trt <- sapply(df, function(m){nrow(m)})[2:4])
#n0 <- as.vector(sample_per_trt <- sapply(df, function(m){nrow(m)})[-(2:4)])
V_t <- apply(post_probs, 1, function(x){which.max(x)})
pt_max <- c(mean(V_t==1),
mean(V_t==2),
mean(V_t==3),
mean(V_t==4)
)
V_0 <- min(sum(pt_max[-1]*((n_t+1)/(n0+1))), max(pt_max[-1]))
norm_v <- sapply(c(V_0, pt_max[-1]), function(r)r*1/sum(c(V_0, pt_max[-1])))
mydatt <- rbind(mydatt, mydat)
}
}
fprobs <- sapply(0:3, function(m){rbeta(K, (alpha_t + sum(mydatt["trt_arm"==m, "y"])),
(beta_t + nrow(mydatt["trt_arm"==m, ]) - sum(mydatt["trt_arm"==m, "y"])))})
sample_per_trt <- table(mydatt$trt_arm)
names(sample_per_trt) <- c("Control 1", "Arm 1", "Arm 3", "Arm 4")
colnames(fprobs) <- Hmisc::Cs(p_0, p_1, p_2, p_3)
trsholdprob <- apply(fprobs[, 2:4], 2, function(r){
mean(r > fprobs[, 1])})#; print(trsholdprob)
return(list(paste("The probability that the best treatment arm",
c(1:3)[trsholdprob==max(trsholdprob)],
"is better than control is",
max(trsholdprob)),
ifelse(max(trsholdprob) > threshold,
"Trial was successsful",
"Trial was not successsful"),
sample_per_trt))
}
mymodel(threshold = .9912, N = 228, K = 1000, design = 2,
alpha_t =.35, beta_t = .65)[[1]]
[1] "The probability that the best treatment arm 3 is better than control is 0.507"
[[2]]
[1] "Trial was not successsful"
[[3]]
Control 1 Arm 1 Arm 3 Arm 4
63 34 64 67
- Replicate the design many (10K) times. Calculate the Type I error.
- Find
$\delta$ for each design (supposing you didn’t already know it). - Replicate the study design assuming treatment effects of
$p_0 = p_1 = p_2 = 0.35$ $p_3 = 0.65$