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Below (in R), I have two INDEPENDENT groups of scores that are binomially distributed. These two groups of scores are known to have different probability of success (i,e., $p_1 \neq p_2$).

Let's now assume two scenarios regarding the probability of success within each group:

(1) The probability of success within each group is the same.

(2) The probability of success within group 1 $\sim Beta(500, 500)$ and that within group 2 $\sim Beta(300, 500).$

Question

To find the differences between these two groups of scores, what statistical test is the best choice?

Here is the R code that provides the scores (i.e., y) for the 2 groups:

set.seed(0)
n1 = 100 ; n2 = 100 ; p1 = .65 ; p2 = .5 ; max.score = 15

y = as.vector(unlist(mapply(FUN = rbinom, n = c(n1, n2), size = c(max.score, max.score), prob = c(p1, p2))))

groups = factor( rep(1:2, times = c(n1, n2)) )
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    $\begingroup$ More context would be helpful, this is pretty generic. For example, does it really make sense in your context to assume all the observations in the same group have exactly the same probability of success? Also, you generate them in pairs, is this meant to be paired data or is it independent groups? Describing the details of your study will help make this a better question and allow us to answer more appropriately. $\endgroup$ – Aaron Jun 15 '17 at 15:35
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    $\begingroup$ It still sounds like a theory problem, which is fine if that's what it is, but if this is real data, you should describe that more fully. The part that I'd be worried about is your assumption that the probability in each group is the same. Is there something about the study that makes this assumption make sense? Otherwise, it should at least be tested, but even if it looks okay, my instinct is that the sample size is small enough to not have a lot of certainty about that from the data so might be better to use a more robust method. $\endgroup$ – Aaron Jun 15 '17 at 15:47
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    $\begingroup$ @Aaron, thanks for your explanation. And why if $p$ be the same in each group (but different across the groups) you said you would be worried? $\endgroup$ – rnorouzian Jun 15 '17 at 16:00
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    $\begingroup$ The standard binomial test is what @COOLSerdash recommended. If the p really is the same, then it doesn't matter that they came from different subjects. About your support for believing that, please edit your question to include those details. $\endgroup$ – Aaron Jun 15 '17 at 16:18
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    $\begingroup$ Well, you keep making assumptions rather than telling about your situation. Why beta distributions? Better to describe your study directly. Plus you don't want to assume you know the parameters of the beta, otherwise there's no analysis to do. :) $\endgroup$ – Aaron Jun 15 '17 at 17:26
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If you're willing to assign a prior Beta distribution on $p_1$ and $p_2$, you could do a Bayesian analysis. The model would be

\begin{align} p_1&\sim \operatorname{Beta}(\alpha_1,\beta_1) \\ p_2&\sim \operatorname{Beta}(\alpha_2,\beta_2) \\ y_1&\sim \operatorname{Bin}(p_1,n_1) \\ y_2&\sim \operatorname{Bin}(p_2,n_2) \end{align}

Where $\alpha$ and $\beta$ are your hyperparameters for $p_1$ and $p_2$, respectively. Here is the example done in R and JAGS. This code can easily be extended to include different $n$. A solution based on conjugacy is presented below.

library(rjags)
library(R2jags)
library(runjags)
library(hdrcde)

## The model

cat("
    model{

    for(i in 1:N) {
      y1[i] ~ dbin(p1, n1)
      y2[i] ~ dbin(p2, n2)
    }

    p1 ~ dbeta(500, 500)
    p2 ~ dbeta(300, 500)

    delta <- p1 - p2
}
", file="jags_model_beta.txt")

## The data

set.seed(0)
n1 = 100 ; n2 = 100 ; p1 = 0.65 ; p2 = 0.5 ; max.score = 15

y = as.vector(unlist(mapply(FUN = rbinom, n = c(n1, n2), size = c(max.score, max.score), prob = c(p1, p2))))

groups = factor( rep(1:2, times = c(n1, n2)) )

data.list <- list(
  y1 = y[groups == 1]
  , y2 = y[groups == 2]
  , n1 = max.score
  , n2 = max.score
  , N = n1
)

## Parameters to store

params <- c(
  "p1"
  , "p2"
  , "delta"
)

## MCMC settings

niter <- 10000 # number of iterations
nburn <- 2000 # number of iterations to discard (the burn-in-period)
nchains <- 5 # number of chains

## Run JAGS

out <- jags(
  data                 = data.list
  , parameters.to.save = params
  , model.file         = "jags_model_beta.txt"
  , n.chains           = nchains
  , n.iter             = niter
  , n.burnin           = nburn
  , n.thin             = 5
  # , inits              = jags.inits
  , progress.bar       = "text")

## Display the results

print(out, 3)

         mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
delta      0.121   0.014   0.093   0.112   0.121   0.131   0.149 1.001  4700
p1         0.584   0.010   0.565   0.578   0.584   0.591   0.604 1.001  8000
p2         0.463   0.010   0.443   0.456   0.463   0.470   0.484 1.001  5200

The delta is the difference between the proportions. The 95% credible interval for the difference is $(0.09; 0.15)$. This does not include $0$ so we have evidence that there is a true difference between proportions. We can also plot the posterior of delta:

jagsfit.mcmc <- as.mcmc(out)
jagsfit.mcmc <- combine.mcmc(jagsfit.mcmc)
hdr.den(as.vector(jagsfit.mcmc[, "delta"]), prob = c(95))

Posterior_density of delta

Solution based on conjugacy

As I explain here, the posterior of $p_1$ and $p_2$ are also Beta distributions if the priors are Beta distributions. Specifically, the posterior distributions are: \begin{align} p_{1}^{\star}&\sim \operatorname{Beta}(\alpha_1 + \sum_{i=1}^{n}x_{1,i},\beta_1 + \sum_{i=1}^{n}N_{1,i} - \sum_{i=1}^{n}x_{1,i}) \\ p_2^{\star}&\sim \operatorname{Beta}(\alpha_2 + \sum_{i=1}^{n}x_{2,i},\beta_2 + \sum_{i=1}^{n}N_{2,i} - \sum_{i=1}^{n}x_{2,i}) \end{align}

Where $x_{1,i}$ and $x_{2,i}$ denote the number of successes for group 1 and 2. $N_{1,i}$ and $N_{2,i}$ denote the sample sizes for group 1 and 2. The solution in R is:

alpha1 <- 500
beta1 <- 500

alpha2 <- 300
beta2 <- 500

post_p1 <- rbeta(1e6, alpha1 + sum(y[groups == 1]), beta1 + n1*15 - sum(y[groups == 1]))
post_p2 <- rbeta(1e6, alpha2 + sum(y[groups == 2]), beta2 + n2*15 - sum(y[groups == 2]))

delta <- post_p1 - post_p2

hist(delta, las = 1, col = "slategrey", freq = FALSE)
(credval <- quantile(delta, c(0.025, 0.975)))
abline(v = credval, col = "red", lwd = 2)

Credible_interval

The 95% credible interval is nearly identical with the one produced with JAGS.

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  • $\begingroup$ Yes, I'd say so. The delta is the direct comparison of the probabilities between the groups. $\endgroup$ – COOLSerdash Jun 15 '17 at 17:47
  • $\begingroup$ Could you please let me know what 15 represents in: beta1 + n1*15? $\endgroup$ – rnorouzian Jun 15 '17 at 18:10
  • $\begingroup$ @rnorouzian The max.score or the sample size. $\endgroup$ – COOLSerdash Jun 15 '17 at 18:55

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