## Posterior predictive checking demo

Checking the assumption of independence in binomial trials (BDA3 p. 147)

ggplot2 is used for plotting, tidyr for manipulating data frames

library(ggplot2)
theme_set(theme_minimal())
library(tidyr)
library(latex2exp)

Data

y <- c(1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0)

Compute test statistic for the data. Test statistic is the number of switches from 0 to 1 or from 1 to 0.

Ty <- sum(diff(y) != 0) + 0.0

Sufficient statistics

n <- length(y)
s <- sum(y)

Compute test statistic for the replicate data.

rb <- function(s, n) {
p <- rbeta(1, s+1, n-s+1)
yr <- rbinom(n, 1, p)
sum(diff(yr) != 0) + 0.0
}
Tyr <- data.frame(x = replicate(10000, rb(s, n)))

Compute posterior predictive p-value

mean(Tyr<=Ty)
## [1] 0.0305

Plot test statistics for the data and replicates. Vertical line corresponds to the original data, and the histogram to the replicate data.

title1 <- 'Binomial example - number of changes?
Pr(T(yrep,theta) <= T(y,theta)|y) = 0.03'
ggplot(data = Tyr) +
geom_histogram(aes(x = x), fill = 'steelblue',
color = 'black', binwidth = 1) +
geom_vline(aes(xintercept = x), data = data.frame(x = Ty),
color = 'red') +
labs(x = TeX(r"(Number of changes in $\textit{y}$ and  $\textit{y}^{ \textrm{rep}}$)"),
y = '', title = title1) +
scale_y_continuous(breaks=NULL)

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