I am interested in the modeling of binary response data in paired observations. We aim to make inference about the effectiveness of a pre-post intervention in a group, potentially adjusting for several covariates and determining whether there is effect modification by a group that received particularly different training as part of an intervention.

Given data of the following form:

`id phase resp 1 pre 1 1 post 0 2 pre 0 2 post 0 3 pre 1 3 post 0`

And a 2×2 contingency table of paired response information:

PreCorrectIncorrectPostCorrectabIncorrectcd

We’re interested in the test of hypothesis: H0:θc=1.

McNemar’s Test gives: Q=(b−c)2b+c∼χ21 under H0 (asymptotically). This is intuitive because, under the null, we would expect an equal proportion of the discordant pairs (b and c) to be favoring a positive effect (b) or a negative effect (c). With the probability of positive case definition defined p=bb+c and n=b+c. The odds of observing a positive discordant pair is p1−p=bc.

On the other hand, conditional logistic regression uses a different approach to test the same hypothesis, by maximizing the conditional likelihood:

L(X;β)=n∏j=1exp(βXj,2)exp(βXj,1)+exp(βXj,2)

where exp(β)=θc.

So, what’s the relationship between these tests? How can one do a simple test of the contingency table presented earlier? Looking at calibration of p-values from clogit and McNemar’s approaches under the null, you’d think they were completely unrelated!

`library(survival) n <- 100 do.one <- function(n) { id <- rep(1:n, each=2) ph <- rep(0:1, times=n) rs <- rbinom(n*2, 1, 0.5) c( 'pclogit' = coef(summary(clogit(rs ~ ph + strata(id))))[5], 'pmctest' = mcnemar.test(table(ph,rs))$p.value ) } out <- replicate(1000, do.one(n)) plot(t(out), main='Calibration plot of pvalues for McNemar and Clogit tests', xlab='p-value McNemar', ylab='p-value conditional logistic regression')`

**Answer**

Sorry, it’s an old issue, I came across this by chance.

There is a mistake in your code for the mcnemar test. Try with:

```
n <- 100
do.one <- function(n) {
id <- rep(1:n, each=2)
case <- rep(0:1, times=n)
rs <- rbinom(n*2, 1, 0.5)
c(
'pclogit' = coef(summary(clogit(case ~ rs + strata(id))))[5],
'pmctest' = mcnemar.test(table(rs[case == 0], rs[case == 1]))$p.value
)
}
out <- replicate(1000, do.one(n))
```

**Attribution***Source : Link , Question Author : AdamO , Answer Author : eusebe*