Relationship between McNemar’s test and conditional logistic regression

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=(bc)2b+cχ21 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 p1p=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)=nj=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')

enter image description here

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))

enter image description here

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

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