I want to assess an intervention to cure a disease. An example is described by the following matrix:
responses <- c("Present", "Absent") matrix(c(101,59,121,33),nrow=2,dimnames = list("Before" = responses,"After" = responses)) After Before Present Absent Present 101 121 Absent 59 33
This example was taken from the McNemar's Wikipedia page. McNemar may be useful to test equality of marginal probabilities for two different exams/tests, however for this particular before-and-after case ignoring a and d seems an incomplete analysis and seems potentially misleading or I am missing something. It is easier to explain with the following example:
M <- matrix(c(1,9,9,81),nrow=2,dimnames = list("Before" = responses,"After" = responses)) M %>% list(.,mcnemar.test(.),chisq.test(.)) [] After Before Present Absent Present 1 9 Absent 9 81 [] McNemar's Chi-squared test data: . McNemar's chi-squared = 0, df = 1, p-value = 1 [] Pearson's Chi-squared test data: . X-squared = 0, df = 1, p-value = 1
Neither the McNemar nor the Chi-square tests reject the null. For McNemar clearly the before and after marginal proportions are equal given that b and c are equal. For the Chi-Square the expected values are equal to the observed, hence the statistic is zero.
However there is a story to be told if we consider a and d and the class proportions before intervention. We can see that from a total of 10 patients with disease Present Before the intervention, 9 changed to Absent, i.e., 90% improved; while from a total of 90 patients initially Absent of disease, only 10% changed to present. This suggests an asymmetry in change.
Which test would support a claim about statistically significant difference in change considering proportions?
How about applying McNemar to the matrix normalised considering initial proportions (i.e., rows sum 100)?
[DISCLAIMER: Assessing whether the intervention was effective or not is hard in this case without a control group, however, being able to justify with stats what the data is telling is meaningful]
EXTRA: The Wikipedia page says that "the null hypothesis of "marginal homogeneity" would mean there was no effect of the treatment". This would be concluded for my later example. However if we replace the second row (Before-Absent) with 1,9, instead of 9,81, McNemar rejects and the conclusion would be the opposite, despite the ratio being the same, the only thing that changed is the sample size of the (Before-Absent) group.
Is the Wikipedia description misleading or am I missing something?