I am advising a small medical study (two groups, treatment is a dummy variable), i.e. a 2x2 contingency table. I'm comparing the value of the Pearson's $\chi^2$ test and a non parametric competitor McNemar's $\chi^2$ test.
The answer brings an additional questions:
they have matched each case in group1 with 4 controls in group2 (matched according to all the variables they deem important, such as sex, age, hemisf) except treatment. Setting aside the question of whether the matching is well done (i.e. whether all important variables have indeed been identified), doesn't the fact that to each case corresponds four controls (i.e. not one) artificially inflate the significance of the results? (they still have to send me part of the data, this is why the table below does not reflect this 4 to 1 ratio).
I obtain the following (very different) results (n=116) (R CrossTable() function).:
[,1] [,2] [1,] 39 9 [2,] 49 19
Statistics for All Table Factors
Pearson's Chi-squared test with Yates' continuity correction
Chi^2 = 0.844691 d.f. = 1 p = 0.3580586
McNemar's Chi-squared test with continuity correction
Chi^2 = 26.22414 d.f. = 1 p = 3.039988e-07
Fisher's Exact Test for Count Data
Sample estimate odds ratio: 1.672924
Alternative hypothesis: true odds ratio is not equal to 1 p = 0.279528 95% confidence interval: 0.6356085 4.692326
The McNemar is the approximate version, but the exact version gives the same conclusions (strong rejection of the null).
My question is: how can i understand such a large difference between $\chi^2$ and McNemar ?