Dear allstaters,
I am wondering whether McNemar test (for paired binary data) and the test of marginal homogeneity (for paired ordinal data) are valid in presence of structural zeroes in the diagonal.
For example, an individual has to choose between two items; he can (1) select the first item, (2) select the second item, or (3) have no preference, but he cannot select both items. If we have N independant patients, we can represent the data as follows:
Selection of Selection of second item:
first item YES NO
YES N11 N12
NO N21 N22
As none of the individuals cannot select the 2 items, N11=0.
N=N12+N21+N22
If we want to compare the probability to select the first item versus the probability to select the second item, McNemar test for paired binry data seems appropriate. But can it deals with the zero in the diagonal?
More interesting is the extension to ordinal data.
For example, an individual has to select the 3 items he prefers in a list of 10 items, then to rank these 3 items (first preferred, second preferred, third preferred). So, each item can be seen as a 4-level ordinal variable taking values
- 4 if the individual selects the item as first preferred,
- 3 if the individual selects the item as second preferred,
- 2 if the individual selects the item as third preferred,
- 1 if the individual does not select the item.
The question is now to compare 2 items of this 10-item list.
If we have N independant patients, we can represent the data as follows:
Item B
1st 2nd 3rd not
Item A selected selected selected selected
1st selected N11 N12 N13 N14
2nd selected N21 N22 N23 N24
3rd selected N31 N32 N33 N34
not selected N41 N42 N43 N44
As the two items cannot be selected at the same position, N11=0, N22=0, and N33=0.
But the two items can be not selected (N44 can be different from 0)
If we want to compare those 2 ordinal variables, the marginal homogeneity test for paired ordinal data seems appropriate. But can it deals with the zero in the diagonal?
I have the feeling that these tests are valid for this sort of endpoints, because I think McNemar and the marginal homogeneity test do not use the data in the diagonal, but if anyone could give me a reference, it would be welcome.
Best regards
Thomas Dumortier
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