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I would like to test independence of several categorical variables for a few datasets. I believe Breslow-day is possible for some of the analyses I want to do, and log-linear may be possible for all, but besides applicability conditions, how are these two analyses different and which is more appropriate for the cases where both are possible?

Experimental participants' responses to a comprehension question were scored as strong or weak in each of two respects: correctness and generality. Participants came from one of two training conditions and were also classified into one of two previous knowledge groups. So I have altogether 4 binary factors: respect, score, training, and group. Score and group were observed, while respect and training were manipulated. Respect is within-subjects, while the others are all between-subjects. In a later experiment I add an additional binary experimental factor, pretraining.

I initially didn't take account for group, so I did a Breslow-day test with respect, score, and training. I found a significant effect, which I interpret as "the effect of training on score differs depending on respect". In particular, one training condition promoted correctness better, while the other promoted abstractness better.

I later wanted to see whether adding group made any difference. As far as I know, I cannot add another factor in Breslow-day. Thus, I tried a log-linear analysis. I first did this with the same 3 factors used in the Breslow-day test, so I could compare the results. Here's R code:

T = table( D$score, D$training, D$respect, dnn=c('Score','Training','Respect') )
loglm(~ Score + Training + Respect, data=T)

This result is NOT significant, which I interpret as meaning the data can be explained completely by main effects of the factors, without any interaction term. That seems to contradict the result of the Breslow-day test, so I'm now not sure which one is more accurate.

Next, I'd like to add the other factors (group, pretraining). Am I right that I MUST use the log-linear analysis in this case?

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