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I've checked multiple threads about handling or visualizing contingency tables, but can't find one that can help my current question. I have a 2x3 contingency table: "group" variable has 3 levels not necessarily ordered, "level" variable has 2 levels (could be 3, but for simplicity, let's say 2). I would like to test independence between group and level variables, and visualize the pairwise relationship between groups as well (group 1 vs 3, 2 vs 3, etc). I'm using vcd::mosaic which runs loglm (which wraps loglin) to visualize individual significance of cells in the contingency table. I like it that the area of each tile reflects the cell frequency and color reflects the statistical significance (residual).

My question is: when I put the full dataset with all 3 groups in loglm, group 3 does not seem to deviate from the null, but when I use the partial datasets with group 1 and 3, or 2 and 3 only, then each pair significantly deviates from the null. What is the best way to perform the test, visualize and interpret the relationship between the 3 groups? Should I use two plots with the pair of groups or one plot with 3 groups? I provide a dummy example dataset and mosaic plot results below:

set.seed(123) 
d <- data.frame(group = c(rep(1,20),rep(2,20),rep(3,20)), 
                level = c(1, rep(0,19),rep(1,19),0, rep(1,10),rep(0,10)))

# 3 groups
tbl123 <- table(d)
vcd::mosaic( ~ group + level, data = tbl123, gp=shading_max, split_vertical=T)  

# 2 group
tbl13 <- table(d[d$group %in% c(1,3), ])
vcd::mosaic( ~ group + level, data = tbl13, gp=shading_max, split_vertical=T)  

tbl23 <- table(d[d$group %in% c(2,3), ])
vcd::mosaic( ~ group + level, data = tbl23, gp=shading_max, split_vertical=T)

enter image description here enter image description here enter image description here

Thanks

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Let me try to answer my question after talking to a few friends. Please feedback/correct where appropriate. Whether to do the loglm test on the full dataset using 3 groups (in which group 3 shows no deviation from null) or on the partial dataset using 2 groups (in which group 3 shows deviation from null) depends on the original practical question at hand. If it makes sense (from the domain knowledge) to do two-group test for complete independence (i.e. group 1 & 3, 2 & 3 or 1 & 2, because the 3 groups arisen due to different causes), then the two-group test is fine. And so it can be visualized with 2x2 mosaic plots above. If the 3 groups are inter-related, e.g. the different eye colors due to the expression of the eye color genes, then should perform the test with the full dataset to get the full picture on the inter-relationship.

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