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I've run a chi-square test of independence on a two-way table with two categories on the rows and four categories on the columns. My $p$-value is below my chosen $\alpha$, so I may conclude that the rows and columns of this two-way table are dependent. I'm having trouble characterizing the dependence among the rows and columns though. How can I make sense of the residuals and what can I say about the probability of being in one of the rows given that I'm in a specific column?

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You should start by converting your table to either row percents or column percents (whichever makes the most sense). Looking at that you may see some obvious similarities and differences.

One rule of thumb is to then look at the contribution of each cell to the overall chi-squared statistic ( $\frac{(O-E)^2}{E}$ ), cells with a value greater than 4 are the ones most likely to be significantly different from others.

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    $\begingroup$ (+1) Mosaic display are helpful to visualize residuals, as you stated in your last sentence. $\endgroup$ – chl Mar 28 '12 at 21:29

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