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We had an exercise in statistics class where we did a small ANOVA analysis. We were comparing how the two levels of a factor influenced the response under different conditions (combinations of other factors). My analysis ended up as a tree where a leaf corresponded to each condition and I could see were/how my factor of interest affected the response. It was quite tedious making sense of the R outputs even in such a small case (and the 'tree' was in my mind).

What techniques are usually used to make a 'factor-level comparison' like this in a more general case with lots of factors and levels? (Intuitive graphs and conclusions) Can it be generalized to ANCOVA?

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What you are asking about is called "interactions", which are frequently visualized and hopefully understood with so-called "interaction plots": you plot the relationship between one independent variable and the fitted observations separately for each level of a factor. If there is no significant interaction, the lines will be parallel - if there is, they will not. Look at ?interaction.plot.

This can readily be extended to ANCOVA. You would plot the continuous covariate on the x axis and use multiple lines linking the covariate to the fit - one line per level of the factor.

We recently had a question about understanding a 3-way interaction between two continuous and one factor variable.

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