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I am having a question regarding a statistical analysis I am conducting. Let's say I have a continuous variable that I measure (Plant Weight) and I have 2 factors with 2 levels (sufficient watering/little watering and fertilizer/no fertilizer). One way to look at this would be a model like this: Plant Weight~watering*fertilizer

I could do a Two-Way Anova and depending on the outcome (e.g. significant interaction), I could do a post hoc test to compare all combinations.

Now I've repeatedly seen people who just combined two factors into one grouping variable with 4 levels like this: Treatment A (water/fertilizer),Treatment B (little water/fertilizer), Treatment C (water/no fertilizer).....

This would be a one-way Anova (Plant Weight~Treatment).

Now if I wanted to know something about the overall influence of the main effects of watering and fertilizer scheme on Plant Weight, I should probably go with the Two-Way Anova, but is there, from a statistical point of view, something wrong with the second way? Is it up to me to make that decision, depending on what I am interested in? In my case, I would like to do a post hoc test that allows for heteroscedasticity (for a data set were a weight function was not sufficient to correct it) but can only deal with 1 factor and not with interaction terms. Would it be correct to do a One-Way Anova and compare each Treatment Group to each other (which would give me exactly what I want to know)?

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Both methods are OK. If you wonder whether the four group variances are equal, an advantage of the one-way approach is that, in R, you could use one-way test, which does not assume equal variances for the levels of the factor. Then, as you say, you can do additional tests if you reject the null hypothesis that means for all levels are equal. You might plan three Welch t tests in advance that would give answers about irrigation levels, fertilizer levels, and interaction. Because these comparisons are part of the design, you should not worry about false discovery if these tests are significant.

An advantage of the two-way design is that comparisons of irrigation levels, fertilizer levels, and and a test for interaction will appear in the original ANOVA table. If there is significant interaction, you will need to be careful to consider whether straightforward irrigation and fertilizer comparisons are meaningful. As far as I know, there is no implementation in R of a two-way ANOVA that allows for unequal variances.

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    $\begingroup$ I have been using the White.adjust=T argument for Two-Way Anovas before and been advised by others that this can be used (it was also discussed here in one long post on heteroscedasticity). It computes coefficient covariances by hccm(). If this is not okay it would be great for me to know. R did not show any warning or error when I used it $\endgroup$
    – Deschain
    Sep 9 '20 at 8:37
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    $\begingroup$ White.adjust=T uses a heteroscedasticity-corrected coefficient covariance matrix fitting the model. This is generally a good idea when you have heteroscedasticity present. $\endgroup$ Sep 9 '20 at 8:50

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