# Why do means&error bars in an ANOVA graph depend on the factors and covariates that define it?

I have a mixed-model design with between-subjects factors Group (treatment, control) and Difficulty (easy, hard), and within-subjects factor Timepoint (pre, post). I want to use my statistical package (Statistica) to plot the dependent variable (memory test score) with separate categories for easy&hard, separate lines for the 2 groups, and repeat such a graph for each time point.

However, I was surprised to see that the means & error bars in the graphs produced by the ANOVA window depend on how the ANOVA is defined: 1) if I do each between-subjects ANOVA separately (once for each level of the within factor, i.e. once for pre and once for post), the means & error bars showing the interaction in each ANOVA are different from the graph showing the 3-way interaction (browen down for pre and post) in the mixed-model 3-way ANOVA 2) if I take IQ as a covariate, I get means & error bars that are again different to the no-covariate case 3) if I enter the covariate into the model crossed with the other factors (e.g. if I look for a "Group X Difficulty X IQ" interaction), I get means & error bars that are yet-again different as compared to when the covariate is added to the model separately, i.e. uncrossed with any other factor.

Why is that? Shouldn't the mean and error bars simply be calculated as the average and SEM (or whatver the error bars represent) respectively? And shouldn't those two statistics be independent of what factors are included in the model (ANOVA) and how they are included (see points 1-3 above)?

If it is in fact legitimate that these things influence the means and error bars, then which graph should I believe? In other words, how should I plot my two between-subjects (Group x Difficulty) interactions: from within the large 3-way (Group x Difficulty x Timepoint) ANOVA or from within each of the 2-way ANOVAs (Group x Difficulty) done separately for pre and for post?

Thanks and sorry for the long question! Please let me know if I need to make it more clear.

• Would very much appreciate getting some expert opinions on this question, thanks a lot in advance! Jun 28, 2014 at 16:29