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I have found a group difference (3 groups) for my dependent variable and now I am supposed to find out if possible covariates influence this finding.

What is the best way to "correct" for 5 different covariates (age, duration of illness, medication, symptom score, gender)?

Shall I run 5 separate ANCOVAs (1 for each covariate) and see if the group difference remains significant or should I better run one single ANCOVA including all covariates?

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Welcome to the site

This is a question about model selection (see the tag with that label) but not completely.

The answer to your question depends on what you want to find out; it also depends on some things about the data.

If you add one variable at a time, you get to see how each variable affects the group differences, but you are then not controlling for other variables. If you run one model with all the variables, you have controlled for them all. However, when you have multiple variables you may run into a) Over fitting (if you don't have a large sample) b) Collinearity (if independent variables are strongly related to each other). In your case, I'd guess there might be colinearity among duration, medication and symptom variables.

However, if those issues are not problems, my inclination is to include all the variables in one model.

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Simpson's paradox would apply to any model that omitted an important covariate.

Simpson's paradox on wikipedia. Pay particular attention to the diagram.

Actually, here's a similar one I made:

enter image description here

The blue line is the effect you might measure for the relationship between y and x, if you ignore the grouping variable indicated by the red-and-black colours. The two upsloping lines are the effect you get if you include the grouping variable. You can see that the blue line generates entirely the wrong impression. This problem could occur if you leave out any important covariate.

There are also many related answers here that you can find with a search

As a result, one-covariate-at-a-time ANCOVA will be unsuitable; each of them may fail to do what you include covariates for.

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