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I expect predictor A to negatively predict my dependent variable, and predictor B to positively predict the dependent variable. Can I include both predictors in a (linear) multiple regression model even though the variables are associated to the dependend variable in different "directions"?

Thank you in advance and apologies for my English.

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Yes. If these relationships with the dependent variable are true, then your coefficient for A will just be negative and the coefficient for B will be positive. You can use the regression to understand the magnitude of the effect A and B have on the dependent variable.

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The only real question here is if A & B are closely correlated to each other (positively or negatively matters not). If they are, then you probably only need one of them.

As long as A & B are relatively independent of each other (i.e. not closely correlated), then yes they will both do well in your multiple regression model.

Including two correlated dependent variables in a multiple regression model won't generally impact its ability to predict outcomes, but it can make interpretation of the correlation coefficients quite difficult.

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