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I am planning a study and am not sure which statistical test I should use. In this study, I have multiple independent variables. Some of these variables are categorical, while others are numerical. I am especially interested in the effects of only one of these variables, which is categorical. I also have three dependent variables, all of which are numerical.

I want to use a test that will allow me to account for the effects of all of my independent variables except for the one I'm interested in so that I can tell whether that one variable has a statistically significant effect. How should I go about doing so?

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Not quite enough information to go on.

But a glm (generalized linear model) should work. If the outcome is continuous then linear regression may be an appropriate glm, if binary then logistic regression. Just include all the variables and the p-value for each variable is adjusted for the effects of the other variables.

For categorical variables many software packages output the p-value for each level of the category rather than the variable as a whole, and so sometimes an extra command is needed depending on software of choice.

Some of this is addressed: Significance of categorical predictor in logistic regression

As well:
How to test the statistical significance for categorical variable in linear regression?
How to interpret insignificant categorical variables for logistic regression

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