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As a part of a research project, I have collected a set of data (2 questionnaires). For each participant I calculate 8 variable (4 by questionnaire). I want to prove that their is a relationship between the first four variables and the second four variables. What should I use? Correlation (Pearson) or Regression or another approach?

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If this is your goal then neither regression nor correlation is right, what you want is probably canonical correlation. . But I am not sure that I am interpreting your question correctly.

Simple and multiple regression involve a single dependent variable and one or more independent variables.

Multivariate regression involves multiple related dependent variables and one or more independent variables.

Correlation is a measure of the relationship between two variables, not labeled dependent or independent.

There are variations of each of these to deal with specific situations.

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