How do you determine if two variables are independent?
Instruction: Explain methods to assess the independence of two variables in a dataset.
Updated
Example Answer
Two variables are independent when their joint distribution factors into their marginal distributions. The way I investigate that depends on the data. For categorical variables, a contingency table and a chi-square independence test may be appropriate, provided the expected counts and sampling design support it. With sparse counts, I would consider an exact test instead.
For numeric variables, a scatterplot and correlation are useful first checks, but zero Pearson correlation does not establish independence. If X is uniform from minus one to one and Y equals X squared, their correlation is zero, yet knowing X determines Y. I would investigate plausible nonlinear relationships too. Finally, failing to reject an independence test means the available data did not establish dependence at that threshold; it is not proof that no dependence exists.
Why this works
The definition, diagnostic choice, and counterexample support different parts of the answer. It avoids treating a low-powered test as confirmation.
Follow-up to prepare for
How would repeated observations from the same person affect an independence test?