# Interpretation of regression results from cross-sectional study after changing the position of outcome, predictor and covariate

I have a set of cross-sectional data have variables: Y (the expected outcome), X (the desire predictor), age, sex and Z (some confounding factors for example).

Originally, I did a regression on Y ~ X + age + sex, and I have high confidence to conclude that X is associated with Y independent of age and sex (results can be provided if necessary).

However, when I introduce the Z into the model, such that: Y ~ X + age + sex + Z, the coefficient of X become insignificant, while Z is significant. I double check that Z is significant if I do Y ~ Z + age + sex. Therefore, I suspected that Z is a strong confounding factor (it might also be mediator and collider, but I cannot be sure due to I do not have longitudinal data) or, indeed, it is a better predictor for Y.

The problem get complicated when I try to do regression on different directions: X ~ Y + age + sex + Z and Z ~ Y + X + age + sex

Interestingly, all predictor variables are statistically significant in the above regressions. If I interpret them separately, it would be "Y (also age, sex and Z) is associated with X independently of other variables within the model" and "Y (also age, sex and X) is associated with Z independently of other variables within the model.

However, based on my previously regression: Y ~ X + age + sex + Z, X is not independently associate with Y when there is Z.

Thus, I would like to know how to correctly interpret the above observation (I could provide the R regression output if necessary). Please also advice on whether it is legit to do such regression by changing the position of variables.

Many thanks!