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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.

I would like to seek your advice on above interpretations.

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!

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You cannot make any causal interpretation about your regressions from cross-sectional data in which you cannot identify the temporal order. Regressions in this scenario only provide partial correlations between variables, which have no causal interpretation, and, especially in this case, no meaningful interpretation broadly. I know this is not the answer you want to hear because you probably collected the data with an interesting question in mind, but without wild assumptions, you cannot make any valid conclusion that would have any substantive relevance (i.e., come close to causality) from your data.

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  • $\begingroup$ I am not trying to have conclusion on the causality. Rather, I would like to found some association traits of the person who are at risk. Therefore, I would like to know how to interpret the regression results (after changing the position of predictor and outcome) and whether it is legit. $\endgroup$ – William Wong May 19 '18 at 11:10
  • $\begingroup$ I think you'll agree that interpretation has a sizable subjective component and will change with greater familiarity with the project. Questions such as "What is the correct interpretation?" or "Is my interpretation legit" don't seem applicable (they don't have a single correct answer). Given your response to @Noah's answer, I suspect what would benefit you is having someone with whom to discuss your analysis. $\endgroup$ – rolando2 May 19 '18 at 12:42
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    $\begingroup$ You say you don't want a causal conclusion, but you have to invoke causal language to explain your results. Confounding, mediation, and collider are all concepts imbued with causality, and the interpretation of your results (no matter who is it aimed for) depends strongly on the status of Z. For example, if Z is a mediator, the interpretation of the decrease in X's effect means you are blocking the effect of X by Z's inclusion (post-treatment bias). X still has an effect but you masked it. (cont.) $\endgroup$ – Noah May 19 '18 at 23:27
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    $\begingroup$ If Z is a confounder, then the interpretation of the decrease in X's effect means that X really has no effect and the association originally observed was due to confounding. These are radically different interpretations but the only way to distinguish between them is by invoking causality, which requires temporal ordering. There is no way to meaningfully interpret your results that could provide any scientific explanation or policy relevance, which is why I urge you not to try to interpret them. $\endgroup$ – Noah May 19 '18 at 23:29
  • $\begingroup$ Thank you for your replies and comments. I agree that it is better for me to also gather some temporal data. But I am just curious about the phenomenon of changing the "position" of variables give somehow different interpretations regarding "associations". On the one hand, I got X is associated with Y independently with age and sex, but not Z; I also got Y is independently associated with X with all the included covariates. Do you think it is a sign that Z is "interesting" and it might be a mediator (though it is not guaranteed as not temporal data were collected)? $\endgroup$ – William Wong May 22 '18 at 17:21

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