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We know that in linear regression, when each coefficient is not significant in multiple regression but significant as a simple regression, it is most likely the reason of Multicollinearity. However how about the inverse case:

each coefficient is significant in multiple regression but not significant as simple regression?

I am not sure if it is possible. If possible, do you know the any of the reason?

I may have a misunderstanding that I always think as significant F test in whole and non significant T test in each coefficient is the unique flag of multicollinearity. So actually above phenomenon(significant in combination and non significant in single) is also a flag, right?

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  • $\begingroup$ Please read up on Simpson's Paradox. Search for that in the search box and you will find tons of great info. Here is a short answer I gave to one question: stats.stackexchange.com/a/478499/141304 $\endgroup$
    – abalter
    Commented Jul 30, 2021 at 19:59
  • $\begingroup$ @abalter it seems still the problem of Multicollinearity? Could you open a question specific to this question? $\endgroup$ Commented Aug 1, 2021 at 17:06
  • $\begingroup$ I don't understand what you mean by "open a question specific to this question." Multicollinearity becomes a problem as you add more variables (multiple regression), when those variables are connected. You are talking about seeing a 0 slope with a single variable but significant slopes as you add covariates. This sort of behavior is to be expected when you add covariates. It is also easily demonstrated in Simpson's Paradox. $\endgroup$
    – abalter
    Commented Aug 2, 2021 at 18:54
  • $\begingroup$ @abalter I know where is my confusion now. Since I always think as significant F test and non significant T test is the unique flag of multicollinearity. So actually above phenomenon is also a flag, right? $\endgroup$ Commented Aug 3, 2021 at 3:08
  • $\begingroup$ Nope. You are way off. The T-test is just the F-test for when you only have one covariate---simple regression. You are confused about what colinearity is. The way to test for collinearity is using the VIF, and this has nothing to do with confounding. I strongly recommend you learn what confounding is, and do as I suggested and read about Simpson's Paradox as it is the simplest way to demonstrate confounding and the power of covariates. stats.stackexchange.com/questions/538773/simpsons-paradox, stats.stackexchange.com/questions/19525/… $\endgroup$
    – abalter
    Commented Aug 3, 2021 at 4:11

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