# How to explain the phenomenon that each coefficient is significant in multiple regression but not significant as simple regression [duplicate]

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?

• 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 Jul 30, 2021 at 19:59
• @abalter it seems still the problem of Multicollinearity? Could you open a question specific to this question? Aug 1, 2021 at 17:06
• 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. Aug 2, 2021 at 18:54
• @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? Aug 3, 2021 at 3:08
• 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/… Aug 3, 2021 at 4:11