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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?

user6703592
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  • 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: https://stats.stackexchange.com/a/478499/141304 – abalter Jul 30 '21 at 19:59
  • @abalter it seems still the problem of Multicollinearity? Could you open a question specific to this question? – user6703592 Aug 01 '21 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. – abalter Aug 02 '21 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? – user6703592 Aug 03 '21 at 03:08
  • 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. https://stats.stackexchange.com/questions/538773/simpsons-paradox, https://stats.stackexchange.com/questions/19525/simpsons-paradox-or-confounding – abalter Aug 03 '21 at 04:11
  • @abalter sorry here “above phenomenon” I mean the Simpson’s paradox, it is just the reason of multicollinearity right? – user6703592 Aug 03 '21 at 04:17
  • In you question, "not sure if it is possible," what does "it" refer to?? It is possible for the F test to be significant while all the t tests are not significant (at the same alpha level) even when the explanatory variables are orthogonal. There are subtle (but essentially trivial) reasons for simple regression p-values to differ from the multiple regression p-values: namely, the error degrees of freedom will differ, so a different Student t distribution will be used to compute the p-values. – whuber Aug 03 '21 at 15:28
  • @whuber I mean `F test to be significant while all the t tests are not significant` is a possible signal of Multicollinearity but not always true. And here `each coefficient is significant in multiple regression but not significant as simple regression` is most likely the result of Multicollinearity, right? – user6703592 Aug 03 '21 at 16:20

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