# What can be said about about significant predictors in simple regression that become insignificant in multiple linear regression?

I have two predictor variables: An indicator variable A and a continuous variable B. My response variable is continuous (and also bounded, have not made it logit for reasons of simplicity).

In simple linear regression, both predictors are significant. When including the two in multiple regression, both become insignificant in an overall significant model.

Other details:

• An interaction variable composed of the two variables is insignificant
• Beta for quantitative variable B is quite different between the two groups of the indicator variable A if I do a stratified analysis with a regression with B (That is, separate the data into the 0 and 1 groups for indicator A, then run regression using B as predictor).
• VIF's are low for the two variables, suggesting no multicolinearity.

Crucial to the theory my study is about is the possibility that the variance explained by B is in large part explained by variable A. It doesn't seem like I am able to justifiably make any such claim.

But what can I say in regards to the impact of A and B on the response and their relationship with one another?

I am happy to provide more details if needed.

Here are the intervals -- FAB.Average is quantitative predictor, the HSSorSTEM is the indicator. My n = 30, but each datapoint is the average of 500+ observations, if that makes any difference.

I'm not quite sure what a joint confidence ellipse is (I work with a knowledgable statistician, but am not myself one), but here was my try at making one: • Can your plot the joint confidence ellipse for the two coefficients? Jul 15, 2016 at 6:50
• @DimitriyV.Masterov -- I tried to fulfill your request as best I can. Jul 15, 2016 at 8:31
• This certainly looks like @Björn's second story is plausible (see here on why). Based on your other added info, I might consider a model where the you have both A, B, and their interaction (if that's not what you are doing already). I would also encourage you to use analytic weights that are inversely proportional to the variance of an observation. This can be helpful if observations represent averages and the weights are the number of elements that gave rise to the average. Jul 15, 2016 at 16:47
• I think if you plot y versus B with the different marker symbols for A where the markers depend on size of the group, you should cleanly see if your story makes sense. Jul 15, 2016 at 16:48