Multiple regression results help

Question

For my first ever research paper I've run a hierarchal multiple linear regression with two predictors and one outcome variable, however I don't understand my results. I've found predictor A to be a significant predictor for my outcome variable alone. However, when both my predictors are in the model, predictor A is not a significant predictor, only predictor B is. How can this be if predictor A was significant in the first model? How does predictor B change how significant predictor A is?

Thank you!

Answer 1

regression coefficients reflect the simultaneous effects of multiple predictors. If the two predictors are inter-dependent (i.e. correlated) the results can differ from single input models.

Answer 2

The tests in multiple regression are "added last" tests. That means they test whether the model significantly improves after including the extra variable in a regression that contains all other predictors.

In your model with no predictors, adding A improves the model, so the test of A is significant in the model with only A.

In a model with A already in the model, adding B improves the model, so the test of B is significant in the model with A and B. But in a model with B already in the model, adding A doesn't improve the model, so the test of A is not significant in the model with A and B. B is doing all the work that A would do, so adding A doesn't improve the model beyond B.

As @IrishStat mentioned, this can occur when A and B are correlated (positively or negatively) with each other. It's a fairly common occurrence in regression modeling. The conclusion you might draw is that A predicts the outcome when B is not in the model (i.e., unavailable), but after including B, A doesn't do much more to predict the outcome. Unfortunately, without more information about the causal structure of your variables, there is little more interpretation available.

Answer 3

To expand a little on @Noah and @IrishStat's answers, in a multiple regression, coefficients for each independent variable/predictor are estimated to obtain the direct effect of each variable, using variation unique to that variable and the variable's correlation with the outcome variable, not using variation shared by predictors. (In technical terms, we are talking about variance and covariance of these variables.) The less unique variation there is, the less significant the estimate will become.

So why, in your example, did you end up with an insignificant predictor A when B was added, and not with a significant predictor A and insignificant predictor B? It is likely because the proportion of variance of predictor A that it has in common with predictor B is larger than the proportion of variance of predictor B that it has in common with predictor A.