# Linear regression very significant βs with multiple variables, not significant alone

Could anyone provide intuition on why for y ~ β1x1 + β2x2 + β3x3, β1 β2 and β3 can be significant in a multiple variable model (p range 7x10-3 to 8x10-4), but the βs are not significant in separate univariate regressions (p range 0.02 to 0.3)?

My intuition is that it has something to do with correlations among the βs, but not quite clear how. Other info in my case:

• variance inflation factors are <1.5 in combined model
• cor(β1, β2) = -0.23, cor(β1, β3) = 0.02, cor(β2, β3) = 0.53
• n=171, so should be enough for 3 coefficients
• The change in estimates from single to multiple variable regression is: β1=-0.03→-0.04, β2=-0.02→-0.05, β3=0.05→0.18 (i.e., directionality does not change but magnitude does)

Thanks!

• Could you clarify what you mean by "alone"? Are you referring to separate univariate regressions? – whuber Nov 19 '18 at 17:04
• @whuber separate univariate regressions. Thank you for clarifying, I edited my post – Felix Richter Nov 19 '18 at 18:15

This looks like the phenomenon of the suppressor variable. Such a variable has low correlation with the outcome (which seems to be the case here for some of them since you quote p = 0.02 to 0.3 for the individual $$\beta$$) but does correlate with one or more of the other predictors (which also seems to be true from the correlations you quote). Including the suppressor variable in the model then accounts for some of the variance in the other predictor(s) which was uncorrelated with the outcome hence making them appear at a higher level of statistical significance. It has suppressed that variance, hence the name.