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)