Question 9: If we have only height in the regression model, height is a highly significant predictor. If we have both height and height squared, neither term is significant. Why is this? What can we do about it?

Height and height squared are likely to be highly correlated. In fact, r = 0.999. This means that a wide range of possible coefficients for height and height squared will predict head circumference almost as well and the confidence intervals for the coefficients are very wide.

We can subtract the mean height, 1701.5 mm, before we square height:

Predictor Coefficient         P     95% CI
intercept 395 <0.001 332 to 457
height 0.0969 <0.001 0.0566 to 0.137
height minus mean squared -0.000104 0.4         -0.000349 to 0.000142
sex 7.23 0.07 -0.70 to 15.12

Only the coefficient, P value and confidence interval for height and for the intercept have changed and height is now a significant predictor again, with estimate and confidence interval almost identical to those in the model without height squared. This was 0.0957 (95% CI 0.0555 to 0.1360, P<0.001).

Continue with Exercise: Head circumference, sex, and height.