cubic models as they allow for a more flexible estimation of the curve while incurring only slightly more complexity in the estimated functional form. The fractional polynomial model was of the form y = α + β1xa + β2xb + ε, where a and b are chosen from the set of fractional powers defined as: (−3, −2.5, −2, −1.5, −1, −.5, 0, .5, 1, 1.5, 2, 2.5, 3) and α, β1, and β2 are regression parameters. All parameters were chosen simultaneously to maximize explained variance in the outcome. The shape of the prediction equation from the fractional polynomial regression models differed little whether the models were fitted on the raw or the aggregated data.
