.
Accurate measurement of body fat can be expensive and time consuming. Good models to predict body fat accurately using standard measurements are still useful in many contexts. A study was conducted to predict body fat using 13 simple body measurements on 251 men. For each subject, the percentage of body fat as measured using an underwater weighing technique, age, weight, height, and 10 body circumference measurements were recorded. Further details on this study are available in [331, 354]. These data are available from the website for this book. The goal of this problem is to compare and contrast several multivariate smoothing methods applied to these data.
a.
Using a smoother of your own choosing, develop a back fitting algorithm to fit an additive model to these data as described in Section 12.1.1. Compare the results of the additive model with those from a multiple regression model.
b.
Use any available software to estimate models for these data, using five methods: (1) the standard multiple linear regression (MLR) model, (2) an additive model (AM), (3) projection pursuit regression (PPR), (4) the alternating conditional expectations (ACE) procedure, and (5) the additivity and variance stabilization (AVAS) approach.
i.
For MLR, AM, ACE, and AVAS, plot the
kth estimated coordinate smooth against the observed values of the
kth predictor for
k
= 1, . . . ,
13. In other words, graph the values of ˆ
sk(x
ik) versus
x
ik
for
i
= 1, . . . ,
251 as in. For PPR, imitate by plotting each component smooth against the projection coordinate. For all methods, include the observed data points in an appropriate way in each graph. Comment on any differences between the methods.
ii.
Carry out leave-one-out cross-validation analyses where the
ith cross-validated residual is computed as the difference between the
ith observed response and the
ith predicted response obtained when the model is fitted omitting the
ith data
point from the dataset. Use these results to compare the predictive performance of MLR, AM, and PPR using a cross-validated residual sum of squares similar to (11.16).