Applied Regression Analysis

The file “BODY_FAT.TXT” contains data adapted from a dataset posted by Roger W. Johnson (Dept. of Mathematics & Computer Science, South Dakota School of Mines & Technology). Johnson also provided the following references, that might be useful to you:

• Bailey, Covert (1994). Smart Exercise: Burning Fat, Getting Fit, Houghton-Mifflin Co., Boston, pp. 179-186.
• Behnke, A.R. and Wilmore, J.H. (1974). Evaluation and Regulation of Body Build and Composition, Prentice-Hall, Englewood Cliffs, N.J.
• Siri, W.E. (1956), "Gross composition of the body", in Advances in Biological and Medical Physics, vol. IV, edited by J.H. Lawrence and C.A. Tobias, Academic Press, Inc., New York.
• Katch, Frank and McArdle, William (1977). Nutrition, Weight Control, and Exercise, Houghton Mifflin Co., Boston.
• Wilmore, Jack (1976). Athletic Training and Physical Fitness: Physiological Principles of the Conditioning Process, Allyn and Bacon, Inc., Boston.

The dataset is about a sample of 252 men and contains the following variables:
• Density: density of the body, determined from underwater weighing
• SiriBFperc: percentage of body fat, calculated as a function of the Density according to Siri’s equation: (495/Density) – 450.
• Over45: indicator for age group (0: up to 45 years, 1: over 45)
• Weight: weight (lbs)
• Height: height (inches)
• NeckC: neck circumference (cm)
• ChestC: chest circumference (cm)
• AbdomenC: abdomen circumference (cm)
• HipC: hip circumference (cm)
• ThighC: thigh circumference (cm)
• KneeC: knee circumference (cm)
• AnkleC: ankle circumference (cm)
• BicepsC: biceps circumference (cm)
• ForearmC: forearm circumference (cm)
• WristC: wrist circumference (cm)

The most accurate way of calculating the percentage of body fat is the one provided by Siri’s equation (495/Density) – 450, which requires a measurement of Density via weighting under water. This measurement is expensive and unpractical. On the other hand, age and all the body measurements listed above are easy to obtain. Thus, we want to understand if we can reliably describe and predict the percentage of body fat using these other variables, through a regression model.

In the dataset, for age we only have a binary indicator separating men below and above 45 years. On the other hand, all the body measurements are continuous variables.

Notice also that body fat percentage is a quantity bound to vary between 0 and 100. This could cause problems when using linear regression models, because we could actually estimate mean levels or predict values smaller than 0 or larger than 100 on certain predictors ranges. However we can safely use linear regression as far as we move in ranges of the predictors where the fitted values are well above 0 and well below 100.

Throughout the semester, the “Application” component in each of the homework sets will consist of employing various modeling, inference and diagnostics techniques learned in class on these data – with the final aim of producing a satisfactory regression model for the percentage of body fat based on all or on a subsample of the available predictor variables.

NOTE: DENSITY (THE VARIABLE ON WHICH SIRI’S EQUATION IS BASED) WILL NOT BE USE AS A PREDICTOR.

When preparing the “Application” part of each homework, make sure that:
• It does not exceed 5 pages, including figures and tables.
• The answer to each question is divided in two parts, one devoted to technical details and outputs, and one devoted to interpretation of the results. The former can contain R output (only the relevant part of it!) and technical answers to the question. The latter should resemble a short report you would write for a client, i.e. be concise and informative but not contain technical terms, and not assume the reader has statistical knowledge.

Keep in mind that some erroneous values were detected in this data set
• Density is given to you so that you can verify whether there were mistakes in the calculation of Percentage of body fat through Siri’s formula.
• There may be some obvious measurement errors in Height, and in other predictor variables.

When performing the analyses required for each homework set, you can remove, or present results with and without, a few units (men) that appear to carry erroneous measurements. Never remove more than 10 units, and always provide an appropriate justification for removing units if you do (report which units you removed).

Remember that a regression model should be satisfactory in three aspects:
• variability explanation;
• diagnostics on the model assumptions;
• parsimony and interpretability.

Applied Regression Analysis The file “BODY_FAT.TXT” contains dat...