MATH 204
Principles of Statistics 2

Make-up Midterm Project
For Sicheng Qian
The attached data set, titled Qian.csv, contains estimates of the percentage of body fat determined by
underwater weighing and various body circumference measurements for 225 men.

As accurate measurement of body fat is inconvenient/costly, it is desirable to have easy methods of
estimating body fat that are not inconvenient/costly. Multiple regression can be useful to this end.

The objective of your analysis is to use the variables listed on p. 2 to construct a model which has the
best possible predictive power while being interpretable and parsimonious.

Your report should describe your approach to model construction and validation including, e.g., the
selection of (possibly transformed) variables through stepwise regression methods, parameter
estimation, and validation of the model assumptions through suitable residual plots. Your conclusions
should also be stated clearly in terms of the original scientific question.

The report, which should be approximately 20 pages long in single-spaced, 12-point size format, should
be conceived as a continuous text intended for a general audience interested in the scientific question.
It can and should include appropriate graphics in support of the main points, but it should not include
code nor consist only of graphs and tables without explanations.

Look on the Web for general advice on statistical reporting before proceeding.

Because this project serves as a make-up midterm worth 50% of your final grade, you are expected to
devote a substantial amount of time and effort to its completion, and to do the work alone. A statement
to this effect must be included in the report.

All aspects of the report will be graded, i.e., both its contents (quality, accuracy, and relevance of the
statistical analysis) and its form (choice of material, language, form, etc.). I will grade the report myself
and the score will be final.

Good luck!


Christian Genest
www.math.mcgill.ca/cgenest/

List of variables available to you
Density determined from underwater weighing (g/cm3)
Percent body fat from Siri’s (1956) equation
Age (years)
Weight (pounds)
Height (inches)
Neck circumference (cm)
Chest circumference (cm)
“Abdomen 2” circumference (cm)
Hip circumference (cm)
Knee circumference (cm)
Ankle circumference (cm)
Biceps (extended) circumference (cm)
Forearm circumference (cm)

Additional Details

A variety of popular health books suggest that the readers assess their health, at least in part, by
estimating their percentage of body fat. In Bailey (1994), for instance, the reader can estimate body fat
from tables using their age and various skin-fold measurements obtained by using a caliper. Other texts
give predictive equations for body fat using body circumference measurements (e.g., abdominal
circumference) and/or skin-fold measurements. See, for instance, Behnke and Wilmore (1974), pp. 66-
67; Wilmore (1976), p. 247; or Katch and McArdle (1977), pp. 120-132.

The measurement standards are those listed in Benhke and Wilmore (1974), pp. 45-48 where, for
instance, the “Abdomen 2” circumference is measured “laterally, at the level of the iliac crests, and
anteriorly, at the umbilicus.”
Data such as those included in your data set were used to produce the predictive equations for lean
body weight given in the abstract “Generalized body composition prediction equation for men using
simple measurement techniques,” K.W. Penrose, A.G. Nelson, A.G. Fisher, FACSM, Human Performance
Research Center, Brigham Young University, Provo, Utah 84602 as listed in Medicine and Science in
Sports and Exercise, vol. 17, no. 2, April 1985, p. 189.

Percentage of body fat for an individual can be estimated once body density has been determined.
Folks, e.g., Siri (1956), assume that the body consists of two components — lean body tissue and fat
tissue. Letting

D = Body Density (gm/cm3),

A = proportion of lean body tissue,

B = proportion of fat tissue (A + B = 1),

a = density of lean body tissue (gm/cm3),

b = density of fat tissue (gm/cm3),

we have

D = 1/[(A/a) + (B/b)].

Solving for B we find

B = (1/D) ´ [ab/(a - b)] - [b/(a - b)].

Using the estimates a = 1.10 gm/cm3 and b = 0.90 gm/cm3 (see Katch and McArdle (1977), p. 111 or
Wilmore (1976), p. 123) we come up with “Siri’s equation,” viz.

Percentage of Body Fat (i.e., 100 ´ B) = 495/D - 450.

Volume, and hence body density, can be accurately measured a variety of ways. The technique of
underwater weighing “computes body volume as the difference between body weight measured in air
and weight measured during water submersion. In other words, body volume is equal to the loss of
weight in water with the appropriate temperature correction for the water’s density” (Katch and
McArdle (1977), p. 113). Using this technique,

Body Density = WA/[(WA - WW)/cf - LV],

where

WA = Weight in air (kg)

WW = Weight in water (kg)

cf = Water correction factor (= 1 at 39.2 degrees F as one-gram of water occupies exactly one cm3 at
this temperature, = .997 at 76-78 degrees F)

LV = Residual Lung Volume (liters).

See Katch and McArdle (1977), p. 115. Other methods of determining body volume are given in Behnke
and Wilmore (1974), p. 22 ff.
References

Bailey, C. (1994). Smart Exercise: Burning Fat, Getting Fit. Houghton-Mifflin Co., Boston, MA, pp. 179-
186.

Behnke, A.R. and Wilmore, J.H. (1974). Evaluation and Regulation of Body Build and Composition.
Prentice-Hall, Englewood Cliffs, NJ.

Katch, F. and McArdle, W. (1977). Nutrition, Weight Control, and Exercise. Houghton Mifflin Co., Boston,
MA.

Siri, W.E. (1956). “Gross composition of the body,” in Advances in Biological and Medical Physics, vol. 4,
edited by J.H. Lawrence and C.A. Tobias, Academic Press, Inc., New York.

Wilmore, J. (1976). Athletic Training and Physical Fitness: Physiological Principles of the Conditioning
Process. Allyn and Bacon, Inc., Boston, MA.