2. Through Thick and Thin By:Mark Bergman Thomas Bursey Jay LaPorte Paul Miller Aaron Sinz

3. Measurement Methods 1. Hydrostatic (Underwater) Weighing 2. Skin Fold Measurements 3. Ultrasound Measurements

4. Through Thick and Thin (Statistical Model) I. Introduction II. Siri’s Equation and Data III. Elements of Regression Analysis IV. Regression Analysis of Body Fat Data V. Demonstrations VI. Conclusion

5. Body Density Body Density = WA/[(WA-WW)/c.f. - LV] WA = Weight in air (kg) WW = Weight in water (kg) c.f. = Water correction factor (=1 at 39.2 deg F as one-gram of water occupies exactly one cm^3 at this temperature, =.997 at 76-78 deg F) LV = Residual Lung Volume (liters)

6. Proportion of Fat Tissue D = Body Density (gm/cm^3) A = Proportion of lean body tissue B = Proportion of fat tissue(A + B =1) a = Density of lean body tissue (gm/cm^3) b = Density of fat tissue (gm/cm^3)

7. Proportion of Fat Tissue D = 1/[(A/a) + (B/b)] B = (1/D)*[a*b/(a-b)]-[b/(a-b)]

8. Estimates a =1.10 gm/cm^3 and b =0.90 gm/cm^3 Percentage of Body Fat = 495 /D - 450 Siri’s Equation

9. Elements of Regression Analysis Simple Regression Simple Regression Multiple Regression Multiple Regression

10. Elements of Regression Analysis Regression Assumptions 1. The population satisfies the equation 2. The true residuals are mutually independent 3. The true residuals all have the same variance 4. The true residuals all have a normal distribution with mean zero

11. Elements of Regression Analysis Sum of Squares Sum of Squares Mean of Squares Mean of Squares Coefficient of Determination Coefficient of Determination

12. Elements of Regression Analysis F-Ratio F-Ratio T-Ratio T-Ratio

13. The Best Predictor For Simple Regression Using Excel Simple Regression Abdomen Circumference

15. The Worst Predictor For Simple Regression Using Excel Ankle Circumference Simple Regression

17. Single Predictors from Best to worst 1. Abdomen Circumference (R^2 =.6617) 2. Chest Circumference (R^2 =.4937) 3. Hip Circumference (R^2 =.3909) 4. Weight (R^2 =.3751) 5. Thigh Circumference (R^2 =.3132) 6. Knee Circumference (R^2 =.2587) 7. Biceps (extended) Circumference (R^2 =.2433) 8. Neck Circumference (R^2 =.2407) 9. Forearm Circumference (R^2 =.1306) 10. Wrist Circumference (R^2 =.1201) 11. Age (R^2 =.0849) 12. Height (R^2 =.0800) 13. Ankle Circumference (R^2 =.0707)

18. Best Single Predictor Equation And The Average Percent Difference From The Given Data y =.6313(abdomen) – 39.28 Average Difference = 3.9163

19. Multiple Regression Using SPSS

20. Multiple Regression And The Affects of Removing a Predictor 1.All Predictors (R^2 =.749) 2. Abdomen Circumference (R^2 =.620) 3. Chest Circumference (R^2 =.749) 4. Hip Circumference (R^2 =.749) 5. Weight (R^2 =.746) 6. Thigh Circumference (R^2 =.746) 7. Knee Circumference (R^2 =.749) 8. Biceps (extended) Circumference (R^2 =.748) 9. Neck Circumference (R^2 =.745) 10. Forearm Circumference (R^2 =.744) 11. Wrist Circumference (R^2 =.739) 12. Age (R^2 =.745) 13. Height (R^2 =.748) 14. Ankle Circumference (R^2 =.748)

22. The Best Predictors Using The Percent Of Significance 1. Abdomen Circumference (Sig. =.000) 2. Wrist Circumference (Sig. =.003) 3. Forearm Circumference (Sig. =.024) 4. Neck Circumference (Sig. =.044) 5. Age (Sig. =.056) 6. Weight (Sig. =.100) 7. Thigh Circumference (Sig. =.103) 8. Hip Circumference (Sig. =.156) 9. Biceps (extended) Circumference (Sig. =.290) 10. Ankle Circumference (Sig. =.433) 11. Height (Sig. =.469) 12. Chest Circumference (Sig. =.810) 13. Knee Circumference (Sig. =.950)

23. The Best Three Predictor Models For Multiple Regression Top Three: 1.Abdomen Circumference, Wrist Circumference, Weight (R^2 =.728) 2.Weight, Abdomen Circumference, Neck Circumference (R^2 =.724) 3.Abdomen Circumference, Weight, Height (R^2 =.721)

24. Best Multiple Predictor Equation And The Average Percent Difference From The Given Data Average Difference = 3.58 body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.930

25. Body Fat Demonstration Using the best model from our Regression Analysis body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.93

26. The Best 3 Predictors are the Abdomen Weight Wrist Measuring the Predictors Abdomen and Wrist are measured in Centimeters (cm) Weight is measured in pounds

27. Measuring the Abdomen Make sure that the heels are together before applying the tapeline. Then measure approximately 3” below the waistline. Measure the abdomen circumference (cm).

28. Measuring the Weight Weight should be taken with an accurate weighing scale. Record the persons weight in pounds.

29. Measuring the Wrist Measurement should be taken between hand and wrist bone. Measure the wrist circumference (cm).

30. Calculating the Body Fat % Body fat = A (.975) – W (.114) – P(1.245) – 27.93 A = abdomen circumference (cm) P = wrist circumference (cm) W = weight (lbs)

31. What Does This Mean ? The normal range for men is 15-18% AgeExcellentGoodFair Poor 19-2410.8 %14.9 %19.0 % 23.3 % 25-2912.8 %16.5 %20.3 %24.4 % 30-3414.5 %18.0 %21.5 %25.2 % 35-3916.1 %19.4 %22.6 %26.1 % 40-4417.5 %20.5 %23.6 %26.9 % 45-4918.6 %21.5 %24.5 %27.6 % 50-5419.8 %22.7 %25.6 %28.7 % 55-5920.2 %23.2 %26.2 %29.3 % 60 +20.3 %23.5 %26.7 %29.8 %

32. References Dr. Steve Deckelman Dr. Steve Deckelman A Course in Mathematical Modeling A Course in Mathematical Modeling –By Douglas Mooney and Randall Swift http://lib.stat.cmu.edu/datasets/bodyfat http://lib.stat.cmu.edu/datasets/bodyfat