Through Thick and Thin

By: Mark Bergman

Thomas Bursey

Jay LaPorte

Paul Miller

Aaron Sinz

Measurement MethodsMeasurement Methods

1.1. Hydrostatic (Underwater) Hydrostatic (Underwater) WeighingWeighing

2.2. Skin Fold MeasurementsSkin Fold Measurements

3.3. Ultrasound MeasurementsUltrasound Measurements

Through Thick and ThinThrough Thick and Thin(Statistical Model)(Statistical Model)

I.I. IntroductionIntroduction

II.II. Siri’s Equation and DataSiri’s Equation and Data

III.III. Elements of Regression Elements of Regression AnalysisAnalysis

IV.IV. Regression Analysis of Body Regression Analysis of Body Fat DataFat Data

V.V. DemonstrationsDemonstrations

VI.VI. ConclusionConclusion

Body DensityBody 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)

Proportion of Fat TissueProportion of Fat Tissue

D = Body Density (gm/cm^3)D = Body Density (gm/cm^3)

A = Proportion of lean body tissueA = Proportion of lean body tissue

B = Proportion of fat tissue(A + B =1)B = Proportion of fat tissue(A + B =1)

a = Density of lean body tissue (gm/cm^3)a = Density of lean body tissue (gm/cm^3)

b = Density of fat tissue (gm/cm^3)b = Density of fat tissue (gm/cm^3)

Proportion of Fat TissueProportion of Fat Tissue

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

B = (1/D)*[a*b/(a-b)]-[b/(a-b)]B = (1/D)*[a*b/(a-b)]-[b/(a-b)]

Estimates Estimates a =1.10 gm/cm^3 and a =1.10 gm/cm^3 and

b =0.90 gm/cm^3b =0.90 gm/cm^3

Percentage of Body Fat = 495 /D - 450Percentage of Body Fat = 495 /D - 450

Siri’s EquationSiri’s Equation

Elements of Regression AnalysisElements of Regression Analysis

Simple RegressionSimple Regression

Multiple RegressionMultiple Regression

xbby 10

nnxbxbby ....110

Elements of Regression AnalysisElements of Regression AnalysisRegression AssumptionsRegression Assumptions

1.1. The population satisfies the equationThe population satisfies the equation

2.2. The true residuals are mutually independentThe true residuals are mutually independent

3.3. The true residuals all have the same varianceThe true residuals all have the same variance

4.4. The true residuals all have a normal distribution The true residuals all have a normal distribution with mean zerowith mean zero

xBBy 10

Elements of Regression AnalysisElements of Regression Analysis

Sum of SquaresSum of Squares

Mean of SquaresMean of Squares

Coefficient of DeterminationCoefficient of Determination

totalrestotal SSSSSSR /)(2

resresres

regregreg

dfSSMS

dfSSMS

/

/

2)( yySS itotal

Elements of Regression AnalysisElements of Regression Analysis

F-RatioF-Ratio

T-RatioT-Ratio

ii seBt /ˆ

resreg MSMSF /

The Best Predictor For Simple Regression Using Excel

Simple Regression

Abdomen Circumference

Abdomeny = 0.6313x - 39.28

R2 = 0.6617

0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Percent Body Fat

Ab

dom

en C

ircu

mfe

ren

ce

Series1

Linear (Series1)

The Worst Predictor For Simple Regression Using Excel

Ankle Circumference

Simple Regression

Ankle Cirumferencey = 1.3133x - 11.189

R2 = 0.0707

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40

Ankle Cirumference

Per

cen

t Bo

dy F

at

Series1

Linear (Series1)

Single Predictors from Best to worst1. 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)

Best Single Predictor Equation And The Average Percent Difference From

The Given Data

y = .6313(abdomen) – 39.28

Average Difference = 3.9163

Multiple Regression Using SPSS

Model Summary

.866a .749 .736 4.307Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), WRIST, AGE, HEIGHT, ANKLE,FOREARM, ABDOMEN, BICEPS, KNEE, NECK, THIGH,CHEST, HIP, WEIGHT

a.

Coefficientsa

-17.775 17.361 -1.024 .307

5.840E-02 .033 .088 1.791 .075

-9.01E-02 .054 -.316 -1.682 .094

-7.20E-02 .096 -.032 -.750 .454

-.467 .233 -.136 -2.008 .046

-2.61E-02 .099 -.026 -.263 .793

.961 .087 1.239 11.078 .000

-.215 .146 -.184 -1.471 .142

.237 .144 .149 1.643 .102

2.610E-02 .242 .008 .108 .914

.170 .222 .034 .767 .444

.191 .172 .069 1.116 .266

.444 .199 .107 2.227 .027

-1.620 .535 -.180 -3.027 .003

(Constant)

AGE

WEIGHT

HEIGHT

NECK

CHEST

ABDOMEN

HIP

THIGH

KNEE

ANKLE

BICEPS

FOREARM

WRIST

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: BODYFATa.

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)

ALL 1 0.749AGE 2 0.745WEIGHT 3 0.746HEIGHT 4 0.748NECK 5 0.745CHEST 6 0.749ABDOMEN 7 0.620HIP 8 0.749THIGH 9 0.746KNEE 10 0.749ANKLE 11 0.748BICEPS 12 0.748FORARM 13 0.744WRIST 14 0.739

Removing Predictors

0.0000.2000.400

0.6000.800

1 2 3 4 5 6 7 8 9 10 11 12 13 14

The Best Predictors Using The Percent Of Significance1. 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)

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)

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

Body Fat DemonstrationBody Fat Demonstration

Using the best model from our Regression Using the best model from our Regression AnalysisAnalysis

body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.93

The Best 3 Predictors are the

• Abdomen

• Weight

• Wrist

Measuring the PredictorsMeasuring the Predictors

Abdomen and Wrist are measured in Centimeters (cm)

Weight is measured in pounds

Measuring the AbdomenMeasuring the AbdomenMake sure that the heels are together before applying the tapeline.

Then measure approximately 3” below the waistline.

Measure the abdomen circumference (cm).

Measuring the WeightMeasuring the WeightWeight should be taken with an accurate weighing scale.

Record the persons weight in pounds.

Measuring the WristMeasuring the Wrist

Measurement should be taken between hand and wrist bone.

Measure the wrist circumference (cm).

Calculating the Body Fat %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)

What Does This Mean ?What Does This Mean ?The normal range for men is 15-18%

Age Excellent Good Fair Poor19-24 10.8 % 14.9 % 19.0 % 23.3 % 25-29 12.8 % 16.5 % 20.3 % 24.4 %30-34 14.5 % 18.0 % 21.5 % 25.2 %35-39 16.1 % 19.4 % 22.6 % 26.1 %40-44 17.5 % 20.5 % 23.6 % 26.9 %45-49 18.6 % 21.5 % 24.5 % 27.6 %50-54 19.8 % 22.7 % 25.6 % 28.7 %55-59 20.2 % 23.2 % 26.2 % 29.3 %60 + 20.3 % 23.5 % 26.7 % 29.8 %

ReferencesReferences Dr. Steve DeckelmanDr. Steve Deckelman A Course in Mathematical ModelingA Course in Mathematical Modeling

– By Douglas Mooney and Randall By Douglas Mooney and Randall SwiftSwift

http://lib.stat.cmu.edu/datasets/bodyfathttp://lib.stat.cmu.edu/datasets/bodyfat