Through Thick and Thin
By: Mark Bergman
Thomas Bursey
Jay LaPorte
Paul Miller
Aaron Sinz
Measurement Methods
1.
2.
3.
Hydrostatic (Underwater)
Weighing
Skin Fold Measurements
Ultrasound Measurements
Through Thick and Thin
(Statistical Model)
I.
II.
III.
IV.
V.
VI.
Introduction
Siri’s Equation and Data
Elements of Regression
Analysis
Regression Analysis of Body
Fat Data
Demonstrations
Conclusion
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)
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)
Proportion of Fat Tissue
D = 1/[(A/a) + (B/b)]
B = (1/D)*[a*b/(a-b)]-[b/(a-b)]
Siri’s Equation
Estimates a =1.10 gm/cm^3 and
b =0.90 gm/cm^3
Percentage of Body Fat = 495 /D - 450
Elements of Regression Analysis
Simple Regression
y = b0 + b1 x
Multiple Regression
y = b0 + b1 x1 + .... + bn xn
Elements of Regression Analysis
Regression Assumptions
1.
The population satisfies the equation
y = B0 + B1 x + ∈
2.
3.
4.
The true residuals are mutually independent
The true residuals all have the same variance
The true residuals all have a normal distribution
with mean zero
Elements of Regression Analysis
Sum of Squares
SStotal =
( yi − y )
2
Mean of Squares
MS reg = SS reg / df reg
MS res = SS res / df res
Coefficient of Determination
R 2 = ( SStotal − SS res ) / SStotal
Elements of Regression Analysis
F-Ratio
F = MS reg / MS res
T-Ratio
t = Bˆ i / sei
Simple Regression
The Best Predictor For Simple Regression Using
Excel
Abdomen Circumference
Abdomen
y = 0.6313x - 39.28
R2 = 0.6617
60
Abd o m e n Cir cu m fe r e n ce
50
40
Series1
30
Linear (Series1)
20
10
0
0
20
40
60
80
Percent Body Fat
100
120
140
160
Simple Regression
The Worst Predictor For Simple Regression Using Excel
Ankle Circumference
Ankle Cirumference
y = 1.3133x - 11.189
R2 = 0.0707
50
45
40
Percent Bo dy F at
35
30
Series1
25
Linear (Series1)
20
15
10
5
0
0
5
10
15
20
Ankle Cirumference
25
30
35
40
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)
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
Coefficients
Model
1
Unstandardized
Coefficients
B
Std. Error
-17.775
17.361
5.840E-02
.033
-9.01E-02
.054
-7.20E-02
.096
-.467
.233
-2.61E-02
.099
.961
.087
-.215
.146
.237
.144
2.610E-02
.242
.170
.222
.191
.172
.444
.199
-1.620
.535
(Constant)
AGE
WEIGHT
HEIGHT
NECK
CHEST
ABDOMEN
HIP
THIGH
KNEE
ANKLE
BICEPS
FOREARM
WRIST
a
Standardi
zed
Coefficien
ts
Beta
t
-1.024
1.791
-1.682
-.750
-2.008
-.263
11.078
-1.471
1.643
.108
.767
1.116
2.227
-3.027
.088
-.316
-.032
-.136
-.026
1.239
-.184
.149
.008
.034
.069
.107
-.180
Sig.
.307
.075
.094
.454
.046
.793
.000
.142
.102
.914
.444
.266
.027
.003
a. Dependent Variable: BODYFAT
Model Summary
Model
1
a.
R
.866a
Adjusted
R Square
R Square
.749
Predictors: (Constant), WRIST, AGE, HEIGHT, ANKLE,
FOREARM, ABDOMEN, BICEPS, KNEE, NECK, THIGH,
CHEST, HIP, WEIGHT
Std. Error of
the Estimate
.736
4.307
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
AGE
WEIGHT
HEIGHT
NECK
CHEST
ABDOMEN
HIP
THIGH
KNEE
ANKLE
BICEPS
FORARM
WRIST
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.749
0.745
0.746
0.748
0.745
0.749
0.620
0.749
0.746
0.749
0.748
0.748
0.744
0.739
Removing Predictors
0.800
0.600
0.400
0.200
0.000
1
2
3
4
5
6
7
8
9 10 11 12 13 14
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)
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
body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.930
Average Difference = 3.58
Body Fat Demonstration
Using the best model from our Regression
Analysis
body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.93
Measuring the Predictors
The Best 3 Predictors are the
• Abdomen
• Weight
• Wrist
Abdomen and Wrist are measured in
Centimeters (cm)
Weight is measured in pounds
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).
Measuring the Weight
Weight should be taken with an accurate
weighing scale.
Record the persons weight in pounds.
Measuring the Wrist
Measurement should be taken between hand
and wrist bone.
Measure the wrist circumference (cm).
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 ?
The normal range for men is 15-18%
Age
Excellent
Good
Fair
Poor
References
Dr. Steve Deckelman
A Course in Mathematical Modeling
– By Douglas Mooney and Randall
Swift
http://lib.stat.cmu.edu/datasets/bodyfat
!
"
#
$!
&
%
&
('' )
' (
&
%"
''
'
()
*
)
()
*
+
() *
) $)
$
(
&
&
() *
) , !&
()
(
*
'
('' )
&
'
#
)
'
&
&
.
)
)
#
&
.
&
)
+
)
x(n + 1) = Rx(n) − P
!
./ '
/#
0
12/
& '

→ x(n + 1) − x(n) 
→
rx ( n )
x(n + 1) − x(n) = rx(n) − p
x(n + 1) = (1 + r ) x(n) − P
R = 1+ r
x(n + 1) = Rx(n) − P
P
)
x(n + 1) = Rx(n) − P
x(0) = x(0)
x(1) = Rx(0) − P
x(2) = Rx(1) − P
= R[ Rx(0) − P ] − P
= R 2 x(0) − RP − P
= R x(0) − P ( R + 1)
2
)
x(2) = R 2 x(0) − P( R + 1)
x(3) = Rx(2) − P
= R[ R x(0) − P( R + 1)] − P
2
= R 3 x(0) − RP( R + 1) − P
= R x(0) − P[ R( R + 1) + 1]
3
= R x(0) − P( R + R + 1)
3
2
)
x(3) = R 3 x(0) − P( R 2 + R + 1)
x(4) = Rx(3) − P
= R[ R 3 x(0) − P( R 2 + R + 1)] − P
= R x(0) − RP( R + R + 1) − P
4
2
= R x(0) − P[ R( R + R + 1) + 1]
4
2
= R x(0) − P( R + R + R + 1)
4
3
2
)
,
'
x(1) = Rx(0) − P
x(2) = R x(0) − P( R + 1)
2
x(3) = R 3 x(0) − P( R 2 + R + 1)
x(4) = R x(0) − P( R + R + R + 1)
4
3
2
x(n) = R n x(0) − P( R n −1 + R n − 2 + ... + R + 1)
)
n −1
+R
n−2
+ ... + R + 1)
x(n) = R x(0) − P( R
n
n−2
+ ... + R + 1)
3)
S = (R
n −1
+R
'
S=R
n −1
+R
n−2
+ ... + R + 1
RS = R n + R n −1 + ... + R 2 + R
1 + RS = R n + ( R n −1 + ... + R 2 + R + 1)
1 + RS = R n + S
RS − S = R n − 1
S ( R − 1) = R n − 1
Rn −1
S=
,R ≠1
R −1
)
) 3)
-
x(n) = R n x(0) − P( R n −1 + R n − 2 + ... + R + 1)
!
n
−1
R
n
x(n) = R x(0) − P
,R ≠1
R −1
)
)
!)
'
!
)
!
Simulation 1
Balance =
$
yearly interest =
monthly payment = $
Months
0
1
2
3
4
5
10
20
30
40
41
$
$
$
$
$
$
$
$
$
$
$
3,000
18%
100
Balance
3,000.00
2,945.00
2,889.18
2,832.51
2,775.00
2,716.63
2,411.35
1,728.20
935.37
15.27
-
$
$
$
$
$
$
$
$
$
$
$
Interest
45.00
44.18
43.34
42.49
41.63
37.11
27.02
15.30
1.70
-
total payment
$ 4,015.27
total interest
$ 1,015.27
effective interest rate 25.29%
$
$
$
$
$
$
$
$
$
$
$
Payment
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
15.27
-
Simulation 2
Balance
$
yearly interest
monthly payment $
Months
0
1
2
3
4
5
10
20
30
40
50
60
61
62
$
$
$
$
$
$
$
$
$
$
$
$
$
$
4,000
18%
100
Balance
4,000.00
3,960.00
3,919.40
3,878.19
3,836.36
3,793.91
3,571.89
3,075.05
2,489.45
1,829.28
1,052.69
151.41
53.69
-
$
$
$
$
$
$
$
$
$
$
$
$
$
$
Interest
60.00
59.40
58.79
58.17
57.55
54.26
46.92
38.40
28.51
17.03
3.72
2.27
-
total payment $
6,153.69
total interest $
2,153.69
effective interest rate 35.00%
$
$
$
$
$
$
$
$
$
$
$
$
$
$
Payment
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
53.69
-
)' &
('
4
.
)
&
!
'
&
&
&
#
>#
%
+)
;:::
6=8
<
9:::
678
5
;::
+)
How do we pay off both cards?
3)
b1 , b 2 = the balances on both cards
where : b1 = 3000, b 2 = 5000
r1 , r2 = the interest rates on both cards
where : r1 = .18, r2 = .12
F = the available funds in your budget
where : F = 300
P = interest charged next month
P = r1 ( b1 − x ) + r 2 ( b2 − y )
4
#
Question:
What is the most effective way to pay of the two
credit card balances?
Answer:
Pay the card with the highest interest rate.
!
'
&
&
&
#
>#
%
+)
6
6
<
7
7
5
+
+)
)'
5 )
'
"
!
)
)
"
!
) #
3)
y=F − x
f ( x) = r1 ( b1 − x) + r 2 ( b2 − ( F − x))
f ( x ) = r 1 b1 − r 1 x + r 2 b 2 − r 2 F + r 2 x
f ( x) = ( r 2 − r1 ) x + r1 b1 + r 2 ( b2 − F )
+
) #
f ( x) = ( r 2 − r1 ) x + r1 b1 + r 2 ( b2 − F )
f (x)
x
Do not attempt this in the real world.
Why?
Your credit card company will charge you late
fees in the real world.
Alternative:
Consolidate your credit cards with a home
equity loan or low interest credit card.
Good Debt
Examples of Good Debt
Education
House
Land
Example: We will use is taking a loan out for an
Applied Math degree.
Assumptions
After 108 months ( 5 years after you graduate)
Till retirement at age of 65
Your Math Degree Pays
•And 758,648 ahead of the
associate degree grad.
•As you can see you
come out 911,616 of the
high school grad
•Well worth your 26,000
in loans.
Conclusions
There is a right time to go into debt.
Just think before you act.
Do the Math.
And make you good investments.
)
('' )
&
) )
) )
"
?
&
*
&.
)
Through Thick and Thin
By: Mark Bergman
Thomas Bursey
Jay LaPorte
Paul Miller
Aaron Sinz
Measurement Methods
1.
2.
3.
Hydrostatic (Underwater)
Weighing
Skin Fold Measurements
Ultrasound Measurements
Through Thick and Thin
(Statistical Model)
I.
II.
III.
IV.
V.
VI.
Introduction
Siri’s Equation and Data
Elements of Regression
Analysis
Regression Analysis of Body
Fat Data
Demonstrations
Conclusion
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)
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)
Proportion of Fat Tissue
D = 1/[(A/a) + (B/b)]
B = (1/D)*[a*b/(a-b)]-[b/(a-b)]
Siri’ s Equation
Estimates a =1.10 gm/cm^3 and
b =0.90 gm/cm^3
Percentage of Body Fat = 495 /D - 450
Elements of Regression Analysis
Simple Regression
y = b0 + b1 x
Multiple Regression
y = b0 + b1 x1 + .... + bn xn
Elements of Regression Analysis
Regression Assumptions
1.
The population satisfies the equation
y = B0 + B1 x + ∈
2.
3.
4.
The true residuals are mutually independent
The true residuals all have the same variance
The true residuals all have a normal distribution
with mean zero
Elements of Regression Analysis
Sum of Squares
SStotal =
( yi − y )
2
Mean of Squares
MS reg = SS reg / df reg
MS res = SS res / df res
Coefficient of Determination
R 2 = ( SStotal − SS res ) / SStotal
Elements of Regression Analysis
F-Ratio
F = MS reg / MS res
T-Ratio
t = Bˆ i / sei
Simple Regression
The Best Predictor For Simple Regression Using
Excel
Abdomen Circumference
Abdomen
y = 0.6313x - 39.28
R2 = 0.6617
60
Abd o m e n Cir cu m fe r e n ce
50
40
Series1
30
Linear (Series1)
20
10
0
0
20
40
60
80
Percent Body Fat
100
120
140
160
Simple Regression
The Worst Predictor For Simple Regression Using Excel
Ankle Circumference
Ankle Cirumference
y = 1.3133x - 11.189
R2 = 0.0707
50
45
40
Percent Bo dy F at
35
30
Series1
25
Linear (Series1)
20
15
10
5
0
0
5
10
15
20
Ankle Cirumference
25
30
35
40
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)
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
Coefficients
Model
1
Unstandardized
Coefficients
B
Std. Error
-17.775
17.361
5.840E-02
.033
-9.01E-02
.054
-7.20E-02
.096
-.467
.233
-2.61E-02
.099
.961
.087
-.215
.146
.237
.144
2.610E-02
.242
.170
.222
.191
.172
.444
.199
-1.620
.535
(Constant)
AGE
WEIGHT
HEIGHT
NECK
CHEST
ABDOMEN
HIP
THIGH
KNEE
ANKLE
BICEPS
FOREARM
WRIST
a
Standardi
zed
Coefficien
ts
Beta
t
-1.024
1.791
-1.682
-.750
-2.008
-.263
11.078
-1.471
1.643
.108
.767
1.116
2.227
-3.027
.088
-.316
-.032
-.136
-.026
1.239
-.184
.149
.008
.034
.069
.107
-.180
Sig.
.307
.075
.094
.454
.046
.793
.000
.142
.102
.914
.444
.266
.027
.003
a. Dependent Variable: BODYFAT
Model Summary
Model
1
a.
R
.866a
Adjusted
R Square
R Square
.749
Predictors: (Constant), WRIST, AGE, HEIGHT, ANKLE,
FOREARM, ABDOMEN, BICEPS, KNEE, NECK, THIGH,
CHEST, HIP, WEIGHT
Std. Error of
the Estimate
.736
4.307
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
AGE
WEIGHT
HEIGHT
NECK
CHEST
ABDOMEN
HIP
THIGH
KNEE
ANKLE
BICEPS
FORARM
WRIST
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.749
0.745
0.746
0.748
0.745
0.749
0.620
0.749
0.746
0.749
0.748
0.748
0.744
0.739
Removing Predictors
0.800
0.600
0.400
0.200
0.000
1
2
3
4
5
6
7
8
9 10 11 12 13 14
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)
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
body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.930
Average Difference = 3.58
Body Fat Demonstration
Using the best model from our Regression
Analysis
body fat = abdomen (.975) – weight (.114) – wrist (1.245) – 27.93
Measuring the Predictors
The Best 3 Predictors are the
• Abdomen
• Weight
• Wrist
Abdomen and Wrist are measured in
Centimeters (cm)
Weight is measured in pounds
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).
Measuring the Weight
Weight should be taken with an accurate
weighing scale.
Record the persons weight in pounds.
Measuring the Wrist
Measurement should be taken between hand
and wrist bone.
Measure the wrist circumference (cm).
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 ?
The normal range for men is 15-18%
Age
Excellent
Good
Fair
Poor
References
Dr. Steve Deckelman
A Course in Mathematical Modeling
– By Douglas Mooney and Randall
Swift
http://lib.stat.cmu.edu/datasets/bodyfat