Multiple Regression Analysis of Body Fat Percent and B.M.I(Body Mass Index)
Kyung Yun
1. Introduction of Body Density data.
This is a comprehensive dataset that lists estimates of the percentage of body fat determined by underwater weighing and various body circumference measurements for 252 men. These data were generously supplied by Dr. A. Garth Fisher who gave permission to freely distribute the data and use for non-commercial purposes. These data are 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.(http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_BMI_Regression#References)
2. Theories about BMI and Body Fat Percentage
Some experts tout BMI(body mass index) as the most accurate and simple way to determine the effect of weight on your health. In fact, most recent medical research uses BMI as an indicator of someone’s health status and disease risk. BMI from 18.5 up to 25 may indicate optimal weight, a BMI lower than 18.5 suggests the person is underweight, a number from 25 up to 30 may indicate the person is overweight, and a number from 30 upwards suggests the person is obese. However, some debate about which values on the BMI scale the thresholds for ‘underweight’, ‘overweight’ and ‘obese’ should be set.
Meanwhile, in September 2000, the American Journal of Clinical Nutrition published a study showing that body-fat percentage may be a better measure of your risk of weight-related diseases than BMI.(http://www.webmd.com/diet/body-fat-measurement). Percentage of body fat for an individual can be estimated by the Siri’s formula(1956) once body density has been determined.
Which one is a better measure of body fat? This question remains to be studied further. Instead, the main goal of this analysis is to find out the relationships between ‘Body Fat Percentage’ and other predictors as well as ‘Body Mass Index’ and others.
3. The variables of Body Density data.
The body density dataset includes the following 15 variables listed from left to right:
Density : Density determined from underwater weighing
Fat : Percent body fat from Siri’s (1956) equation
Age : Age (years)
Weight : Weight (kg)
Height : Height (cm)
Neck : Neck circumference (cm)
Chest: Chest circumference (cm)
Abdomen : Abdomen circumference (cm)
Hip : Hip circumference (cm)
Thigh : Thigh circumference (cm)
Knee : Keee circumference (cm)
Ankle : Ankle circumference (cm)
Biceps : Biceps (extended) circumference (cm)
Forearm : Forearm circumference (cm)
Wrist : Wrist circumference (cm)
New variable MassIndex will be created as the ratio of Weight(kg) to Height(m) squared for the measure of Body Mass Index in addition to the above variables. Model Selection process and regression for both response variables; Fat and MassIndex will be compared to explain which measure explains this data better.
4.Reading Body Density Data and Creating a new dataset
library(car)
library(stargazer)
library(Zelig)bmi <- read.table("K:/QC/Soc/SOC712/Homework/BMI.txt",header=TRUE)
summary(bmi)## Density Fat Age Height
## Min. :0.995 Min. : 0.00 Min. :22.00 Min. : 74.93
## 1st Qu.:1.041 1st Qu.:12.47 1st Qu.:35.75 1st Qu.:173.35
## Median :1.055 Median :19.20 Median :43.00 Median :177.80
## Mean :1.056 Mean :19.15 Mean :44.88 Mean :178.18
## 3rd Qu.:1.070 3rd Qu.:25.30 3rd Qu.:54.00 3rd Qu.:183.51
## Max. :1.109 Max. :47.50 Max. :81.00 Max. :197.49
## Weight Neck Chest Abdomen
## Min. : 53.75 Min. :31.10 Min. : 79.30 Min. : 69.40
## 1st Qu.: 72.12 1st Qu.:36.40 1st Qu.: 94.35 1st Qu.: 84.58
## Median : 80.06 Median :38.00 Median : 99.65 Median : 90.95
## Mean : 81.16 Mean :37.99 Mean :100.82 Mean : 92.56
## 3rd Qu.: 89.36 3rd Qu.:39.42 3rd Qu.:105.38 3rd Qu.: 99.33
## Max. :164.72 Max. :51.20 Max. :136.20 Max. :148.10
## Hip Thigh Knee Ankle
## Min. : 85.0 Min. :47.20 Min. :33.00 Min. :19.1
## 1st Qu.: 95.5 1st Qu.:56.00 1st Qu.:36.98 1st Qu.:22.0
## Median : 99.3 Median :59.00 Median :38.50 Median :22.8
## Mean : 99.9 Mean :59.41 Mean :38.59 Mean :23.1
## 3rd Qu.:103.5 3rd Qu.:62.35 3rd Qu.:39.92 3rd Qu.:24.0
## Max. :147.7 Max. :87.30 Max. :49.10 Max. :33.9
## Biceps Forearm Wrist
## Min. :24.80 Min. :21.00 Min. :15.80
## 1st Qu.:30.20 1st Qu.:27.30 1st Qu.:17.60
## Median :32.05 Median :28.70 Median :18.30
## Mean :32.27 Mean :28.66 Mean :18.23
## 3rd Qu.:34.33 3rd Qu.:30.00 3rd Qu.:18.80
## Max. :45.00 Max. :34.90 Max. :21.40The above table shows the Body Density Dataset having 15 variables and 252 observations.
Next step is to create new variable MassIndex using mutate function indplyr package. As a result, 16 variables are in the data named bmi2.
library(dplyr)
bmi2 <- mutate(bmi, MassIndex = Weight/(Height/100)^2)names(bmi2)## [1] "Density" "Fat" "Age" "Height" "Weight"
## [6] "Neck" "Chest" "Abdomen" "Hip" "Thigh"
## [11] "Knee" "Ankle" "Biceps" "Forearm" "Wrist"
## [16] "MassIndex"To select significant predictors for the response variable ‘Fat’, linear regression is performed.
The below table shows that only ‘Density’ has a significant linear relationship with ‘Fat’.
summary(bmi.mod1 <- lm(Fat ~ Density + Age + Height + Weight + Neck + Chest + Abdomen + Hip + Thigh + Knee + Ankle + Biceps + Forearm + Wrist, data = bmi2))##
## Call:
## lm(formula = Fat ~ Density + Age + Height + Weight + Neck + Chest +
## Abdomen + Hip + Thigh + Knee + Ankle + Biceps + Forearm +
## Wrist, data = bmi2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.4357 -0.3724 -0.1275 0.2156 15.1474
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.500e+02 1.071e+01 42.005 <2e-16 ***
## Density -4.112e+02 8.258e+00 -49.796 <2e-16 ***
## Age 1.259e-02 9.626e-03 1.308 0.192
## Height -3.142e-03 1.120e-02 -0.281 0.779
## Weight 2.217e-02 3.520e-02 0.630 0.529
## Neck -2.846e-02 6.938e-02 -0.410 0.682
## Chest 2.678e-02 2.936e-02 0.912 0.363
## Abdomen 1.857e-02 3.175e-02 0.585 0.559
## Hip 1.917e-02 4.343e-02 0.441 0.659
## Thigh -1.676e-02 4.303e-02 -0.389 0.697
## Knee -4.639e-03 7.162e-02 -0.065 0.948
## Ankle -8.568e-02 6.576e-02 -1.303 0.194
## Biceps -5.505e-02 5.087e-02 -1.082 0.280
## Forearm 3.386e-02 5.953e-02 0.569 0.570
## Wrist 7.345e-03 1.617e-01 0.045 0.964
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.274 on 237 degrees of freedom
## Multiple R-squared: 0.9781, Adjusted R-squared: 0.9768
## F-statistic: 756.3 on 14 and 237 DF, p-value: < 2.2e-16Again, linear regression is performed to select significant predictors for the response variable ‘MassIndex’. As a result, the predictors; Height, Weight,Neck, Chest, Abdomen, Thigh, and Knee are chosen for their significance.
summary(bmi.mod2 <- lm(MassIndex ~ Density + Age + Height + Weight + Neck + Chest + Abdomen + Hip + Thigh + Knee + Ankle + Biceps + Forearm + Wrist, data = bmi2))##
## Call:
## lm(formula = MassIndex ~ Density + Age + Height + Weight + Neck +
## Chest + Abdomen + Hip + Thigh + Knee + Ankle + Biceps + Forearm +
## Wrist, data = bmi2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.3523 -2.0685 -0.3381 1.6979 28.2505
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 229.605482 30.330244 7.570 8.26e-13 ***
## Density -6.343269 23.380600 -0.271 0.786393
## Age -0.007158 0.027253 -0.263 0.793040
## Height -1.029726 0.031703 -32.480 < 2e-16 ***
## Weight 1.154493 0.099651 11.585 < 2e-16 ***
## Neck -0.515661 0.196416 -2.625 0.009219 **
## Chest -0.344475 0.083132 -4.144 4.76e-05 ***
## Abdomen -0.330356 0.089894 -3.675 0.000294 ***
## Hip -0.146461 0.122946 -1.191 0.234743
## Thigh -0.325452 0.121819 -2.672 0.008072 **
## Knee 0.654171 0.202773 3.226 0.001432 **
## Ankle -0.246928 0.186169 -1.326 0.185996
## Biceps -0.272801 0.144028 -1.894 0.059431 .
## Forearm 0.100265 0.168550 0.595 0.552497
## Wrist -0.088536 0.457726 -0.193 0.846792
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.608 on 237 degrees of freedom
## Multiple R-squared: 0.8655, Adjusted R-squared: 0.8575
## F-statistic: 108.9 on 14 and 237 DF, p-value: < 2.2e-165. Rregressions with 2 subsets made by dplyr package.
Based on the above result, 2 different subsets of data are selected. First subset ‘subbmi1 has Fat(as response variable) and density after removing all other variables. Second subset ’subbmi2’ has variables: MassIndex(as response variable) and Height, Weight,Neck, Chest, Abdomen, Thigh, and Knee as predictors.
subbmi1 <- select(bmi2, Fat, Density)
names(subbmi1)## [1] "Fat" "Density"subbmi2 <- select(bmi2, MassIndex, Height, Weight,Neck, Chest, Abdomen, Thigh, Knee)
names(subbmi2)## [1] "MassIndex" "Height" "Weight" "Neck" "Chest" "Abdomen"
## [7] "Thigh" "Knee"Then regressions results for 2 different subsets and response variables are compared as follows.(Hlavac 2014) The table shows that FAT~Density regression model has higher R-sqaure value than MassIndex regression model. Multiple R-squared value is 0.976 which implies that approximately 97.6 % of the vairiability of the dependant variable is explained by the fitted regression line. MassIndex regression has 86.2% R-squre. The weighted combination of the 7 predictor variables explained approximately 86.2% of the variance of the dependent variable in the MassIndex regression.
Density=-434.360 implies that Fat will be expected to decrease by 434.360 percent for an additonal density unit. Another resonse variable MassIndex has negative linear relationship with Height, Neck, Chest, Abdomen, and Thigh but postive linear relationship with Weight and Knee.
library(stargazer)
bmi.m1 <- lm(Fat ~ Density, data =subbmi1)
bmi.m2 <- lm(MassIndex ~ Height+Weight+Neck+Chest+ Abdomen+Thigh+Knee, data =subbmi2)
stargazer(bmi.m1, bmi.m2, title="Comparison of 2 Regression outputs",type="html", single.row=TRUE)| Dependent variable: | ||
| Fat | MassIndex | |
| (1) | (2) | |
| Density | -434.360*** (4.334) | |
| Height | -1.017*** (0.030) | |
| Weight | 1.040*** (0.075) | |
| Neck | -0.537*** (0.172) | |
| Chest | -0.337*** (0.080) | |
| Abdomen | -0.317*** (0.060) | |
| Thigh | -0.391*** (0.098) | |
| Knee | 0.597*** (0.189) | |
| Constant | 477.650*** (4.576) | 206.708*** (10.790) |
| Observations | 252 | 252 |
| R2 | 0.976 | 0.862 |
| Adjusted R2 | 0.976 | 0.858 |
| Residual Std. Error | 1.307 (df = 250) | 3.606 (df = 244) |
| F Statistic | 10,044.030*** (df = 1; 250) | 217.073*** (df = 7; 244) |
| Note: | p<0.1; p<0.05; p<0.01 | |
6. Discussion of the results
Both Body Mass Index (BMI) and Body Fat Percent are measures of the body fat. For the Body Density Data 2 different response variables are used to identify the relationsip between each response variable and their own significant predictors to determine which body fat measure represents this dataset better. The result of this analysis shows that Body Fat percent(‘Fat’) has a significant negative linear relationship only with the variable Denstiy since the fomula for Body Fat Percent is 
and Body Density 
, where
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)
Meanwhile, Body Mass Index(‘MassIndex’) has strong linear relationships with multiple predictors since Body Mass Index can be calculated by 
and Height and Weight are also correlated with other measurements such as Abdomen, Thigh, et el.
Accurate measurement of body fat is inconvenient or costly and it is desirable to have easy methods of estimating body fat that are cost-effective and convenient. However, most of the body measures except for the Body Fat Percent and the Body Density are relatively easier to obtain for data collection. Body Mass Index is also calculated simply from weight and height of an individual. Conclusively, Body Mass Index is a more conveniet and less costly method for Body fat measures since it is strongly correlated with most body measures except Body Fat Percent and Density. One plausible problem with Body Mass Index(‘MassIndex’) regression is that there is a possiblity of multicollinearity among predictor variables since all body measures are highly correlated.(Fox and Weisberg 2011)
However, What is the better or more accurate measure of human body fat remains to be answered with further medical reseaches.
References
Fox, John, and Harvey Sanford Weisberg. 2011. An R Companion to Applied Regression. 2nd ed. Thousand Oaks, CA: Sage Publications.
Hlavac, Marek(2014). 2014. stargazer:LaTex Code and ASCII Text for Well-Formatted Regression and Summary Staistics Tables. http://CRAN.R-project.org/package=stargazer.