Modeling Body Measurements

Data Source:

Heinz G, Peterson LJ, Johnson RW, Kerk CJ. 2003. Exploring Relationships in Body Dimensions. Journal of Statistics Education 11(2). http://jse.amstat.org/v11n2/datasets.heinz.html

Problem Statement

The purpose of this investigation is to find the variation in data when I compare a certain variable against males and females.
I have chosen the variable hgt(Respondent’s height in centimeters) for analysis.
I am going to inspect the heights of males and females separately and depict the contrast by checking if the data falls in a normal distribution.
I have conducted the Shapiro-Wilk Test to see if the variable hgt is normally distributed.

Load Packages

library(readr) # Useful for importing data
library(dplyr) # Useful for data manipulation

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

#library(magrittr) # Useful for Forward-Pipe operator
library(ggplot2)

Data

Imported the body measurements data and subset the data frame to contain only the variable of choice- hgt and sex.

data <- read_csv("bdims.csv")

## Rows: 507 Columns: 25
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (25): bia.di, bii.di, bit.di, che.de, che.di, elb.di, wri.di, kne.di, an...
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

data_subset <- data[c("hgt","sex")]

Summary Statistics

Conducted the Shapiro-Wilk normality test on the raw data- variable hgt(without splitting data according to sex). The data is normal if the p-value is above 0.05. In our case, the p-value is 0.01045 which means our variable is not normally distributed. I will rerun this test after splitting the variable by sex.
Defined the variable sex as a factor and defined labels- Male and Female. Confirmed by checking the structures(str function).
Created separate data frames for storing data for males and females.
Used summary function to calculate the descriptive statistics for males and females separately.
I have used the Graphical method - Density curve to assess our data and confirm it is normally distributed. Plotted the density curves for male and female heights in red and blue respectively.
Reran the Shapiro-Wilk normality test and now our variable is normally distributed since the p-value is above 0.05.

# This is a chunk for Summary Statistics section. 

shapiro.test(data_subset$hgt)

## 
##  Shapiro-Wilk normality test
## 
## data:  data_subset$hgt
## W = 0.99233, p-value = 0.01045

data_subset$sex = factor(data_subset$sex,
                  levels = c("1","0"),
                  labels = c("Male","Female"))
str(data_subset)

## tibble [507 × 2] (S3: tbl_df/tbl/data.frame)
##  $ hgt: num [1:507] 174 175 194 186 187 ...
##  $ sex: Factor w/ 2 levels "Male","Female": 1 1 1 1 1 1 1 1 1 1 ...

males <- data_subset %>% filter(sex=="Male")
females <- data_subset %>% filter(sex=="Female")
summary(males)

##       hgt            sex     
##  Min.   :157.2   Male  :247  
##  1st Qu.:172.9   Female:  0  
##  Median :177.8               
##  Mean   :177.7               
##  3rd Qu.:182.7               
##  Max.   :198.1

summary(females)

##       hgt            sex     
##  Min.   :147.2   Male  :  0  
##  1st Qu.:160.0   Female:260  
##  Median :164.5               
##  Mean   :164.9               
##  3rd Qu.:169.5               
##  Max.   :182.9

m1 <- males$hgt
f1 <- females$hgt

d1 <- density(m1)
d2 <- density(f1)
plot(d1,col="blue",main="Density curve for Male Height",xlab="Male Height (in cm)")
polygon(d1,col = "red", border="black")

plot(d2,col="red",main="Density curve for Female Height",xlab="Female Height (in cm)")
polygon(d2,col = "blue", border="black")

shapiro.test(m1)

## 
##  Shapiro-Wilk normality test
## 
## data:  m1
## W = 0.99358, p-value = 0.3716

shapiro.test(f1)

## 
##  Shapiro-Wilk normality test
## 
## data:  f1
## W = 0.99283, p-value = 0.2437

Distribution Fitting

Plotted two histograms for male and female heights in red and blue respectively with the normal distribution overlay depicted by the purple curve. The density curves are displayed in black.
The male height is almost normally distributed, with a peak in the middle and fairly symmetrical.
The female height is approximately normally distributed, with a peak roughly in the middle.
Since the points in the QQ-normal plot roughly lie on a straight diagonal line, I can infer that the data is normally distributed.

# This is a chunk for Distribution Fitting section. 

h1<- hist(m1, breaks=12,density=20, col="red", xlab="Male Height (in cm)",main="Histogram of Male height", prob=TRUE,ylim=c(0,0.08))
lines(density(m1))
xfit<-seq(min(m1),max(m1),length=40)
yfit<-dnorm(xfit,mean=mean(m1),sd=sd(m1))
lines(xfit, yfit, col="purple",lwd=2)

r_norm_male<-rnorm(length(m1)*2,m=mean(m1),sd=sd(m1))
plot(ecdf(r_norm_male),main="Empirical cumulative distribution function(Male Heights)")

standard_norm_male<-(r_norm_male-mean(r_norm_male))/sd(r_norm_male)
qqnorm(standard_norm_male) 
abline(0,1)

h2<-hist(f1, breaks=15,density=20, col="blue", xlab="Female Height (in cm)",main="Histogram of Female height", prob=TRUE,ylim=c(0,0.10))
lines(density(f1))
xfit<-seq(min(f1),max(f1),length=40)
yfit<-dnorm(xfit,mean=mean(f1),sd=sd(f1))
lines(xfit, yfit, col="purple",lwd=2)

r_norm_female<-rnorm(length(f1)*2,m=mean(f1),sd=sd(f1))
plot(ecdf(r_norm_female),main="Empirical cumulative distribution function(Female Heights)")

standard_norm_female<-(r_norm_female-mean(r_norm_female))/sd(r_norm_female)
qqnorm(standard_norm_female) 
abline(0,1)

Interpretation

The variable ‘hgt’ almost fits the normal distribution for males and females. I have determined this by running the Shapiro-Wilk normality test, along with QQ-plot also by plotting the histogram with normal distribution overlay and a density curve. The insight gained from this investigation is that males are slightly taller than females.