Introduction

• Human Body measurements are used to evaluate health trends in various populations.
• This study examines the relationship between chest diameter of a person and their height.
• It’s obvious to state that if a person is taller, they have a broader chest.
• Generally, it is expected that they have a positive linear relationship.
• But are the results statistically significant?

Problem Statement

• Do height of a person influence their chest size?
• Firstly, Summary Statistics, visualisation and appropriate test are applied to examine the relationship.
• Secondly, a correlation analysis is conducted to identify if there is any relationship or not.
• Subsequently, a linear regression analysis is applied to the variable of interest to investigate whether the outcome is statistically significant or not.
• Finally, the results are interpreted using statistics and visualisation.

Data

• The dataset presented here includes the correspondence between body build, weight, and girths in a group of physically active young men and women, most of whom were within the normal weight range.

• The measurement is taken on the 247 men and 260 women in the dataset. These were primarily individuals in their twenties and early thirties, with a scattering of older men and women, all physically active (several hours of exercise a week)

• Variable of interest
  • Chest Diameter (A numerical vector, respondent’s chest diameter in centimetres, measured at nipple level, mid-expiration.)
  • Height ( A numerical vector, respondent’s height in centimetres.)
• As this investigation aims to analyze the relationship between chest diameter and height of a person.

• Data Source: Heinz G, Peterson LJ, Johnson RW, Kerk CJ. 2003. Exploring Relationships in Body Dimensions. Journal of Statistics Education 11(2).

Descriptive Statistics

Looking at the summary statistics, it is evident that a person chest diameter is 27.97 on average with a range from 25.65 to 29.95. Similarly, for the height of a person, the average is recorded as 171.1 with a range from 163.8 to 177.8.

summary_stats <- data %>% summary()

knitr::kable(summary_stats)

Data Visualisation I

Glancing at the histogram, the distribution of both chest diameter and height are relatively normal.

ggplot(data,aes(che.di))+geom_histogram(bins=30)+ ylab('Frequency')+ggtitle('Histogram of Chest Diameter')
ggplot(data,aes(hgt))+geom_histogram(bins=30)+ ylab('Frequency')+ggtitle('Histogram of Height')

Data Visualisation II

ggplot(data,aes(y=che.di))+geom_boxplot(fill='red')+ggtitle('Summary of Chest Diameter')
ggplot(data,aes(y=hgt))+geom_boxplot(fill='yellow')+ggtitle('Summary of Height')

Data Visulisation III

ggplot(data,aes(x=hgt,y=che.di))+geom_point(color='#0072B2')+ggtitle('Height Vs. Chest Diameter')

Hypothesis Testing I

Correlation Analysis

Hypothesis Generation

\(H_0\): There is no correlation between chest diameter and height of a person.

\(H_a\): There is significant correlation between chest diameter and height of a person

Mathematically,

\(H_0: r = 0\)

\(H_a: r \ne0\)

Test Result I

library(Hmisc)
library(psychometric)
corr<-as.matrix(dplyr::select(data,che.di,hgt)) #Create a matrix of the variables to be correlated
rcorr(corr, type = "pearson")

##        che.di  hgt
## che.di   1.00 0.63
## hgt      0.63 1.00
## 
## n= 507 
## 
## 
## P
##        che.di hgt
## che.di         0 
## hgt     0

r=cor(data$che.di,data$hgt)
CIr(r = r, n = 50, level = .95)

## [1] 0.4222205 0.7707495

A Pearson’s correlation was calculated to measure the strength of the linear relationship between chest diameter and height. The positive correlation was statistically significant, r=.63, p<.001, 95% CI [0.422, .770]

Hypothesis Testing II

Linear Regression Analysis

Hypothesis Generation

\(H_0\): The data do not fit the linear regression model

\(H_a\): The data fit the linear regression model

Mathematically,

\(H_0: \alpha = 0\)

\(H_a: \alpha \ne0\)

Test Result II

model <- lm(che.di~hgt,data=data)
model %>% summary()

## 
## Call:
## lm(formula = che.di ~ hgt, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.3102 -1.4326 -0.0696  1.4168  6.8929 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -3.2947     1.7319  -1.902   0.0577 .  
## hgt           0.1827     0.0101  18.082   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.138 on 505 degrees of freedom
## Multiple R-squared:  0.393,  Adjusted R-squared:  0.3918 
## F-statistic:   327 on 1 and 505 DF,  p-value: < 2.2e-16

model %>% confint()

##                  2.5 %    97.5 %
## (Intercept) -6.6972252 0.1079121
## hgt          0.1628512 0.2025541

Model Visualisation

Model Validation

plot(model)

Discussion

Major Findings

• A linear regression model was fitted to predict the dependent variable, Chest Diameter, using measures of height in centimetre as a single predictor.
• Before fitting the regression, a scatter plot estimating the bivariate relationship between chest diameter and height was inspected.
• The scatter plot demonstrated evidence of a positive linear relationship. Other non-linear trends were ruled out.
• The overall regression model was statistically significant, F(1,505)=327, p<.001, moreover explained 39.3% of the variability in chest diameter, R2=.393.
• The estimated regression equation was che.di=-3.295+.183∗ hgt.
• The positive slope for height was statistically significant, b=0.183, t(505)=18.09, p<.001, 95% CI [0.163, 0.203].
• Final inspection of the residuals supported normality and homoscedasticity.

References

• James Baglin 2 April, 2019, Viewed 15 May 2020 https://astral-theory-157510.appspot.com/secured/MATH1324_Module_09.html#linear_regression
• Viewed 14 May https://stackoverflow.com/questions/15633714/adding-a-regression-line-on-a-ggplot
• Viewed 15 May https://www.calvin.edu/~rpruim/courses/m343/F12/RStudio/LatexExamples.html
• Viewed 16 May http://www.cookbook-r.com/Graphs/Colors_(ggplot2)/
• Grete Heinz,Louis J. Peterson,Carter J. Kerk,Journal of Statistics Education Volume 11, Number 2 (2003),Viewed May 15 http://jse.amstat.org/v11n2/datasets.heinz.html?fbclid=IwAR30lqQs0TtbCa0e3LGmubpSSzOLE_tY03IBObsdmq3EFQq1S4ZNrRLz_4w