Problem Statement

• The purpose of experiment is to compare obesity(Fat) level between two Gender - Male and Female, to conclude which gender has more risk of health problems.
• Obesity depend on height, weight and Age ratio(Body Mass Index).
• To compare obesity, we will use two-sample t-test(Independent samples t-test) by checking boxplot, normality, p-value, significant difference between mean of gender.

Data Collection Process

• For experiment, We have used public domain obesity data and it was collected using Risk Surveillance System with various stratification category like Education, Income, Gender, Age, Ethnicity for different parameter Locality of US, timeline.
• We have created sample data by choosing year(2011) and Staritification Category(Gender) from the full dataset.
• DataSource: https://www.kaggle.com/adu47249/obesity-stats

# DataSource: https://www.kaggle.com/adu47249/obesity-stats
init_dataset <- read.csv("obesity.csv",stringsAsFactors = FALSE)
dataset <- init_dataset %>% filter(StratificationCategory==c("Gender"))  %>% select(-c("Age.years.","Sample_Size","Income","Race.Ethnicity"))
knitr::kable(head(dataset,5L))

Summary of Data (Field/Variables)

# Initial Data Structure of dataset
str(dataset)

## 'data.frame':    3814 obs. of  7 variables:
##  $ YearStart             : int  2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 ...
##  $ YearEnd               : int  2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 ...
##  $ LocationAbbr          : chr  "AL" "AL" "AL" "AL" ...
##  $ LocationDesc          : chr  "Alabama" "Alabama" "Alabama" "Alabama" ...
##  $ Data_Value            : num  32.3 31.8 39 30.5 40.1 47.9 26.6 22.3 45.4 39.6 ...
##  $ Gender                : chr  "Male" "Female" "Male" "Female" ...
##  $ StratificationCategory: chr  "Gender" "Gender" "Gender" "Gender" ...

1. YearStart, YearEnd : Start Year, End Year
2. LocationAbbr,Desc : Location Abbrevation, Desc
3. Data_Value : Obesity index
4. Gender : Gender(Male/Female)
5. StratificationCategory : Data Stratification Category(Age/Income/Ethnicity/Gender)

Data Manipulation for experiment (Filter)

• Gender should be factor(Male, Female)
• Dataset had many column like LocationId, Income, Age etc which are removed by selecting expected column.

1. Gender
2. Data_Value(Obesity Index)

# Convert Gender to factor
dataset$Gender <- factor(dataset$Gender,levels=c("Male","Female"),labels=c("Male","Female"),ordered = TRUE)
# Select relevant column (Data_Value and Gender)
df <- dataset %>% filter(YearStart == 2011) %>% select(c("Gender","Data_Value"))
# Filtered data frame
str(df)

## 'data.frame':    936 obs. of  2 variables:
##  $ Gender    : Ord.factor w/ 2 levels "Male"<"Female": 1 2 1 2 2 1 1 2 1 2 ...
##  $ Data_Value: num  32.3 31.8 39 30.5 40.1 47.9 26.6 22.3 45.4 39.6 ...

Data Manipulation (Outliers) cntd..

• Identifying missing values and outliers, Replace the outlier with median
• Show boxplot(Ignore outliers because it is below 5%)

# Male-Female dataframe
df_male <- df %>% filter(Gender == "Male")
n_male <- nrow(df_male)
male_median <- median(df_male$Data_Value)
df_female <- df %>% filter(Gender == "Female")
n_female <- nrow(df_female)
female_median <- median(df_female$Data_Value)

Data Manipulation (Outliers) cntd..

# Identify Outliers
box <- boxplot(Data_Value~Gender,df,xlab="Gender",ylab="Obesity")

Data Manipulation (Outliers) cntd..

# replacing outliers with median of each group
df$Data_Value <- ifelse(df$Data_Value %in% box$out, ifelse(df$Gender=="Female", female_median,male_median), df$Data_Value)
boxplot(Data_Value~Gender,df,xlab="Gender",ylab="Obesity")

Decsriptive Statistics Summary

• Summary table of data (Mean, Median, Quantile, Standard Deviation, Missing data) by each gender

summary_tab <- df %>% group_by(Gender) %>% summarise(Min = min(Data_Value,na.rm = TRUE),
                                      Q1 = quantile(Data_Value,probs = .25,na.rm = TRUE),
                                      Median = median(Data_Value, na.rm = TRUE),
                                      Q3 = quantile(Data_Value,probs = .75,na.rm = TRUE),
                                      Max = max(Data_Value,na.rm = TRUE),
                                      Mean = mean(Data_Value, na.rm = TRUE),
                                      SD = sd(Data_Value, na.rm = TRUE),
                                      n = n(),
                                      Missing = sum(is.na(Data_Value)))
knitr::kable(summary_tab,digits=round(1))

Hypothesis Testing (Normality test)

• Test Normality by plotting QQPlot for Obesity by Male and Female

# Male qqPlot
rnorm(df_male$Data_Value) %>% qqPlot()

## [1] 153 398

Hypothesis Testing (Normality test) cntd..

# Female qqPlot
rnorm(df_female$Data_Value) %>% qqPlot(col="red")

## [1] 108 102

Hypothesis Testing (Normality test) cntd..

• Plot Histogram and density curve for obesity by Male-Female. By seeing plots, we can prove “data is normal”.

# Density curve for Male
xfit_male<-seq(min(df_male$Data_Value),max(df_male$Data_Value),length=length(df_male$Data_Value))
yfit_male<-dnorm(xfit_male,mean=mean(df_male$Data_Value),sd=sd(df_male$Data_Value))
h_male <- hist(df_male$Data_Value,col="blue",main="Male obesity",breaks=10,xlab="Obesity")
yfit_male <- yfit_male*diff(h_male$mids[1:2])*length(df_male$Data_Value)
lines(xfit_male, yfit_male, lwd=2)

Hypothesis Testing (Normality test) cntd..

# Density curve for Female
xfit_female<-seq(min(df_female$Data_Value),max(df_female$Data_Value),length=length(df_female$Data_Value))
yfit_female<-dnorm(xfit_female,mean=mean(df_female$Data_Value),sd=sd(df_female$Data_Value))
h_female <- hist(df_female$Data_Value,col="red",main="Female obesity",breaks=10,xlab="Obesity")
yfit_female <- yfit_female*diff(h_female$mids[1:2])*length(df_female$Data_Value)
lines(xfit_female, yfit_female, lwd=2)

Hypothesis Testing (Variance Check) cntd..

• Homogeneity of variance: We have used LeveneTest to identify equal variance between male and female. \[H_0: \mu_1^2 = \mu_2^2 \] \[H_A: \mu_1^2 \ne \mu_2^2 \]

Ho: Both Gender has equal variance

Ha: Both Gender has unqual variance

# compare the variances of male and female obesity
leveneTest(Data_Value~Gender,data=df)

• Here P < 0.05, so we need to reject Ho. It means, male and female has unequal variance.

Hypothesis Testing (Two-sample t-test) cntd..

• Two-sample t-test - Assuming Unequal Variance, because leveneTest had p-value < 0.05.

\[H_0: \mu_1 - \mu_2 = 0 \]

\[H_A: \mu_1 - \mu_2 \ne 0 \]

Ho: Both Gender has equal obesity

Ha: Both Gender has unqual obesity

\[t = \frac{\overline{x_1} - \overline{x_2}}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]

\[df = \frac{({\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}})^2}{{\frac{({\frac{s_1^2}{n_1}})^2}{n_1-1}}+{\frac{({\frac{s_2^2}{n_2}})^2}{n_2-1}}}\]

Hypothesis Testing (Two-sample t-test - Unequal variance) cntd..

# unequal variance t-test
res <- t.test(Data_Value~Gender,data=df,var.equal = FALSE, alternative = "two.sided")
res

## 
##  Welch Two Sample t-test
## 
## data:  Data_Value by Gender
## t = 13.179, df = 784.52, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  6.082871 8.212000
## sample estimates:
##   mean in group Male mean in group Female 
##             33.67778             26.53034

# t value
round(res$statistic,2)

##     t 
## 13.18

Hypothesis Testing (Two-sample t-test - Unequal variance) cntd..

# df` degree of freedom
round(res$parameter)

##  df 
## 785

# p-value
res$p.value

## [1] 5.637934e-36

# conf.int
round(res$conf.int,2)

## [1] 6.08 8.21
## attr(,"conf.level")
## [1] 0.95

Discussion

• Based on the result, you can say: at 95% confidence level, there is significant difference (p-value < 0.05) of the two means. So, we have to reject the null hypothesis.
• The maximum difference of the mean can be as low as 6.08 and as high as 8.12.
• As we have to reject null hypothesis, Male and female has equal obesity.
• Population mean of male(33.68) is higher than female(26.53).
• As result, we can conclude - “Male has more obesity than female”.

References

• Data Source: https://www.kaggle.com/adu47249/obesity-stats