RPubs link information

• The link to the published page is:
• https://rpubs.com/bohanh123/RMITAAA2

Introduction

• Each year, 17.9 million people die of cardiovascular diseases
• 31% of global deaths
• Main triggers are:
  • tobacco use
  • unhealthy diet
  • physical inactivity
  • alcohol abuse

Problem Statement

• Aim of this investigation:
  • Find the relationship of mean serum sodium between male and female
• Used statistic techniques:
  • Levene’s test
  • t-test

Data

• Data set used for investigation is Heart Failure Prediction
• From Chicco and Jurman(2020)
• Contains 299 observations and 13 variables
• Key variables:
  • sex: categorical variable indicationg the sex of the patients
    • originally 0 and 1
    • converted to “Female” and “Male” for better presentation.
  • serum_sodium: level of serum sodium in the blood (mEq/L)

Descriptive Statistics and Visualisation

• Statistical features

Decsriptive Statistics Cont.

• Visualisation - Histogram

Decsriptive Statistics Cont.

• Visualisation - Q-Q plot

The distribution is approximately normal.

Decsriptive Statistics Cont.

• Visualisation - Box plot

Outliers present

Decsriptive Statistics Cont.

• To handle outliers, capping method is used.

\[ If\ x_i \gt \textrm{Upper fence},\ then \ x_i=\ \textrm{95% quantile} \\ If\ x_i \lt \textrm{Lower fence},\ then \ x_i=\ \textrm{5% quantile} \] Where

\[ \textrm{Upper fence} = Q_3 + 1.5*IQR \\ \textrm{Lower fence} = Q_1 - 1.5*IQR \]

Hypothesis Testing - Levene’s test

We assume homogeneity of variance, then alternative hypothesis is heterogeneity of variance.

\[ H_0: \sigma^2_1 = \sigma^2_2 \\ H_A: \sigma^2_1 \neq \sigma^2_2 \] where \(\sigma_1^2\) and \(\sigma_2^2\) refer to the population variance respectively.

To test this, we perform the Levene’s test of equal variance for serum sodium between males and females.

Hypothesis Testing - Levene’s test

leveneTest(serum_sodium ~ sex, data=heart.df) %>% knitr::kable()

As \(p \gt 0.05\), the Levene’s test is not statistically significant.We fail to reject \(H_0\). It is safe to assume equal variance.

Hypthesis Testing - Two-sample t-test

According to Central Limit Theorem, since the sample sizes of each group are greater than 30, we can proceed with a two-sample t-test.

Now we assume that the average for male and female are the same. Alternative hypothesis is that they are different.

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

Where \(\mu_1\) and \(\mu_2\) refer to the average respectively.

To test this, we perform two-sample t-test assuming equal variance and two-sided hypothesis test.

Hypthesis Testing - Two-sample t-test

t.test(serum_sodium ~ sex, data=heart.df, var.equal=TRUE, alternative="two.sided")

## 
##  Two Sample t-test
## 
## data:  serum_sodium by sex
## t = -1.0605, df = 297, p-value = 0.2898
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.4009807  0.4198319
## sample estimates:
##   mean in group Male mean in group Female 
##             136.6237             137.1143

We get the test statistic \(t=-1.0605\).

Assuming \(\alpha = 0.05\) and a two-tailed test, \(t*\) is calculated as

qt(p = 0.025, df = 297)

## [1] -1.967984

Hypthesis Testing - Two-sample t-test

The test statistic \(t = -1.0605\) is less extreme than \(t* = -1.967984\), which means we fail to reject \(H_0\).

According to p-value method, \(p = 0.2898 > 0.05\), we also failed to reject \(H_0\).

From the R report, the 95% CI of the difference (-0.4906) is [-1.4009807, 0.4198319], which captures \(H_0\). We cannot reject \(H_0\).

Discussion

• A two-sample t-test was used to test for a significant difference between the mean serum sodium of male patients and female patients.

• From a visualisation of Q-Q plot, the distributions are both approximately normal.

• A Levene’s test of homogeneity of variance proved that equal variance can be safely assumed.

• Using central limit theorem, we made sure that t-test could be applied.

• The result of the two-sample t-test assuming equal variance showed there was no statistically significant difference between the average of male patients and female patients, \(t(df=297)=-1.06\), \(p=0.29\), 95%CI for the difference [-1.40, 0.42].

• The result of the investigation sugests that there is no significant difference between the serum sodium averages of females and males.

References

• Chicco D., Jurman G. 2020, Machine learning can predict survival of patients with heart failure from serum creatinine and ejection fraction alone, BMC Medical Informatics and Decision Making 20, 16(2020), https://doi.org/10.1186/s12911-020-1023-5.