Descriptive Statistics

Bivariate Analysis

Introduction

This is our first lab when we are considering 2 dimensions and instead of calculating univariate statistics by groups (or factors) of other variable - we will measure their common relationships based on co-variance and correlation coefficients.

*Please be very careful when choosing the measure of correlation! In case of different measurument scales we have to recode one of the variables into weaker scale.

It would be nice to add some additional plots in the background. Feel free to add your own sections and use external packages.

Data

This time we are going to use a typical credit scoring data with predefined “default” variables and personal demografic and income data. Please take a look closer at headers and descriptions of each variable.

Scatterplots

First let’s visualize our quantitative relationships using scatterplots.

You can also normalize the skewed distribution of incomes using log:

Thank to using log of income is much easier to see the relation of the variables in the plot, because the plot is more clear.

We can shortly sum up what we can read from those plots:

Most of the people have education on level 1 and 2 only few people have education at level 5,

In not each case the level of the higher level of education is related with higher income

Lowest incomes are in age group 20 - 30, but the highest ones in group 35 - 55

We can see a relation between income and years of employment, people with bigger experience earn more then people with less experience

People with higher level of eductaion have rather less experience then people with lower levels

We can add an estimated linear regression line:

## `geom_smooth()` using formula = 'y ~ x'

## `geom_smooth()` using formula = 'y ~ x'

Scatterplots by groups

We can finally see if there any differences between risk status:

We can also see more closely if there any differences between those two distributions adding their estimated density plots:

From this plot we can say that older people have more years of employment which is connected with the fact that the higher age he bigger experience

People with higher education have rather smaller amount of years of employment then people with lower levels of education

We can also put those plots together:

Scatterplots with density curves

We can also see more closely if there any differences between those two distributions adding their estimated density plots:

We can see that there is the biggest density between age 25 and 30 and 0-5 years of employment, in this group of people we can see almost eac level of education

Correlation coefficients - Pearson’s linear correlation

Ok, let’s move to some calculations. In R, we can use the cor() function. It takes three arguments and the method: cor(x, y, method) For 2 quantitative data, with all assumptions met, we can calculate simple Pearson’s coefficient of linear correlation:

Ok, what about the percentage of the explained variability?

So as we can see almost ??? of total log of incomes’ variability is explained by differences in age. The rest (???) is probably explained by other factors.

Partial and semipartial correlation

The partial and semi-partial (also known as part) correlations are used to express the specific portion of variance explained by eliminating the effect of other variables when assessing the correlation between two variables.

Partial correlation holds constant one variable when computing the relations to others. Suppose we want to know the correlation between X and Y holding Z constant for both X and Y. That would be the partial correlation between X and Y controlling for Z.

Semipartial correlation holds Z constant for either X or Y, but not both, so if we wanted to control X for Z, we could compute the semipartial correlation between X and Y holding Z constant for X.

Suppose we want to know the correlation between the log of income and age controlling for years of employment. How highly correlated are these after controlling for tenure?

**There can be more than one control variable.

How can we interpret the obtained partial correlation coefficient? What is the difference between that one and the semi-partial coefficient:

Rank correlation

For 2 different scales - like for example this pair of variables: income vs. education levels - we cannot use Pearson’s coefficient. The only possibility is to rank also incomes… and lose some more detailed information about them.

First, let’s see boxplots of income by education levels.

Now, let’s see Kendal’s coefficient of rank correlation (robust for ties).

Point-biserial correlation

Let’s try to verify if there is a significant relationship between incomes and risk status. First, let’s take a look at the boxplot:

## Warning: Continuous x aesthetic
## ℹ did you forget `aes(group = ...)`?

If you would like to compare 1 quantitative variable (income) and 1 dychotomous variable (default status - binary), then you can use point-biserial coefficient:

## 
##  Pearson's product-moment correlation
## 
## data:  bank$income and bank$default
## t = -1.8797, df = 698, p-value = 0.06056
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.144313546  0.003149751
## sample estimates:
##         cor 
## -0.07096966

The point-biserial correlation analysis between income and default status shows a weak negative correlation, but this correlation is not statistically significant (p-value is greater than 0.05).

Nonlinear correlation - eta coefficient

If you would like to check if there are any nonlinearities between 2 variables, the only possibility (beside transformations and linear analysis) is to calculate “eta” coefficient and compare it with the Pearson’s linear coefficient.

## [1] 0.07096966

The eta coefficient is 0.07096966, the same as the previously calculated Pearson’s correlation coefficient. This indicates that the relationship between income and default status is linear and weak.

Correlation matrix

We can also prepare the correlation matrix for all quantitative variables stored in our data frame.

We can use ggcorr() function:

As you can see - the default correlation matrix is not the best idea for all measurement scales (including binary variable “default”).

That’s why now we can perform our bivariate analysis with ggpair with grouping.

Correlation matrix with scatterplots

Here is what we are about to calculate: - The correlation matrix between age, log_income, employ, address, debtinc, creddebt, and othdebt variable grouped by whether the person has a default status or not. - Plot the distribution of each variable by group - Display the scatter plot with the trend by group

## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation idioms with `aes()`.
## ℹ See also `vignette("ggplot2-in-packages")` for more information.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## `geom_smooth()` using formula = 'y ~ x'

Qualitative data

In case of two variables measured on nominal or ordinal&nominal scale - we are forced to organize so called “contingency” table with frequencies and calculate some kind of the correlation coefficient based on them. This is so called “contingency analysis”.

Let’s consider one example based on our data: verify, if there is any significant correlation between education level and credit risk.

EducAndDefaultTable <- table(bank$educ, bank$default)

EducAndDefaultTable

##    
##       0   1
##   1 293  79
##   2 139  59
##   3  57  30
##   4  24  14
##   5   4   1

ContingencyTable <- chisq.test(EducAndDefaultTable)

## Warning in chisq.test(EducAndDefaultTable): Chi-squared approximation may be
## incorrect

print(ContingencyTable)

## 
##  Pearson's Chi-squared test
## 
## data:  EducAndDefaultTable
## X-squared = 11.492, df = 4, p-value = 0.02155

The p-value is 0.02155 and this is less than the common significance level of 0.05. This suggests that there is a statistically significant association between education level and credit risk in the dataset.

Exercise 1. Contingency analysis.

Do you believe in the Afterlife? https://nationalpost.com/news/canada/millennials-do-you-believe-in-life-after-life A survey was conducted and a random sample of 1091 questionnaires is given in the form of the following contingency table:

##         Believe
## Gender   Yes  No
##   Female 435 375
##   Male   147 134

Our task is to check if there is a significant relationship between the belief in the afterlife and gender. We can perform this procedure with the simple chi-square statistics and chosen qualitative correlation coefficient (two-way 2x2 table).

## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  dane
## X-squared = 0.11103, df = 1, p-value = 0.739

##         Believe
## Gender         Yes        No
##   Female 0.3987168 0.3437214
##   Male   0.1347388 0.1228231

As you can see we can calculate our chi-square statistic really quickly for two-way tables or larger. Now we can standardize this contingency measure to see if the relationship is significant.

## [1] 0.01218871

Exercise 2. Contingency analysis for the ‘Titanic’ data.

Let’s consider the titanic dataset which contains a complete list of passengers and crew members on the RMS Titanic. It includes a variable indicating whether a person did survive the sinking of the RMS Titanic on April 15, 1912. A data frame contains 2456 observations on 14 variables.

The website http://www.encyclopedia-titanica.org/ offers detailed information about passengers and crew members on the RMS Titanic. According to the website 1317 passengers and 890 crew member were aboard.

8 musicians and 9 employees of the shipyard company are listed as passengers, but travelled with a free ticket, which is why they have NA values in fare. In addition to that, fare is truely missing for a few regular passengers.

# your answer here
titanic[titanic == ""] <- NA
titanic <- titanic %>% drop_na()

titanic$Status <- as.factor(titanic$Status)
titanic$Class...Department <- as.factor(titanic$Class...Department)

ContingencyTableOfClassAndSurvivors <- table(titanic$Class...Department, titanic$Status)
chi_squared_test <- chisq.test(ContingencyTableOfClassAndSurvivors)

Phi(ContingencyTableOfClassAndSurvivors)

## [1] 0.4068214

ContCoef(ContingencyTableOfClassAndSurvivors)

## [1] 0.3768313

CramerV(ContingencyTableOfClassAndSurvivors)

## [1] 0.4068214

TschuprowT(ContingencyTableOfClassAndSurvivors)

## [1] 0.3420946

mosaicplot(ContingencyTableOfClassAndSurvivors)

barplot(ContingencyTableOfClassAndSurvivors)

The contingency table suggests that passenger survival on the Titanic was significantly influenced by the class they were traveling in. Passengers in first class had the highest likelihood of surviva Third-class passengers faced the highest mortality rate.