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:

We can add an estimated linear regression line:

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

Scatterplots by groups

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

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

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

## Instalowanie pakietu w 'C:/Users/magda/AppData/Local/R/win-library/4.3'
## (ponieważ 'lib' nie jest określony)

## pakiet 'ggExtra' został pomyślnie rozpakowany oraz sumy MD5 zostały sprawdzone
## 
## Pobrane pakiety binarne są w
##  C:\Users\magda\AppData\Local\Temp\Rtmp23SdA1\downloaded_packages

## Warning: pakiet 'ggExtra' został zbudowany w wersji R 4.3.3

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:

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:

## [1] "Pearson's linear correlation:  0.574346033083121"

Ok, what about the percentage of the explained variability?

## [1] "Percentage of the explained variability:  32.9873365718318 %"

So as we can see almost 32,99% of total log of incomes’ variability is explained by differences in age. The rest (67,1%) 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.

## [1] "Partial correlation:  0.319426338025541"

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

## [1] "Semi-partial correlation:  0.22037110188406"

Partial correlation coefficient calculated for the log of income and age controlling for years of employment equals 0,319. It means that after including influence of years of employment there exist positive and moderate relationship between age and logarithm of income. So after removing the effects of years of employment, these two are still moderately correlated.

On the other hand, the semi-partial correlation between log income and age, controlling for years of employment for log income equals 0,220. Therefore after taking into account the effect of years of employment on the logarithm of income, there is a weaker but still positive relationship between age and the logarithm of income. In other words, age is moderately correlated with log income, after removing the effect of years of employment on log income only.

Finally, comparing these two coefficients we can see that controlling for years of employment has a greater impact on logarithm of income than on age, which is visible in the lower value of the semi-partial correlation coefficient compared to the partial correlation.

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.

In these box-plots we see that people with the highest education generally earn the most. In case of lower education levels the difference between them isn’t as significant.

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

## [1] "Kendall's rank correlation coefficient:  0.158"

Kendall’s rank correlation coefficient is in range -1 to 1, where 1 inform us about perfect positive correlation, -1 about perfect negative correlation and 0 about no correlation. Value 0,158 suggest weak correlation between income and age.

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:

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

## [1] "Point-biserial correlation coefficient (income): 0.071"

## [1] "Point-biserial correlation coefficient (log income): 0.135"

We can see that this correlation is positive and weak, since the absolute value is below 0,2.

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] "Eta coefficient (income ~ default): 0.0709696566192232"

## [1] "Eta coefficient (log income ~ default): 0.135225798931301"

The values of eta coefficient and Pearsons coefficient are the same, so there is no nonlinear correlation.

Correlation matrix

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

We can use ggcorr() function:

## Warning in ggcorr(bank, method = c("everything", "pearson"), label = TRUE):
## data in column(s) 'ed', 'default', 'def', 'educ' are not numeric and were
## ignored

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

## `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`.

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.

bank$ed <- as.factor(bank$ed)
bank$default <- as.factor(bank$default)

contingency_table <- table(bank$ed, bank$default)
print(contingency_table)

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

chisq.test(contingency_table)

## Warning in chisq.test(contingency_table): Aproksymacja chi-kwadrat może być
## niepoprawna

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

prop.table(contingency_table)

##    
##               0           1
##   1 0.418571429 0.112857143
##   2 0.198571429 0.084285714
##   3 0.081428571 0.042857143
##   4 0.034285714 0.020000000
##   5 0.005714286 0.001428571

cramers_v <- CramerV(contingency_table)
print(paste("Cramer's V:", round(cramers_v, 3)))

## [1] "Cramer's V: 0.128"

The results suggest that there is a relationship between education level and credit risk, although it is weak (the value of Cramer’s V is low). However, due to the low p-value, we can consider this relationship to be statistically significant.

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).

## Call: cohen.kappa1(x = x, w = w, n.obs = n.obs, alpha = alpha, levels = levels, 
##     w.exp = w.exp)
## 
## Cohen Kappa and Weighted Kappa correlation coefficients and confidence boundaries 
##                   lower estimate upper
## unweighted kappa -0.043    0.011 0.065
## weighted kappa   -0.043    0.011 0.065
## 
##  Number of subjects = 1091

## 
##  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

## [1] 0.0121878

## [1] 0.01218871

## [1] 0.01218871

The unweighted Cohen Kappa value ranges from -0.043 to 0.065, suggesting that there is only minimal above-random agreement between belief in an afterlife and gender. Similarly the Chi-square test indicates that there is no statistically significant relationship between belief in the afterlife and gender (p > 0.05). Furthermore, all from coefficients Phi, Cont, Cramer’s V and Tschuprow’s T are around 0,012, which suggest very weak relationship between this belief and gender.

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[complete.cases(titanic$Fare.Price), ]

contingency_table_titanic <- table(titanic$Status, titanic$Crew.or.Passenger.)


print(contingency_table_titanic)

##           
##            Crew Passenger
##             214        35
##   Survivor  211       500
##   Victim    679       817

prop.table(contingency_table_titanic)

##           
##                  Crew  Passenger
##            0.08713355 0.01425081
##   Survivor 0.08591205 0.20358306
##   Victim   0.27646580 0.33265472

chisq.test(contingency_table_titanic)

## 
##  Pearson's Chi-squared test
## 
## data:  contingency_table_titanic
## X-squared = 236.24, df = 2, p-value < 2.2e-16

Phi(contingency_table_titanic)

## [1] 0.3101466

ContCoef(contingency_table_titanic)

## [1] 0.2962265

CramerV(contingency_table_titanic)

## [1] 0.3101466

TschuprowT(contingency_table_titanic)

## [1] 0.2608012

mosaicplot(contingency_table_titanic)

barplot(contingency_table_titanic)

The result of the chi-square test indicates that there is a statistically significant relationship between status (survival or death) and crew or passenger affiliation (p < 0.05).
Coefficients values estimates around 0,3, which means that there exist moderate correlation between these two variables. Based on this analysis we can say that belonging to the crew or passengers had a significant impact on survival during the Titanic disaster. Passengers had higher chance of surviving than the crew members.