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.

#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:

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

## TableGrob (2 x 2) "arrange": 3 grobs
##   z     cells    name           grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]

Scatterplots with density curves

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

## Warning: The dot-dot notation (`..scaled..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(scaled)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

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] 0.574346

## [1] -0.02677729

#the percentage of the explained variability

## [1] 0.3298734

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.

## [1] "Partial correlation: 0.319426338025541"

## [1] "Semipartial correlation: 0.601846726471143"

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).

## [1] 0.1577567

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] -0.07096966

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] 0.9684572

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 geom_text(data = textData, aes(label = !!as.name("diagLabel")), :
## Ignoring unknown parameters: `type`

## Scale for fill is already present.
## Adding another scale for fill, which will replace the existing scale.

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.

## [[1]]

## 
## [[2]]

## 
## [[3]]

## 
## [[4]]

## 
## [[5]]

## 
## [[6]]

## 
## [[7]]

## Warning in warn_if_args_exist(list(...)): Extra arguments: "aes_string" are
## being ignored.  If these are meant to be aesthetics, submit them using the
## 'mapping' variable within ggpairs with ggplot2::aes or ggplot2::aes_string.

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.

# Tworzenie tablicy kontyngencji
contingency_table <- table(bank$ed, bank$default)

# Wyświetlenie tablicy kontyngencji
print("Contingency Table:")

## [1] "Contingency Table:"

print(contingency_table)

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

# Wykonanie testu chi-kwadrat
chi_square_test <- chisq.test(contingency_table)

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

# Wyświetlenie wyników testu
print("Chi-Square Test Results:")

## [1] "Chi-Square Test Results:"

print(chi_square_test)

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

The chi-square test results indicate that there is a significant association between education level and default status, as the p-value (0.02155) is less than the significance level of 0.05

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

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.

##                          Status Disembarked.at Home.Country Age Year.of.Birth
## DE GRASSE, Mr J.           <NA>      Cherbourg               NA            NA
## EVANS, Miss                <NA>      Cherbourg               NA            NA
## MULLEN,                    <NA>      Cherbourg               NA            NA
## WOTTON, Mr Henry Swaffin   <NA>      Cherbourg               54          1858
## BRAND, Mr                  <NA>      Cherbourg               NA            NA
## FLETCHER, Miss N.          <NA>      Cherbourg               NA            NA
##                          Crew.or.Passenger. Gender Class...Department
## DE GRASSE, Mr J.                  Passenger   Male          2nd Class
## EVANS, Miss                       Passenger Female          2nd Class
## MULLEN,                           Passenger Female          2nd Class
## WOTTON, Mr Henry Swaffin          Passenger   Male          1st Class
## BRAND, Mr                         Passenger   Male          1st Class
## FLETCHER, Miss N.                 Passenger Female          1st Class
##                             Embarked     Job               Job.details
## DE GRASSE, Mr J.         Southampton                                  
## EVANS, Miss              Southampton                                  
## MULLEN,                  Southampton                                  
## WOTTON, Mr Henry Swaffin Southampton Butcher Butcher's Shop Proprietor
## BRAND, Mr                Southampton                                  
## FLETCHER, Miss N.        Southampton                                  
##                          Ticket.Number Fare.Price Fare_GBP Fare_today
## DE GRASSE, Mr J.                   761         P1      1.0     82.110
## EVANS, Miss                         88         P1      1.0     82.110
## MULLEN,                            404         P1      1.0     82.110
## WOTTON, Mr Henry Swaffin            86     P1 10s      1.5    123.165
## BRAND, Mr                            8     P1 10s      1.5    123.165
## FLETCHER, Miss N.                  405     P1 10s      1.5    123.165
##                                                                                        Profile.on.Encyclopedia.Titanica
## DE GRASSE, Mr J.                                http://www.encyclopedia-titanica.org/titanic-biography/j-de-grasse.html
## EVANS, Miss                                           http://www.encyclopedia-titanica.org/titanic-biography/evans.html
## MULLEN,                                              http://www.encyclopedia-titanica.org/titanic-biography/mullen.html
## WOTTON, Mr Henry Swaffin http://www.encyclopedia-titanica.org/titanic-cross-channel-passenger/henry-swaffin-wotton.html
## BRAND, Mr                                             http://www.encyclopedia-titanica.org/titanic-biography/brand.html
## FLETCHER, Miss N.                                http://www.encyclopedia-titanica.org/titanic-biography/n-fletcher.html

Statistics number

## [1] "Phi value"

## [1] 0.4703662

## [1] "Contingency Coeficient"

## [1] 0.4256325

## [1] "Cramer Value"

## [1] 0.4703662

## [1] "Tschuprow"

## [1] 0.4703662

Association between survival status and passenger class is small according to Coeficient. Thanks to Plot we can see that Famale had better chance of survaving.