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.

## [1] 0.574346

##    estimate      p.value statistic   n gp  Method
## 1 0.2775388 4.816032e-28  10.97912 700  1 kendall

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

We can add an estimated linear regression line:

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:

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] 0.574346

Ok, what about the percentage of the explained variability?

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

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.

##    estimate      p.value statistic   n gp  Method
## 1 0.3194263 4.805085e-18  8.899323 700  1 pearson

##    estimate      p.value statistic   n gp  Method
## 1 0.2203711 3.899134e-09  5.964597 700  1 pearson

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

##    estimate      p.value statistic   n gp  Method
## 1 0.3194263 4.805085e-18  8.899323 700  1 pearson

##    estimate      p.value statistic   n gp  Method
## 1 0.2203711 3.899134e-09  5.964597 700  1 pearson

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] 1.725159e-05

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, label = TRUE): data in column(s) '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.

#contingency table
contingency_table <- table(bank$ed, bank$debtinc)
print(contingency_table)

##    
##     0.4 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.6 1.7 1.8 1.9 2.1 2.2 2.3 2.4 2.5
##   1   1   1   0   0   2 0   2   4   1   0   2   2   1   1   3   0   1   3   3
##   2   0   1   1   1   1 1   1   0   1   2   2   1   0   3   1   1   0   2   1
##   3   0   0   0   0   0 0   0   0   0   0   0   1   0   0   0   1   2   1   1
##   4   0   0   0   0   0 0   0   0   0   0   1   1   0   1   0   0   0   0   1
##   5   0   0   0   0   0 0   0   0   0   0   0   0   0   0   0   0   0   0   0
##    
##     2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5
##   1   4   1   1   4 2   1   3   2   1   1   4   4   0   0 2   3   3   0   5   5
##   2   1   0   2   1 2   2   0   0   2   1   0   3   1   1 1   1   1   3   2   3
##   3   1   0   2   0 1   1   0   1   0   0   0   1   0   0 1   1   1   0   2   2
##   4   0   0   0   1 0   0   0   0   0   0   1   0   0   0 0   0   0   1   0   0
##   5   0   0   0   0 0   1   0   0   0   0   0   0   0   0 0   0   0   0   0   0
##    
##     4.6 4.7 4.8 4.9 5 5.1 5.2 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6
##   1   1   4   6   4 5   1   3   5   3   3   3   1   2 3   3   3   2   4   3   3
##   2   2   0   1   1 1   1   1   2   0   3   1   4   0 0   1   1   3   2   1   2
##   3   0   1   0   1 0   0   0   2   1   0   1   1   0 2   1   0   1   1   0   2
##   4   0   0   0   0 1   0   0   0   1   0   1   0   0 0   0   0   0   1   1   1
##   5   0   0   0   0 0   0   0   0   0   0   0   0   1 0   0   0   0   0   0   0
##    
##     6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8 8.1 8.2 8.3 8.4 8.5 8.6
##   1   5   1   3 3   2   4   2   2   3   3   1   1   2 6   1   2   2   4   2   1
##   2   2   4   1 2   1   4   2   0   0   2   1   2   2 0   0   1   1   0   2   3
##   3   1   0   0 0   0   0   0   0   0   0   3   0   1 0   0   0   0   1   0   0
##   4   1   0   0 0   0   0   0   0   0   0   1   0   1 1   0   0   0   0   1   0
##   5   0   0   0 0   0   0   0   0   0   0   0   1   0 0   1   0   0   0   0   0
##    
##     8.7 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4
##   1   0   1   4 4   5   1   2   0   2   3   4   3   1  2    2    2    2    2
##   2   1   0   0 1   1   0   1   0   4   2   2   0   0  0    1    1    1    1
##   3   1   0   0 0   0   1   2   1   0   0   0   0   0  0    0    0    0    2
##   4   0   0   0 0   0   0   0   0   0   0   0   1   0  1    0    0    0    1
##   5   0   0   0 0   0   0   0   1   0   0   0   0   0  0    0    0    0    0
##    
##     10.5 10.6 10.7 10.8 10.9 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12
##   1    5    3    1    1    2  1    1    3    4    1    1    1    2    0    0  3
##   2    0    0    0    0    0  0    2    0    0    2    4    0    0    1    1  0
##   3    1    0    0    1    0  1    0    0    0    0    0    0    0    0    0  1
##   4    0    1    0    0    0  1    1    0    0    0    1    0    0    0    0  0
##   5    0    0    0    0    0  0    0    0    0    0    0    0    0    0    0  0
##    
##     12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13 13.1 13.2 13.3 13.4 13.5
##   1    1    1    1    1    1    2    1    2    3  2    1    2    0    0    1
##   2    1    1    0    2    2    0    0    1    1  2    0    3    2    1    1
##   3    1    1    0    0    0    0    0    0    1  1    0    2    1    1    0
##   4    1    0    0    1    0    0    1    0    0  0    0    0    0    0    0
##   5    0    0    0    0    0    0    0    0    0  0    0    0    0    0    0
##    
##     13.6 13.7 13.8 13.9 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 15 15.1 15.2
##   1    1    2    2    1  2    1    1    2    3    0    1    2    3  3    1    1
##   2    1    0    0    0  0    0    0    1    1    0    0    1    1  4    0    0
##   3    1    0    1    0  0    1    1    0    1    1    1    1    0  0    1    0
##   4    0    0    0    0  0    0    0    0    0    0    0    0    0  0    0    0
##   5    0    0    0    0  0    0    0    0    0    0    0    0    0  0    0    0
##    
##     15.3 15.4 15.6 15.7 16 16.1 16.3 16.4 16.5 16.6 16.7 16.9 17 17.1 17.2 17.3
##   1    0    1    2    0  1    2    3    1    1    1    1    1  2    1    0    3
##   2    0    2    0    0  1    2    0    2    0    0    0    1  0    1    1    1
##   3    1    0    0    1  2    0    0    0    1    0    0    0  0    0    0    0
##   4    0    0    0    1  0    1    0    0    1    0    0    0  0    0    0    2
##   5    0    0    0    0  0    0    0    0    0    0    0    0  0    0    0    0
##    
##     17.4 17.5 17.6 17.8 18 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.1
##   1    1    1    0    2  0    1    0    0    0    1    2    0    1    2    0
##   2    1    0    0    0  1    2    1    1    2    0    0    3    0    1    1
##   3    0    0    1    0  0    0    0    0    0    0    0    0    0    0    0
##   4    0    0    1    0  0    0    0    0    1    0    0    0    0    0    0
##   5    0    0    0    0  0    0    0    0    0    0    0    0    0    0    0
##    
##     19.2 19.4 19.5 19.6 19.7 19.8 19.9 20 20.1 20.2 20.4 20.5 20.9 21 21.2 21.3
##   1    1    1    0    1    1    2    1  1    0    1    0    1    0  1    0    2
##   2    1    0    1    0    0    0    0  0    1    0    1    0    1  0    1    0
##   3    1    0    0    0    0    1    3  0    1    0    0    0    0  0    0    0
##   4    0    0    0    0    0    0    0  0    0    0    0    0    0  0    0    0
##   5    0    0    0    0    0    0    0  0    0    0    0    0    0  0    0    0
##    
##     21.4 21.7 22.3 22.7 22.9 23 23.1 23.3 23.6 23.7 23.8 24.2 24.4 24.5 24.6
##   1    1    0    1    0    0  0    3    0    0    1    2    0    1    1    0
##   2    0    1    0    1    1  0    2    1    1    0    2    0    0    0    0
##   3    0    0    0    0    0  1    0    0    0    0    0    0    0    0    1
##   4    0    0    0    0    0  0    0    0    0    0    0    1    0    0    0
##   5    0    0    0    0    0  0    0    0    0    0    0    0    0    0    0
##    
##     24.7 25.1 25.2 25.3 25.4 25.6 25.7 26 26.3 26.5 27.1 27.7 28.5 28.9 29.7
##   1    1    1    0    1    1    1    1  0    0    1    1    1    1    2    0
##   2    0    1    0    0    0    0    0  1    0    0    0    1    0    0    0
##   3    0    0    1    0    0    0    0  0    1    1    0    0    0    0    1
##   4    0    0    0    0    0    0    0  0    0    0    0    0    0    0    0
##   5    0    0    0    0    0    0    0  0    0    0    0    0    0    0    0
##    
##     30.1 30.6 30.7 30.8 32.5 33.3 33.4 35.3 36.6 41.3
##   1    1    1    1    1    1    1    0    0    0    1
##   2    0    0    0    0    0    0    0    0    0    0
##   3    0    0    0    0    0    0    0    0    1    0
##   4    0    0    0    0    0    0    1    1    0    0
##   5    0    0    0    0    0    0    0    0    0    0

#chi-squared test of independence
chi_squared_test <- chisq.test(contingency_table)

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

print(chi_squared_test)

## 
##  Pearson's Chi-squared test
## 
## data:  contingency_table
## X-squared = 879.06, df = 920, p-value = 0.8298

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.

# Clean the data by replacing empty strings with NA and dropping all NA values
titanic[titanic == ""] <- NA
titanic <- na.omit(titanic)

# Recoding variables as factors for categorical analysis
titanic$Status <- as.factor(titanic$Status)
titanic$Gender <- as.factor(titanic$Gender)
titanic$Embarked <- as.factor(titanic$Embarked)

# Plot the distribution of Age by Survival Status
library(ggplot2)
ggplot(titanic, aes(x = Age, fill = Status)) + 
    geom_histogram(binwidth = 5, position = "fill", color = "black") +
    labs(y = "Proportion", x = "Age", fill = "Survival Status", title = "Survival by Age Group")

# Contingency tables and Chi-squared tests
# 1. Class and Survival
table_Class_Survived <- table(titanic$Class, titanic$Status)
chi_Class_Survived <- chisq.test(table_Class_Survived)
print(table_Class_Survived)

##            
##             Survivor Victim
##   1st Class       50     53
##   2nd Class       11     83
##   3rd Class       33    341

print(chi_Class_Survived)

## 
##  Pearson's Chi-squared test
## 
## data:  table_Class_Survived
## X-squared = 94.503, df = 2, p-value < 2.2e-16

# 2. Gender and Survival
table_Gender_Survived <- table(titanic$Gender, titanic$Status)
chi_Gender_Survived <- chisq.test(table_Gender_Survived)
print(table_Gender_Survived)

##         
##          Survivor Victim
##   Female       35     36
##   Male         59    441

print(chi_Gender_Survived)

## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  table_Gender_Survived
## X-squared = 60.863, df = 1, p-value = 6.12e-15

# 3. Embarkation point and Survival
table_Embarked_Survived <- table(titanic$Embarked, titanic$Status)
chi_Embarked_Survived <- chisq.test(table_Embarked_Survived)

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

print(table_Embarked_Survived)

##              
##               Survivor Victim
##   Belfast            0      1
##   Cherbourg         30     55
##   Queenstown         9     48
##   Southampton       55    373

print(chi_Embarked_Survived)

## 
##  Pearson's Chi-squared test
## 
## data:  table_Embarked_Survived
## X-squared = 26.195, df = 3, p-value = 8.68e-06