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

## [1] 0.1577567

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

## Warning: Using size for a discrete variable is not advised.

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:

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

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

## Warning: The following aesthetics were dropped during statistical transformation: size.
## ℹ This can happen when ggplot fails to infer the correct grouping structure in
##   the data.
## ℹ Did you forget to specify a `group` aesthetic or to convert a numerical
##   variable into a factor?

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

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] 0.574346

Ok, what about the percentage of the explained variability?

## [1] 32.98734

So as we can see almost 33% of total log of incomes’ variability is explained by differences in age. The rest 67% 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] 0.3194263

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

## [1] 0.1867239

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:

## 
##  Pearson's product-moment correlation
## 
## data:  bank$income and as.numeric(bank$def)
## 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

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

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

## `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$education <- factor(bank$educ)
bank$def <- factor(bank$def)

contingency_table <- table(bank$education, bank$def)

chi_square_test <- chisq.test(contingency_table)

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

contingency_table

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

chi_square_test

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

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$Status <- ifelse(titanic$Status == "Victim" | titanic$Status == "Survivor" , titanic$Status, "Disembarked before")

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

titanic$Fare_GBP[is.na(titanic$Fare_GBP)] <- 0

survived_class <- table(titanic$Status, titanic$Class_Department)
survived_gender <- table(titanic$Status, titanic$Gender)
survived_embarked <- table(titanic$Status, titanic$Embarked)

survived_class

##                     
##                      1st Class 2nd Class 3rd Class Deck Crew Engineering Crew
##   Disembarked before        26         9         0        33              179
##   Survivor                 201       119       180        43               71
##   Victim                   123       166       528        23              253
##                     
##                      Restaurant Staff Victualling Crew
##   Disembarked before                0                2
##   Survivor                          3               94
##   Victim                           66              337

survived_gender

##                     
##                      Female Male
##   Disembarked before     15  234
##   Survivor              359  352
##   Victim                130 1366

survived_embarked

##                     
##                      Belfast Cherbourg Queenstown Southampton
##   Disembarked before     217         0          0          32
##   Survivor                43       154         44         470
##   Victim                 155       118         79        1144

1st class passengers had the highest survival rate(201), followed by 2nd class(119), and lastly 3rd class(180), which had the lowest survival rate. This indicates a strong association between class and survival rate, probably higher-class passengers had better access to lifeboats.

A higher proportion of females survived (359) compared to males (352)(if you watched titanic you know who survived and who did not :D),and there were more male victims (1366) than female victims (130). The number of males who disembarked before the sinking (234) is significantly higher than females (15).

Among the crew, deck crew had a higher survival rate compared to other crew types. The engineering crew had a particularly high death rate, likely due to their roles which kept them below decks during the disaster.

Most survivors embarked at Southampton (470). The highest number of victims also embarked from Southampton (1144).

chisq_class <- chisq.test(survived_class)
chisq_gender <- chisq.test(survived_gender)
chisq_embarked <- chisq.test(survived_embarked)

chisq_class

## 
##  Pearson's Chi-squared test
## 
## data:  survived_class
## X-squared = 819.13, df = 12, p-value < 2.2e-16

chisq_gender

## 
##  Pearson's Chi-squared test
## 
## data:  survived_gender
## X-squared = 552.06, df = 2, p-value < 2.2e-16

chisq_embarked

## 
##  Pearson's Chi-squared test
## 
## data:  survived_embarked
## X-squared = 1071.5, df = 6, p-value < 2.2e-16

The chi-square statistic indicates a highly significant result. We can say that there’s a strong association between class and survival on the titanic.

Same thing here. We can say that there’s a strong association between gender and survival on the titanic. Women and kids were obviously the first ones on the lifeboats.

Here too. The survival rate varied significantly among passengers who embarked from different ports.

ggplot(titanic, aes(x = Class...Department, fill = Status)) +
  geom_bar(position = "fill") +
  labs(title = "Survival by Class/Department", y = "Proportion", x = "Class/Department")

ggplot(titanic, aes(x = Gender, fill = Status)) +
  geom_bar(position = "fill") +
  labs(title = "Survival by Gender", y = "Proportion", x = "Gender")

ggplot(titanic, aes(x = Embarked, fill = Status)) +
  geom_bar(position = "fill") +
  labs(title = "Survival by Embarked Location", y = "Proportion", x = "Embarked Location")