Pearson Correlation

The Pearson correlation method is usually used as a primary check for the relationship between two variables.

The coefficient of correlation, , is a measure of the strength of the linear relationship between two variables and . It is computed as follow:

$$\rho = \frac{cov(x,y)}{\sigma_x\sigma_y}$$

with

$\sigma_x=\Sigma(x-\bar{x})^2$, i.e., standard deviation of the variable $x$ $\sigma_y=\Sigma(y-\bar{y})^2$, i.e., standard deviation of the variable $y$

The correlation ranges between -1 and 1.

• A value of near or equal to 0 implies little or no linear relationship between $x$ and $y$.
• In contrast, the closer comes to 1 or -1, the stronger the linear relationship.

We can compute the t-test as follow and check the distribution table with a degree of freedom equals to:

$$t=\frac{\rho}{\sqrt{1-\rho^2}}\sqrt{n-2}$$

Spearman Rank Correlation

A rank correlation sorts the observations by rank and computes the level of similarity between the rank. A rank correlation has the advantage of being robust to outliers and is not linked to the distribution of the data. Note that, a rank correlation is suitable for the ordinal variable.

Spearman’s rank correlation is always between -1 and 1 with a value close to the extremity indicates strong relationship. It is computed as follow:

$$\rho = \frac{cov(rg_x,rg_y)}{\sigma_{rg_x}\sigma_{rg_y}}$$

with stated the covariances between ranks. The denominator calculates the standard deviations.

In R, we can use the cor() function. It takes three arguments, , and the method.

cor(x, y, method)

Arguments:

• x: First vector
• y: Second vector
• method: The formula used to compute the correlation. Three string values:
  • “pearson”
  • “kendall”
  • “spearman”

An optional argument can be added if the vectors contain missing value: use = “complete.obs”

We will use the BudgetUK dataset. This dataset reports the budget allocation of British households between 1980 and 1982. There are 1519 observations with ten features, among them:

• wfood: share food share spend
• wfuel: share fuel spend
• wcloth: budget share for clothing spend
• walc: share alcohol spend
• wtrans: share transport spend
• wother: share of other goods spend
• totexp: total household spend in pound
• income total net household income
• age: age of household
• children: number of children

library(dplyr)
PATH <-"https://raw.githubusercontent.com/guru99-edu/R-Programming/master/british_household.csv"
data <- read.csv(PATH) %>%
  filter(income < 500) %>%
  mutate(log_income = log(income),
         log_totexp = log(totexp),
         children_fac = factor(children, order = TRUE, labels = c("No", "Yes"))) %>%
  select(-c(X, children, totexp, income))
glimpse(data)

## Rows: 1,516
## Columns: 10
## $ wfood        <dbl> 0.4272, 0.3739, 0.1941, 0.4438, 0.3331, 0.3752, 0.2568, …
## $ wfuel        <dbl> 0.1342, 0.1686, 0.4056, 0.1258, 0.0824, 0.0481, 0.0909, …
## $ wcloth       <dbl> 0.0000, 0.0091, 0.0012, 0.0539, 0.0399, 0.1170, 0.0453, …
## $ walc         <dbl> 0.0106, 0.0825, 0.0513, 0.0397, 0.1571, 0.0210, 0.0153, …
## $ wtrans       <dbl> 0.1458, 0.1215, 0.2063, 0.0652, 0.2403, 0.0955, 0.0227, …
## $ wother       <dbl> 0.2822, 0.2444, 0.1415, 0.2716, 0.1473, 0.3431, 0.5689, …
## $ age          <int> 25, 39, 47, 33, 31, 24, 46, 25, 30, 41, 48, 24, 28, 31, …
## $ log_income   <dbl> 4.867534, 5.010635, 5.438079, 4.605170, 4.605170, 4.2484…
## $ log_totexp   <dbl> 3.912023, 4.499810, 5.192957, 4.382027, 4.499810, 4.2484…
## $ children_fac <ord> Yes, Yes, Yes, Yes, No, No, No, No, No, No, Yes, No, Yes…

Code Explanation

• We first import the data and have a look with the glimpse() function from the dplyr library.
• Three points are above 500K, so we decided to exclude them.
• It is a common practice to convert a monetary variable in log. It helps to reduce the impact of outliers and decreases the skewness in the dataset.

We can compute the correlation coefficient between income and wfood variables with the “pearson” and “spearman” methods.

cor(data$log_income, data$wfood, method = "pearson")

## [1] -0.2466986

cor(data$log_income, data$wfood, method = "spearman")

## [1] -0.2501252

Correlation Matrix

The bivariate correlation is a good start, but we can get a broader picture with multivariate analysis. A correlation with many variables is pictured inside a correlation matrix. A correlation matrix is a matrix that represents the pair correlation of all the variables.

The cor() function returns a correlation matrix. The only difference with the bivariate correlation is we don’t need to specify which variables. By default, R computes the correlation between all the variables.

Note that, a correlation cannot be computed for factor variable. We need to make sure we drop categorical feature before we pass the data frame inside cor().

A correlation matrix is symmetrical which means the values above the diagonal have the same values as the one below. It is more visual to show half of the matrix.

We exclude children_fac because it is a factor level variable. cor does not perform correlation on a categorical variable.

# the last column of data is a factor level. We don't include it in the code
mat_1 <- as.dist(round(cor(data[,1:9]), 2))
mat_1

##            wfood wfuel wcloth  walc wtrans wother   age log_income
## wfuel       0.11                                                  
## wcloth     -0.33 -0.25                                            
## walc       -0.12 -0.13  -0.09                                     
## wtrans     -0.34 -0.16  -0.19 -0.22                               
## wother     -0.35 -0.14  -0.22 -0.12  -0.29                        
## age         0.02 -0.05   0.04 -0.14   0.03   0.02                 
## log_income -0.25 -0.12   0.10  0.04   0.06   0.13  0.23           
## log_totexp -0.50 -0.36   0.34  0.12   0.15   0.15  0.21       0.49

Code Explanation

• cor(data): Display the correlation matrix
• round(data, 2): Round the correlation matrix with two decimals
• as.dist(): Shows the second half only

Significance level

The significance level is useful in some situations when we use the pearson or spearman method. The function rcorr() from the library Hmisc computes for us the p-value. We can download the library with install.packages("Hmisc").

The rcorr() requires a data frame to be stored as a matrix. We can convert our data into a matrix before to compute the correlation matrix with the p-value.

library("Hmisc")
data_rcorr <- as.matrix(data[, 1:9])

mat_2 <- rcorr(data_rcorr)
# mat_2 <- rcorr(as.matrix(data)) returns the same output

The list object mat_2 contains three elements:

• r: Output of the correlation matrix
• n: Number of observation
• P: p-value

We are interested in the third element, the p-value. It is common to show the correlation matrix with the p-value instead of the coefficient of correlation.

p_value <- round(mat_2[["P"]], 3)
p_value

##            wfood wfuel wcloth  walc wtrans wother   age log_income log_totexp
## wfood         NA 0.000  0.000 0.000  0.000  0.000 0.365      0.000          0
## wfuel      0.000    NA  0.000 0.000  0.000  0.000 0.076      0.000          0
## wcloth     0.000 0.000     NA 0.001  0.000  0.000 0.160      0.000          0
## walc       0.000 0.000  0.001    NA  0.000  0.000 0.000      0.105          0
## wtrans     0.000 0.000  0.000 0.000     NA  0.000 0.259      0.020          0
## wother     0.000 0.000  0.000 0.000  0.000     NA 0.355      0.000          0
## age        0.365 0.076  0.160 0.000  0.259  0.355    NA      0.000          0
## log_income 0.000 0.000  0.000 0.105  0.020  0.000 0.000         NA          0
## log_totexp 0.000 0.000  0.000 0.000  0.000  0.000 0.000      0.000         NA

Code Explanation

• mat_2[[“P”]]: The p-values are stored in the element called P
• round(mat_2[[“P”]], 3): Round the elements with three digits

Visualize Correlation Matrix

A heat map is another way to show a correlation matrix. The GGally library is an extension of ggplot2. Currently, it is not available in the conda library. We can install directly in the console. install.packages("GGally").

The library includes different functions to show the summary statistics such as the correlation and distribution of all the variables in a matrix.

The ggcorr() function has lots of arguments. We will introduce only the arguments we will use in the tutorial:

ggcorr(df, method = c("pairwise", "pearson"),
  nbreaks = NULL, digits = 2, low = "#3B9AB2",
  mid = "#EEEEEE", high = "#F21A00",
  geom = "tile", label = FALSE,
  label_alpha = FALSE)

Basic heat map

The most basic plot of the package is a heat map. The legend of the graph shows a gradient color from - 1 to 1, with hot color indicating strong positive correlation and cold color, a negative correlation.

library(GGally)
ggcorr(data)

Code Explanation

• ggcorr(data): Only one argument is needed, which is the data frame name. Factor level variables are not included in the plot.

ggcorr(data,
    nbreaks = 6,
    low = "steelblue",
    mid = "white",
    high = "darkred",
    geom = "circle")

ggcorr(data,
    nbreaks = 6,
    label = TRUE,
    label_size = 3,
    color = "grey50")

ggpair(df, columns = 1: ncol(df), title = NULL,
    upper = list(continuous = "cor"),
    lower = list(continuous = "smooth"),
    mapping = NULL)     

library(ggplot2)
ggpairs(data, columns = c("log_totexp", "log_income", "age", "wtrans"), title = "Bivariate analysis of revenue expenditure by the British household", upper = list(continuous = wrap("cor",
        size = 3)),
    lower = list(continuous = wrap("smooth",
        alpha = 0.3,
        size = 0.1)),
    mapping = aes(color = children_fac))

ggpairs(data, columns = c("log_totexp", "log_income", "age", "wtrans"),
    title = "Bivariate analysis of revenue expenditure by the British household",
    upper = list(continuous = wrap("cor",
            size = 3),
        mapping = aes(color = children_fac)),
    lower = list(
        continuous = wrap("smooth",
            alpha = 0.3,
            size = 0.1))
)

Summary

We can summarize the function in the table below: