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

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

**Output:**

## 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

### 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 from conda and copy the code to paste it in the terminal:

conda install -c r r-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

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

**Output:**

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

## 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")