July 26, 2023
Correlation is a statistical measure that describes the direction and strength of two variables’ relationship.
It is used to determine whether two variables have a relationship and how closely are the two variables are related.
The Pearson’s \(r\), or correlation coefficient, ranges from -1 to 1.
A value of -1 denotes a perfect negative correlation, in which one variable decreases as the other increases.
A value of 1 denotes a perfect positive correlation, in which both variables increase or decrease simultaneously.
A value of 0 indicates that there is no relationship between the two variables.
Correlation analysis can prove beneficial in analyzing different aspects of Marketing such as customer behavior, advertising effectiveness and customer satisfaction.
Customer Behavior: Correlation analysis can be used to understand the relationships between customer behavior, such as product purchase history, and demographic data, such as age and income.
Advertising effectiveness: Correlation analysis can be used to better understand the relationship between advertising spend and sales. This can assist marketers in determining the most effective advertising channels and optimizing their advertising budget.
Customer Satisfaction: Correlation analysis can help understand the connection between customer satisfaction and product features. This can help marketers understand what features are most important to their customers and inform product design decisions.
Studying numerous facets of finance, such as portfolio diversification, risk management, asset pricing, and credit risk management, might benefit from correlation analysis.
Portfolio diversification: Correlation can describe how closely related are the assets in a portfolio to one another. Investors can reduce their exposure to risk and diversify their holdings by knowing the correlation between various assets. If two assets have a high positive correlation, investment in both assets may not offer much diversification benefit. The benefits of diversification become larger, in case the correlation between two assets is low or negative.
Risk management: Correlation can be used to assess the interaction between different risk indicators including interest rates, market volatility, and credit risk. By understanding the relationship between these components, investors may more effectively control their risk exposure and safeguard their portfolios from potential losses.
Credit risk management: The relationship between various credit risk parameters, such as borrower creditworthiness and loan performance, could be measured using correlation in credit risk management. Lenders can more accurately analyse the risk of potential borrowers and make more informed lending decisions by being aware of the association between these attributes.
When researching different facets of organisational behaviour, such as employee engagement, diversity and inclusion, organisational culture, and job performance, correlation analysis could prove beneficial.
Employee engagement: The relationship between many elements that affect employee engagement, such as job satisfaction, employee motivation, and organisational culture, can be measured using correlation. Companies can discover the major factors that influence employee engagement and make positive interventions to enhance employee engagement by a deeper understanding of the relationship between these variables.
Diversity and inclusion: The relationship between diversity and inclusion variables and their influence on organisational outcomes, such as employee performance, customer satisfaction, and profitability, can be measured using correlation. Companies may create diversity and inclusion policies and initiatives that are more effective by deeply recognizing the relationship between these elements.
Job performance: Correlation analysis could quantify the relationship between many elements that affect worker motivation, skill level, and training. Companies can identify the primary influences on work performance and create more effective performance management programs and strategies by understanding the relationship between these variables.
\[ s = \sqrt(∑(x_i - x_m)^2 / n) \]
\(x_i\) is an individual data value
\(x_m\) is the mean of the data set
\(n\) is the number of observations in the data set
# Load the mtcars data set
data(mtcars)
# Calculate the mean and standard deviation of the "mpg"
mpg_mean <- mean(mtcars$mpg)
mpg_sd <- sd(mtcars$mpg)
# Print the results
cat("Mean:", round(mpg_mean, 2), "\n")Mean: 20.09 cat("Standard deviation:", round(mpg_sd, 2), "\n")Standard deviation: 6.03 Covariance is a measure of the relationship between two variables and the extent to which they vary together.
It is a value that describes the direction of the relationship between two variables.
The formula for covariance (cov) is:
\[ cov = (∑(x_i - x_m) * (y_i - y_m)) / (n-1) \]
# Load the mtcars dataset
data(mtcars)
# Calculate the covariance between the mpg and wt variables
cov(mtcars$mpg, mtcars$wt)[1] -5.116685The Pearson’s Correlation Coefficient, a statistical metric commonly referred to as Pearson’s r, describes the degree and direction of a linear relationship between two variables.
The Pearson’s Correlation Coefficient ranges from -1 to 1, where:
\[ r = ∑((x_i - x_m) * (y_i - y_m)) / \sqrt((∑(x_i - x_m)^2) * (∑(y_i - y_m)^2)) \]
# Load the mtcars dataset
data(mtcars)
# Calculate the correlation between the mpg and wt variables
cor(mtcars$mpg, mtcars$wt)[1] -0.8676594# Load the mtcars dataset
data(mtcars)
# relationship between the mpg and wt variables
plot(mtcars$mpg, mtcars$wt, xlab = "MPG", ylab = "Wt")
The correlation coefficients between two variables are displayed in a table called a correlation matrix.
It is a symmetric matrix where the upper and lower triangles hold the correlation coefficients between each pair of variables, and the diagonal holds the correlation of each variable with itself, which is always 1.
It is an effective tool for investigating the connections between variables in a dataset.
# Load the mtcars dataset
data(mtcars)
# Calculate the correlation matrix for mpg, wt, hp, drat
cor(mtcars[,c("mpg","wt","hp","drat")]) mpg wt hp drat
mpg 1.0000000 -0.8676594 -0.7761684 0.6811719
wt -0.8676594 1.0000000 0.6587479 -0.7124406
hp -0.7761684 0.6587479 1.0000000 -0.4487591
drat 0.6811719 -0.7124406 -0.4487591 1.0000000# Load the mtcars dataset
data(mtcars)
# scatter plot matrix for mpg, wt, hp, drat
pairs(mtcars[,c("mpg","wt","hp","drat")], pch = 19)
The hypothesis testing process can be used to examine the null hypothesis that there is no significant correlation between two variables using Pearson’s correlation coefficient:
H0: There is no significant correlation between the two variables (the correlation coefficient is zero).
H1: There is a significant correlation between the two variables (the correlation coefficient is not zero).
Reject the null hypothesis if the p-value is less than the significance level.
Fail to reject the null hypothesis if the p-value exceeds the significance level.
cor.test()# Load the mtcars dataset
data(mtcars)
# Perform a hypothesis test for the correlation between mpg and wt
cor.test(mtcars$mpg, mtcars$wt)
Pearson's product-moment correlation
data: mtcars$mpg and mtcars$wt
t = -9.559, df = 30, p-value = 1.294e-10
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.9338264 -0.7440872
sample estimates:
cor
-0.8676594 This will execute a hypothesis test and calculate the Pearson correlation coefficient, returning the test statistic, degrees of freedom, p-value, and correlation coefficient confidence interval.
We can reject the null hypothesis and determine that there is a significant correlation between the two variables if the p-value is smaller than the significance level.
corr.test() function, part of the psych package, can be used to perform hypothesis testing and confidence interval estimation for Pearson’s correlation coefficient.# Load the mtcars dataset
data(mtcars)
# Calculate the correlation between mpg and wt
library(psych)
corr.test(mtcars$mpg,
mtcars$wt,
method = "pearson",
use = "pairwise",
adjust = "none",
ci = TRUE,
alpha = 0.05)Call:corr.test(x = mtcars$mpg, y = mtcars$wt, use = "pairwise", method = "pearson",
adjust = "none", alpha = 0.05, ci = TRUE)
Correlation matrix
[1] -0.87
Sample Size
[1] 32
[1] 0
To see confidence intervals of the correlations, print with the short=FALSE optionThis will return the correlation coefficient, p-value, and confidence interval for the correlation between mpg and wt.
The output will also include information on the sample size, missing values, and adjustment method used.
To use the corr.test() function on a dataframe in R, it is helpful to first select the columns of interest and pass them as arguments to the function.
# Load the mtcars dataset
data(mtcars)
# Select the columns of interest
vars <- c("mpg", "disp", "hp", "wt")
# Calculate the correlation matrix and perform hypothesis tests
library(psych)
corr.test(mtcars[, vars],
method = "pearson",
use = "pairwise",
adjust = "none",
ci = TRUE,
alpha = 0.05)Call:corr.test(x = mtcars[, vars], use = "pairwise", method = "pearson",
adjust = "none", alpha = 0.05, ci = TRUE)
Correlation matrix
mpg disp hp wt
mpg 1.00 -0.85 -0.78 -0.87
disp -0.85 1.00 0.79 0.89
hp -0.78 0.79 1.00 0.66
wt -0.87 0.89 0.66 1.00
Sample Size
[1] 32
Probability values (Entries above the diagonal are adjusted for multiple tests.)
mpg disp hp wt
mpg 0 0 0 0
disp 0 0 0 0
hp 0 0 0 0
wt 0 0 0 0
To see confidence intervals of the correlations, print with the short=FALSE optionIn this example, the names of the relevant columns from the mtcars dataset are contained in a character vector in the vars variable. The mtcars dataset subset containing only these columns is subsequently submitted to the corr.test() function.
The result will be a correlation matrix with rows and columns matching to the relevant variables, as well as a set of hypothesis tests.
The hypothesis tests will report the correlation coefficient, p-value, and confidence range for each correlation, and the matrix will display the pairwise correlations between the variables.
corrgram# Load the mtcars dataset
data(mtcars)
# Select the columns of interest
vars <- c("mpg", "disp", "hp", "wt")
# Calculate the correlation matrix and perform hypothesis tests
library(corrgram)
# Create the corrgram plot
corrgram(mtcars[, vars],
order = TRUE,
lower.panel = panel.ellipse,
upper.panel = panel.cor)Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
corrplot# Load the mtcars dataset
data(mtcars)
library(corrplot)corrplot 0.92 loaded# Select the columns of interest
vars <- c("mpg", "disp", "hp", "wt")
M <- cor(mtcars[, vars])
corrplot(M, type="upper", order="hclust")
corrplot# Load the mtcars dataset
data(mtcars)
library(corrplot)
# Select the columns of interest
vars <- c("mpg", "disp", "hp", "wt")
M <- cor(mtcars[, vars])
corrplot(M, method="number")
Pearson’s correlation coefficient is a parametric measure of the linear relationship between two variables. It relies on several assumptions, including:
Linearity: The two variables’ relationship should be linear. The Pearson’s correlation coefficient may not adequately represent the strength of the association between the variables if the relationship is nonlinear.
Normality: The correlated variables should have a normal distribution. The reliability of Pearson’s correlation coefficient may be compromised if the variables are not normally distributed.
Homoscedasticity: All levels of the variables should have the same variance for the two variables. The Pearson correlation coefficient may be impacted by unequal variance, making the relationship between the variables look stronger or weaker than it actually is.
Independence: The two variables need to be unrelated to one another. The Pearson correlation coefficient may not effectively depict the relationship between the variables if the two variables are not independent.
Outliers: The correlation coefficient might be distorted by outliers, which can affect the outcomes. As a result, it is advised to check the data for outliers prior to determining the Pearson’s correlation coefficient.
Before using Pearson’s correlation coefficient, it is essential to validate these hypotheses to avoid incorrect interpretations of the data. Where these assumptions are not true, nonparametric correlation measurements, such as Spearman’s correlation coefficient, may be more appropriate.
A nonparametric indicator of the relationship between two variables is the Spearman correlation. It evaluates how well a monotonic function, which can only be strictly increasing or decreasing, can describe the relationship between two variables.
It is independent of the distributional assumptions underlying the correlated variables.
Instead, it computes the correlation coefficient based on the ranks of the variables, rather than their actual values.
# Load a dataset
data(mtcars)
# Calculate the Spearman correlation between mpg and wt
cor(mtcars$mpg,
mtcars$wt,
method = "spearman")[1] -0.886422[1]
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