Correlation and Regression with R

Correlation and regression analysis are two fundamental statistical techniques used to examine the relationships between variables. R Programming Language is a programming language and environment for statistical computing and graphics, making it an excellent choice for conducting these analyses.

1. Correlation Analysis

Correlation analysis is a statistical technique used to measure the strength and direction of the relationship between two continuous variables. The most common measure of correlation is the Pearson correlation coefficient. It quantifies the linear relationship between two variables. The Pearson correlation coefficient, denoted as "r" :

Parameters:

• : Correlation coefficient 
• : value first dataset
• : Mean of first dataset
• : value second dataset
• : Mean of second dataset

It can take values between -1 (perfect negative correlation) and 1 (perfect positive correlation), with 0 indicating no linear correlation.

1.1. Correlation using R

We calculate the Pearson correlation between study hours and exam scores to understand the strength of their linear relationship.

• study_hours: A numeric vector storing the number of hours studied.
• exam_scores: A numeric vector storing corresponding exam scores.
• cor: Computes the Pearson correlation coefficient between two numeric vectors.
• correlation: Stores the resulting correlation value which indicates the strength and direction of the linear relationship.

Output:

[1] 0.9569094

1.2. Visualize the data and correlation

We create a scatterplot to visualize the relationship between study hours and exam scores and overlay a regression line to show the trend along with the correlation value.

• plot: Displays a scatterplot of exam scores against study hours.
• main: Sets the title of the scatterplot.
• abline: Adds a straight line to the plot, typically used to display a regression line.
• lm: Fits a linear regression model to the data.
• col: Sets the color of the regression line.
• text: Places the correlation value as text at specified coordinates on the plot.
• paste: Combines text and numeric values into a single string.
• round: Rounds the correlation value to a specified number of decimal places.

Output:

The scatterplot shows a positive linear trend, meaning as study hours increase, exam scores tend to rise. The regression line quantifies this relationship.

2. Regression Analysis

Regression analysis is used to model the relationship between one or more independent variables and a dependent variable. In simple linear regression, there is one independent variable, while in multiple regression there are multiple independent variables. The goal is to find a linear equation that best fits the data.

There are two types of Regression analysis.

1. Simple Linear Regression
2. Multiple Linear Regression

2.1. Simple Linear Regression in R

We use simple linear regression to predict exam scores based on study hours.

• lm: Used to fit a linear regression model.
• summary: Displays detailed regression results including coefficients, standard errors and R-squared values.

Output:

2.2. Visualize the data and regression line

We use a scatterplot to visualize the relationship between study hours and exam scores, then add a regression line to show the model's prediction.

Output:

The results helps us analyze how study hours influence exam scores and provides a quantitative model for prediction.

2.3. Multiple Linear Regression Example in R

We build a multiple linear regression model using the mtcars dataset to predict mpg based on wt, hp, qsec and am.

• data: Loads the specified dataset into memory.
• mtcars: Built-in dataset in R containing specifications of various car models.
• lm: Fits a linear regression model.
• mpg ~ wt + hp + qsec + am: Formula used in the model indicating mpg is predicted by weight, horsepower, quarter mile time and transmission.
• summary: Provides detailed statistics of the regression model including coefficients, R-squared value and p-values.

Output:

2.4. Visualize the data and regression line

We create two plots side by side where one is showing the relationship between car weight and MPG and the other comparing actual and predicted MPG values.

• par: Sets the layout for displaying multiple plots in one graphic window.
• plot: Draws a scatterplot to visualize data points.
• abline: Adds a straight reference or regression line to a plot.
• regression_model: Stores the linear regression result used for prediction.
• predict: Computes predicted values using the fitted regression model.

Output:

The interpretation of coefficients and visualizations is help us to understand the impact of each variable on the dependent variable (MPG).

Difference between Correlation and Regression Analysis

Correlation and regression analysis are both statistical techniques used to explore relationships between variables, but they serve different purposes and provide distinct types of information in R.