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Bivariate analysis is a statistical method used to investigate the relationship between two variables. In R, it can be performed using the built-in functions such as cor(), cov(), and t.test(). These functions can help identify correlations between variables, calculate covariance, and perform a t-test to assess the significance of differences. Examples of these functions, along with their syntax and output, are provided below.
The term bivariate analysis refers to the analysis of two variables. You can remember this because the prefix “bi” means “two.”
The purpose of bivariate analysis is to understand the relationship between two variables
There are three common ways to perform bivariate analysis:
1. Scatterplots
2. Correlation Coefficients
3. Simple Linear Regression
The following example shows how to perform each of these types of bivariate analysis using the following dataset that contains information about two variables: (1) Hours spent studying and (2) Exam score received by 20 different students:
#create data frame df <- data.frame(hours=c(1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8), score=c(75, 66, 68, 74, 78, 72, 85, 82, 90, 82, 80, 88, 85, 90, 92, 94, 94, 88, 91, 96)) #view first six rows of data frame head(df) hours score 1 1 75 2 1 66 3 1 68 4 2 74 5 2 78 6 2 72
1. Scatterplots
We can use the following syntax to create a scatterplot of hours studied vs. exam score in R:
#create scatterplot of hours studied vs. exam score plot(df$hours, df$score, pch=16, col='steelblue', main='Hours Studied vs. Exam Score', xlab='Hours Studied', ylab='Exam Score')
The x-axis shows the hours studied and the y-axis shows the exam score received.
From the plot we can see that there is a positive relationship between the two variables: As hours studied increases, exam score tends to increase as well.
2. Correlation Coefficients
A Pearson Correlation Coefficient is a way to quantify the linear relationship between two variables.
We can use the cor() function in R to calculate the Pearson Correlation Coefficient between two variables:
#calculate correlation between hours studied and exam score received
cor(df$hours, df$score)
[1] 0.891306
The correlation coefficient turns out to be 0.891.
This value is close to 1, which indicates a strong positive correlation between hours studied and exam score received.
3. Simple Linear Regression
Simple linear regression is a statistical method we can use to find the equation of the line that best “fits” a dataset, which we can then use to understand the exact relationship between two variables.
We can use the lm() function in R to fit a for hours studied and exam score received:
#fit simple linear regression model fit <- lm(score ~ hours, data=df) #view summary of model summary(fit) Call: lm(formula = score ~ hours, data = df) Residuals: Min 1Q Median 3Q Max -6.920 -3.927 1.309 1.903 9.385 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 69.0734 1.9651 35.15 < 2e-16 *** hours 3.8471 0.4613 8.34 1.35e-07 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 4.171 on 18 degrees of freedom Multiple R-squared: 0.7944, Adjusted R-squared: 0.783 F-statistic: 69.56 on 1 and 18 DF, p-value: 1.347e-07
The fitted regression equation turns out to be:
Exam Score = 69.0734 + 3.8471*(hours studied)
This tells us that each additional hour studied is associated with an average increase of 3.8471 in exam score.
We can also use the fitted regression equation to predict the score that a student will receive based on their total hours studied.
For example, a student who studies for 3 hours is predicted to receive a score of 81.6147:
- Exam Score = 69.0734 + 3.8471*(hours studied)
- Exam Score = 69.0734 + 3.8471*(3)
- Exam Score = 81.6147
The following tutorials provide additional information about bivariate analysis: