Understanding Simple Linear Regression in R: A Comprehensive Guide
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Chapter 1: Introduction to Simple Linear Regression
Simple linear regression is a statistical approach that helps to describe the connection between a dependent variable (outcome) and an independent variable (predictor). This method is extensively utilized in predictive modeling, and R, equipped with powerful statistical tools, offers a user-friendly interface for conducting this analysis.
In this model, the relationship between the dependent variable ( Y ) and the independent variable ( X ) is represented through a linear equation:
[ Y = beta_0 + beta_1 X + epsilon ]
In this equation, ( beta_0 ) signifies the intercept, which is the anticipated value of ( Y ) when ( X ) is zero; ( beta_1 ) denotes the slope, indicating the change in ( Y ) with a one-unit change in ( X ). A negative slope suggests that as ( X ) increases, ( Y ) tends to decrease, while ( epsilon ) represents the error term. The aim is to estimate the parameters ( beta_0 ) and ( beta_1 ) that best represent the data.
Section 1.1: Conducting Simple Linear Regression in R
Let's delve into a practical example using R. We will examine whether height (in inches) can be forecasted based on age.
Download the data here.
Load the Data:
library(readxl)
age_and_height <- read_excel("ageandheight.xls") # Load the data
view(age_and_height) # Display the data
Exploring the Data
Before executing the regression, it's beneficial to visualize the data to understand the relationship between the variables and to identify any outliers.
Creating Boxplots to Identify Outliers:
par(mfrow=c(1, 2)) # Split the graph area into 2 columns
boxplot(age_and_height$age, main="Age", sub=paste("Outlier rows: ",
boxplot.stats(age_and_height$age)$out)) # Boxplot for age
boxplot(age_and_height$height, main="Height", sub=paste("Outlier rows: ",
boxplot.stats(age_and_height$height)$out)) # Boxplot for height
The boxplots for age and height indicate that there are no outliers in the dataset.
Creating a Scatterplot to Observe Linear Trends:
# Scatterplot
plot(age_and_height$height ~ age_and_height$age, xlab="Age",
ylab="Height", pch=16, col="red")
The scatterplot illustrates a discernible linear relationship between age and height.
Section 1.2: Fitting the Linear Regression Model
Next, we will utilize the lm function to fit a simple linear regression model where height is the dependent variable and age is the independent variable.
# Fit the model
lm_height = lm(height ~ age, data=age_and_height) # Create the linear regression model
summary(lm_height) # Review the results
To visualize the regression line on the plot:
# Plot with regression line
library(ggplot2)
ggplot(age_and_height, aes(x=age, y=height)) +
geom_point() +
geom_smooth(method="lm", se=FALSE, color="blue") +
labs(title="Linear Regression of Age on Height",
x="Age",
y="Height")
Assessing Model Fit
Before interpreting the results and utilizing the model for predictions, we should verify a crucial assumption of linear regression: that the residuals follow a normal distribution. This can be assessed using a QQ plot or a formal test like the Shapiro-Wilk test. Model fit diagnostic ==================================
plot(lm_height$residuals, pch=16, col="red")
qqnorm(lm_height$residuals, pch=16, col="red")
qqline(lm_height$residuals, pch=16, col="red")
shapiro.test(lm_height$residuals)
The points align closely with the straight line, except for two, indicating a satisfactory fit. The p-value from the Shapiro-Wilk test is 0.3, which is greater than 0.05, confirming that the residuals are normally distributed.
Making Predictions
We can leverage our fitted model to make predictions. For instance, let’s predict the height of a child aged 20.5 months. The output reveals the intercept (( alpha )) and slope (( beta )) values for age. For a 20.5-month-old child, ( alpha ) is 64.92 and ( beta ) is 0.635, leading to the prediction of height in centimeters as follows:
[ 64.92 + (0.635 times 20.5) = 77.94 , text{cm} ]
# Extract fitted values
lm_height$fitted.values
plot(age_and_height$height ~ lm_height$fitted.values, xlab="Predicted values",
ylab="Observed values", pch=16, col="red")
# Predict height for a 20.5-month-old child
new_data <- data.frame(age=20.5)
predict(lm_height, new_data) # 77.94 cm
Documentation of the Analysis
The analysis indicates that age is a significant predictor of height, with ( F(1,10) = 880 ), ( p < 0.0001 ). Age accounted for 98.8% of the variability in height (R² = 0.988). The regression equation derived is:
[ text{Height} = 64.92 + (0.635 times text{age}) ]
Chapter 2: Additional Resources
The first video provides a clear explanation of simple linear regression in R, guiding viewers through the concepts and code step-by-step.
The second video offers a detailed, step-by-step tutorial on performing linear regression in R, making it ideal for beginners and those looking to enhance their understanding.