Understanding Linear Regression and Residual Analysis

As a data analyst or machine learning practitioner, you have likely encountered linear regression and its importance in modeling relationships between variables. In this article, we will delve into the world of linear regression, explore how to create scatterplots of residuals, and understand the significance of residual analysis.

Introduction to Linear Regression

Linear regression is a statistical technique used to model the relationship between a dependent variable (y) and one or more independent variables (x). The goal is to find a linear equation that best predicts the value of y based on x. In this article, we will focus on simple linear regression using a linear model.

Key Concepts in Linear Regression

Before diving into residual analysis, let’s briefly review some key concepts:

• Intercept: The intercept (b0) represents the constant or baseline value of the dependent variable.
• Slope: The slope (b1) represents the rate of change between the independent variable and the dependent variable.
• Residuals: Residuals are the differences between observed values and predicted values.

Creating a Scatterplot of Residuals

The question posed in the Stack Overflow post revolves around creating a scatterplot of residuals, with different colors for females and males. To achieve this, we need to assign the residuals from our linear model as a new column in the original data frame.

Assigning Residuals to a New Column

To create such a plot, you should assign the residuals from your model to a new column within the intp.trust data frame:

# Get the residuals from the model
lm.res <- resid(lm.fit)

# Add the residuals as a new column in intp.trust
intp.trust$lm.res <- lm.res

Creating the Scatterplot

Now that we have the residuals assigned to a new column, we can create a scatterplot with ggplot:

## Create a scatterplot of residuals with different colors for females and males
ggplot(intp.trust, aes(x = intp.trust$lm.res, y = intp.trust$v225, color = factor(intp.trust$v225))) +
  geom_point()

However, this code snippet uses v225 to create the plot. To obtain the scatter plot that differentiates between females and males using v225, we must correctly identify whether each data point corresponds to a male or female.

Identifying Males and Females in v225

If you want the scatterplot to show differences for males and females, it is crucial that you understand how v225 was coded. Since it was used as a binary variable (0/1) where 1 meant a female and 0 represented a male, we should convert v225 into a factor and use this factor in our scatterplot.

## Create the scatterplot with correct color coding for males and females
ggplot(intp.trust, aes(x = intp.trust$lm.res, y = intp.trust$v225, color = factor(intp.trust$v225))) +
  geom_point()

Adding a Layer to Highlight Females

To highlight the data points of interest (females) and distinguish them from males in your scatterplot, you can use the alpha parameter:

## Create the scatterplot with correct color coding for males and females
ggplot(intp.trust, aes(x = intp.trust$lm.res, y = intp.trust$v225, color = factor(intp.trust$v225))) +
  geom_point(alpha=0.5) # this makes female points semi-transparent.

Example Code with Dummy Data

Let’s explore an example using dummy data to better illustrate the concepts.

Generating Dummy Data

Here is a simple function that generates dummy data for demonstration purposes:

# Function to generate true values based on gender and x
true_function <- function(x, is_female) {
  ifelse(is_female, 5, 2) +
    ifelse(is_female, -1.5, 1.5) * x +
    rnorm(length(x))
}

set.seed(123)
dat <- data.frame(
  x = runif(200, 1, 5), # Independent variable
  is_female = rbinom(200, 1, .5), # Binary variable for gender (male/female)
)

# Generate y values based on true function
dat$y <- with(dat, true_function(x, is_female))

# Regression model
lm_fit <- lm(y ~ x + as.factor(is_female), data=dat) # Fit the linear regression model

# Calculate residuals from the model
dat$resid <- resid(lm_fit)

# Scatterplot of residuals
ggplot(dat, aes(x=x, y=resid, color=as.factor(is_female))) +
  geom_point() # Plot points for males and females on scatter plot.

This dummy data example highlights how you can create a scatterplot that shows differences between males and females using v225. The use of the alpha parameter in our code allows us to see semi-transparent female points.

Conclusion

To summarize, understanding residual analysis is key when working with linear regression models. By following these steps and creating your own example plots with dummy data, you can gain a deeper insight into how residuals are used in predictive modeling.

This article has walked through the process of assigning residuals from a model to a new column within the original dataset and using ggplot2 to create scatterplots that show differences between genders.

Last modified on 2024-01-04