SST, SSR, and SSE Calculations in R: A Comprehensive Guide

Introduction

This comprehensive guide demonstrates various methods to calculate Sum of Squares components (SST, SSR, and SSE) in R. We’ll explore implementations using base R, tidyverse, and the stats package, providing clear examples and visualizations for each approach.

Mathematical Foundations

Total Sum of Squares (SST)

SST measures the total variation in the dependent variable (y) around its mean. It represents the total amount of variability in the data:

Formula:

\[ SST = \sum(y_i – \bar{y})^2 \]

Where:

• \( y_i \) = each observed value
• \( \bar{y} \) = the mean of all observed values

This value is always positive because it sums the squared differences between the observed values and their mean.

Regression Sum of Squares (SSR)

SSR quantifies the variation explained by the regression model:

Formula:

\[ SSR = \sum(\hat{y}_i – \bar{y})^2 \]

Where:

• \( \hat{y}_i \) = each predicted value
• \( \bar{y} \) = the mean of the observed values

A higher SSR indicates that the regression model explains a large proportion of the variability in the data.

Error Sum of Squares (SSE)

SSE measures the unexplained variation:

Formula:

\[ SSE = \sum(y_i – \hat{y}_i)^2 \]

Where:

• \( y_i \) = each observed value
• \( \hat{y}_i \) = each predicted value

Lower SSE indicates better model fit.

The Fundamental Relationship

\[ SST = SSR + SSE \]

This relationship shows how total variation splits between explained and unexplained components.

Coefficient of Determination (R²)

\[ R^2 = \frac{SSR}{SST} = 1 – \frac{SSE}{SST} \]

Interpretation:

• \( R^2 = 1 \): Perfect model fit
• \( R^2 = 0 \): Model explains no variance
• Higher \( R^2 \) indicates better fit

Implementation in Base R

Base R provides a straightforward way to compute Sum of Squares components (SST, SSR, and SSE) using simple functions and operations. This section walks you through a step-by-step implementation using an example dataset.

Step-by-Step Code

# Step 1: Create the dataset
hours <- c(2, 4, 6, 8, 10)  # Independent variable
scores <- c(65, 75, 85, 90, 95)  # Dependent variable

# Step 2: Fit the regression model
model <- lm(scores ~ hours)  # Linear regression model

# Step 3: Compute the mean of the dependent variable
y_mean <- mean(scores)

# Step 4: Get the predicted values from the model
y_pred <- predict(model)

# Step 5: Calculate SST (Total Sum of Squares)
sst <- sum((scores - y_mean)^2)

# Step 6: Calculate SSR (Regression Sum of Squares)
ssr <- sum((y_pred - y_mean)^2)

# Step 7: Calculate SSE (Error Sum of Squares)
sse <- sum((scores - y_pred)^2)

# Step 8: Compute R-squared
r_squared <- ssr / sst

# Step 9: Print results
cat("SST:", sst, "\n")
cat("SSR:", ssr, "\n")
cat("SSE:", sse, "\n")
cat("R-squared:", r_squared, "\n")

Detailed Explanation

Let's break down each step in the implementation:

• Step 1: Create the dataset

  The variables hours (independent variable) and scores (dependent variable) represent the study hours and test scores, respectively.

• Step 2: Fit the regression model

  The lm() function fits a linear regression model to the data, with scores as the dependent variable and hours as the independent variable.

• Step 3: Compute the mean of the dependent variable

  The mean of the dependent variable scores is calculated using mean(). This value (\( \bar{y} \)) is used in SST and SSR calculations.

• Step 4: Get the predicted values

  The predicted values (\( \hat{y}_i \)) are obtained using the predict() function, which applies the fitted model to the data.

• Step 5: Calculate SST

  SST is computed as the sum of squared differences between the observed values and their mean: \[ SST = \sum(y_i - \bar{y})^2 \]

• Step 6: Calculate SSR

  SSR is the sum of squared differences between the predicted values and the mean of the observed values: \[ SSR = \sum(\hat{y}_i - \bar{y})^2 \]

• Step 7: Calculate SSE

  SSE is the sum of squared differences between the observed and predicted values: \[ SSE = \sum(y_i - \hat{y}_i)^2 \]

• Step 8: Compute \( R^2 \)

  The coefficient of determination (\( R^2 \)) is calculated as: \[ R^2 = \frac{SSR}{SST} \]

• Step 9: Print results

  The results for SST, SSR, SSE, and \( R^2 \) are displayed using the cat() function.

Example Output

Key Takeaways

• Base R provides all the tools needed to compute SST, SSR, and SSE without relying on additional libraries.
• The relationship \( SST = SSR + SSE \) holds true, confirming the calculations.
• High \( R^2 \) values (close to 1) indicate that the model explains most of the variability in the data.

Implementation with Tidyverse

The Tidyverse is a collection of R packages designed for data science workflows, providing an intuitive and pipeline-friendly approach to data manipulation and analysis. In this section, we’ll compute SST, SSR, and SSE using Tidyverse functions.

Step-by-Step Code

# Step 1: Load the required library
library(tidyverse)

# Step 2: Create a tibble with the data
df <- tibble(
  hours = c(2, 4, 6, 8, 10),  # Independent variable
  scores = c(65, 75, 85, 90, 95)  # Dependent variable
)

# Step 3: Fit the regression model
model_tidy <- lm(scores ~ hours, data = df)

# Step 4: Add predictions and components to the tibble
df <- df %>%
  mutate(
    predicted = predict(model_tidy),         # Predicted values
    mean_score = mean(scores),               # Mean of observed values
    sst_comp = (scores - mean_score)^2,      # SST components
    ssr_comp = (predicted - mean_score)^2,   # SSR components
    sse_comp = (scores - predicted)^2        # SSE components
  )

# Step 5: Calculate SST, SSR, SSE, and R-squared
results <- df %>%
  summarise(
    sst = sum(sst_comp),                     # Total Sum of Squares
    ssr = sum(ssr_comp),                     # Regression Sum of Squares
    sse = sum(sse_comp),                     # Error Sum of Squares
    r_squared = ssr / sst                    # Coefficient of Determination
  )

# Step 6: Print results
print(results)

Detailed Explanation

Here’s a breakdown of the Tidyverse workflow:

• Step 1: Load the Tidyverse library

  The tidyverse package includes functions for data manipulation (dplyr) and creating tibbles (tibble), among others. Load it using library(tidyverse).

• Step 2: Create a tibble

  A tibble is a modern version of a dataframe, offering better printing and compatibility with Tidyverse functions. The tibble contains two columns: hours (independent variable) and scores (dependent variable).

• Step 3: Fit the regression model

  The lm() function fits a linear regression model. Here, scores is the dependent variable, and hours is the independent variable.

• Step 4: Add predictions and components

  Using mutate(), new columns are added to the tibble:

  • predicted: Predicted values (\( \hat{y}_i \)) from the regression model.
  • mean_score: Mean of the dependent variable (\( \bar{y} \)).
  • sst_comp: Components for SST (\( (y_i - \bar{y})^2 \)).
  • ssr_comp: Components for SSR (\( (\hat{y}_i - \bar{y})^2 \)).
  • sse_comp: Components for SSE (\( (y_i - \hat{y}_i)^2 \)).
• Step 5: Summarise results

  The summarise() function computes the final values of SST, SSR, SSE, and \( R^2 \) by summing their respective components and performing the required calculations.

• Step 6: Print results

  The print() function displays the calculated values in the console.

Example Output

Key Takeaways

• The Tidyverse provides a pipeline-friendly approach, making the code cleaner and easier to follow.
• Intermediate calculations (e.g., SST, SSR, SSE components) are stored directly in the tibble, allowing for quick inspection or debugging.
• The relationship \( SST = SSR + SSE \) holds true, as verified by the results.
• Using Tidyverse functions like mutate() and summarise() simplifies complex workflows.

Implementation with stats Package

The stats package is a core part of R and includes functions for fitting linear models, performing statistical analyses, and calculating various components of regression analysis. In this section, we’ll compute SST, SSR, SSE, and \( R^2 \) using the stats package.

Step-by-Step Code

# Step 1: Create the dataset
hours <- c(2, 4, 6, 8, 10)  # Independent variable
scores <- c(65, 75, 85, 90, 95)  # Dependent variable

# Step 2: Fit the regression model
model_stats <- lm(scores ~ hours)  # Linear regression model

# Step 3: Extract model summary
summary_stats <- summary(model_stats)

# Step 4: Use ANOVA to calculate SST, SSR, and SSE
anova_results <- anova(model_stats)

sst <- sum(anova_results["Sum Sq"])           # Total Sum of Squares
ssr <- anova_results["Sum Sq"][1, ]          # Regression Sum of Squares
sse <- anova_results["Sum Sq"][2, ]          # Error Sum of Squares

# Step 5: Extract R-squared from the model summary
r_squared <- summary_stats$r.squared

# Step 6: Print results
cat("SST:", sst, "\n")
cat("SSR:", ssr, "\n")
cat("SSE:", sse, "\n")
cat("R-squared:", r_squared, "\n")

Detailed Explanation

Here’s a breakdown of each step in the implementation:

• Step 1: Create the dataset

  The variables hours (independent variable) and scores (dependent variable) represent the study hours and test scores, respectively.

• Step 2: Fit the regression model

  The lm() function from the stats package fits a linear regression model. Here, scores is the dependent variable, and hours is the independent variable.

• Step 3: Extract model summary

  The summary() function provides detailed information about the regression model, including \( R^2 \), coefficients, standard errors, t-values, and p-values.

• Step 4: Use ANOVA to calculate SST, SSR, and SSE

  The anova() function performs an analysis of variance on the fitted model and returns a table with the following key values:

  • Sum Sq: The sum of squares for the regression and residual components.
  • sst: The total sum of squares is the sum of all Sum Sq values.
  • ssr: The first value under Sum Sq, representing the regression sum of squares.
  • sse: The second value under Sum Sq, representing the error sum of squares.
• Step 5: Extract \( R^2 \) from the model summary

  The coefficient of determination (\( R^2 \)) is directly available in the model summary under r.squared.

• Step 6: Print results

  The cat() function displays the calculated values for SST, SSR, SSE, and \( R^2 \) in the console.

Example Output

Key Takeaways

• The anova() function simplifies the computation of SST, SSR, and SSE by providing the necessary components directly.
• The relationship \( SST = SSR + SSE \) is verified through the results.
• High \( R^2 \) values (close to 1) indicate that the model explains most of the variability in the data.
• The stats package is part of R’s base installation, making it a reliable and efficient option for regression analysis.

Visualization

To better understand the components of regression analysis, let's visualize the data points, regression line, and the components SST, SSR, and SSE. Below is an R script to plot the three subplots showing each component with the observed data, the fitted line, and the mean.

# If required, install the gridExtra package
if (!requireNamespace("gridExtra", quietly = TRUE)) {
  install.packages("gridExtra")
}

# Load required libraries
library(ggplot2)
library(gridExtra)

# Step 1: Create dataset
hours <- c(2, 4, 6, 8, 10)
scores <- c(65, 75, 85, 90, 95)
model <- lm(scores ~ hours)
predicted <- predict(model)
mean_score <- mean(scores)

# Step 2: Prepare data for visualization
df <- data.frame(hours, scores, predicted)

# Step 3: Plot SSR
plot_ssr <- ggplot(df, aes(x = hours)) +
  geom_point(aes(y = scores), color = "blue", size = 3, label = "Observed Data") +
  geom_line(aes(y = predicted), color = "red", size = 1, label = "Regression Line") +
  geom_hline(yintercept = mean_score, color = "#006400", linetype = "dashed", label = "Mean Line") +
  geom_segment(aes(xend = hours, y = mean_score, yend = predicted),
               color = "red", linetype = "dotdash", alpha = 0.7) +
  ggtitle("Regression Sum of Squares (SSR)") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5)) +  # Center the title
  ylab("Test Scores")

# Step 4: Plot SSE
plot_sse <- ggplot(df, aes(x = hours)) +
  geom_point(aes(y = scores), color = "blue", size = 3, label = "Observed Data") +
  geom_line(aes(y = predicted), color = "red", size = 1, label = "Regression Line") +
  geom_hline(yintercept = mean_score, color = "#006400", linetype = "dashed", label = "Mean Line") +
  geom_segment(aes(xend = hours, y = predicted, yend = scores),
               color = "blue", linetype = "solid", alpha = 0.7) +
  ggtitle("Error Sum of Squares (SSE)") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5)) +  # Center the title
  ylab("Test Scores")

# Step 5: Plot SST
plot_sst <- ggplot(df, aes(x = hours)) +
  geom_point(aes(y = scores), color = "blue", size = 3, label = "Observed Data") +
  geom_hline(yintercept = mean_score, color = "#006400", linetype = "dashed", label = "Mean Line") +
  geom_segment(aes(xend = hours, y = mean_score, yend = scores),
               color = "#006400", linetype = "dotted", alpha = 0.7) +
  ggtitle("Total Sum of Squares (SST)") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5)) +  # Center the title
  xlab("Study Hours") +
  ylab("Test Scores")

# Step 6: Combine all plots
grid.arrange(plot_ssr, plot_sse, plot_sst, ncol = 1)

Complete R Implementation

This section combines everything covered so far into a single, complete workflow. The script calculates SST, SSR, SSE, and \( R^2 \), visualizes the components, and outputs a clean summary of results. This implementation is modular, making it easy to adapt to other datasets.

# Step 1: Load required libraries
if (!requireNamespace("gridExtra", quietly = TRUE)) {
  install.packages("gridExtra")
}
library(ggplot2)
library(gridExtra)
library(tidyverse)

# Step 2: Define a function for analysis
analyze_regression <- function(data, x_var, y_var) {
  # Fit model
  formula <- as.formula(paste(y_var, "~", x_var))
  model <- lm(formula, data = data)

  # Add predictions and calculations to the data
  data <- data %>%
    mutate(
      predicted = predict(model),
      mean_y = mean(!!sym(y_var)),
      sst_comp = (!!sym(y_var) - mean_y)^2,
      ssr_comp = (predicted - mean_y)^2,
      sse_comp = (!!sym(y_var) - predicted)^2
    )

  # Summarize results
  results <- data %>%
    summarise(
      sst = sum(sst_comp),
      ssr = sum(ssr_comp),
      sse = sum(sse_comp),
      r_squared = ssr / sst
    )

  # Create visualization plots
  plot_ssr <- ggplot(data, aes(x = !!sym(x_var))) +
    geom_point(aes(y = !!sym(y_var)), color = "blue", size = 3) +
    geom_line(aes(y = predicted), color = "red", size = 1) +
    geom_hline(yintercept = mean(data[[y_var]]), color = "#006400", linetype = "dashed") +
    geom_segment(aes(xend = !!sym(x_var), y = mean_y, yend = predicted),
                 color = "red", linetype = "dotdash", alpha = 0.7) +
    ggtitle("Regression Sum of Squares (SSR)") +
    theme_minimal() +
    theme(plot.title = element_text(hjust = 0.5)) +
    ylab(y_var)

  plot_sse <- ggplot(data, aes(x = !!sym(x_var))) +
    geom_point(aes(y = !!sym(y_var)), color = "blue", size = 3) +
    geom_line(aes(y = predicted), color = "red", size = 1) +
    geom_hline(yintercept = mean(data[[y_var]]), color = "#006400", linetype = "dashed") +
    geom_segment(aes(xend = !!sym(x_var), y = predicted, yend = !!sym(y_var)),
                 color = "blue", linetype = "solid", alpha = 0.7) +
    ggtitle("Error Sum of Squares (SSE)") +
    theme_minimal() +
    theme(plot.title = element_text(hjust = 0.5)) +
    ylab(y_var)

  plot_sst <- ggplot(data, aes(x = !!sym(x_var))) +
    geom_point(aes(y = !!sym(y_var)), color = "blue", size = 3) +
    geom_hline(yintercept = mean(data[[y_var]]), color = "#006400", linetype = "dashed") +
    geom_segment(aes(xend = !!sym(x_var), y = mean_y, yend = !!sym(y_var)),
                 color = "#006400", linetype = "dotted", alpha = 0.7) +
    ggtitle("Total Sum of Squares (SST)") +
    theme_minimal() +
    theme(plot.title = element_text(hjust = 0.5)) +
    xlab(x_var) +
    ylab(y_var)

  # Combine all plots
  combined_plot <- grid.arrange(plot_ssr, plot_sse, plot_sst, ncol = 1)

  # Return results
  list(
    model = model,
    results = results,
    plots = combined_plot,
    data = data
  )
}

# Step 3: Example usage
df <- tibble(
  hours = c(2, 4, 6, 8, 10),
  scores = c(65, 75, 85, 90, 95)
)

analysis <- analyze_regression(df, "hours", "scores")
print(analysis$results)

Detailed Explanation

The implementation includes three key components:

• Data Preparation: The function uses mutate() to calculate SST, SSR, and SSE components directly in the data.
• Visualization: Subplots for SST, SSR, and SSE are created using ggplot2, with the regression line, mean line, and appropriate vertical lines to represent components.
• Results Summary: The summarise() function calculates the total SST, SSR, SSE, and \( R^2 \), which are returned as a summary table.

Example Output

Key Takeaways

• This function encapsulates the entire workflow, making it reusable for any dataset.
• Visualizations are modular and highlight the relationships between SST, SSR, and SSE.
• The calculated results verify the relationship \( SST = SSR + SSE \) and provide a clear measure of \( R^2 \).
• The modular approach makes it easy to extend the function with additional analysis or visualizations.

Using car Package

The car (Companion to Applied Regression) package in R provides advanced tools for regression analysis, including ANOVA tables and diagnostic methods. This section demonstrates how to use the car package to compute SST, SSR, and SSE efficiently and generate ANOVA summaries.

Step-by-Step Code

# If required, install the car package
if (!requireNamespace("car", quietly = TRUE)) {
  install.packages("car")
}

# Load the car package
library(car)

# Step 1: Create the dataset
hours <- c(2, 4, 6, 8, 10)
scores <- c(65, 75, 85, 90, 95)

# Step 2: Fit the regression model
model_car <- lm(scores ~ hours)

# Step 3: Compute ANOVA table
anova_table <- Anova(model_car, type = "II")

# Step 4: Extract SST, SSR, and SSE
sst <- sum(anova_table$`Sum Sq`)  # Total Sum of Squares
ssr <- anova_table$`Sum Sq`[1]   # Regression Sum of Squares
sse <- anova_table$`Sum Sq`[2]   # Error Sum of Squares

# Step 5: Compute R-squared
r_squared <- summary(model_car)$r.squared

# Step 6: Print results
cat("SST:", sst, "\n")
cat("SSR:", ssr, "\n")
cat("SSE:", sse, "\n")
cat("R-squared:", r_squared, "\n")

# Step 7: Display the ANOVA table
print(anova_table)

Detailed Explanation

The `car` package simplifies the calculation of sum of squares and provides a more detailed ANOVA output. Here’s a breakdown of the workflow:

• Step 1: Create the dataset

  The variables hours and scores represent the independent and dependent variables, respectively.

• Step 2: Fit the regression model

  The lm() function is used to fit a linear regression model. Here, scores is modeled as a function of hours.

• Step 3: Compute the ANOVA table

  The Anova() function generates an ANOVA table, which includes the sum of squares for each term in the model.

• Step 4: Extract SST, SSR, and SSE

  The total sum of squares (SST) is the sum of all Sum Sq values in the ANOVA table. The regression sum of squares (SSR) corresponds to the first row of the table, while the error sum of squares (SSE) corresponds to the second row.

• Step 5: Compute \( R^2 \)

  The coefficient of determination (\( R^2 \)) is calculated directly from the model summary using summary(model_car)$r.squared.

• Step 6: Print results

  The results for SST, SSR, SSE, and \( R^2 \) are displayed in the console using the cat() function.

• Step 7: Display the ANOVA table

  The print() function outputs the detailed ANOVA table, which provides additional insights into the model terms and their contributions to the overall variability.

Example Output

Understanding the ANOVA Table

The ANOVA table generated by the car package provides a breakdown of the variability in the data. Here’s what each term represents:

• Sum Sq:

  The sum of squares for each term in the model. For example:

  • hours: Regression Sum of Squares (SSR), the variation explained by the predictor variable.
  • Residuals: Error Sum of Squares (SSE), the variation unexplained by the model.

  The total sum of squares (SST) is the sum of these values: \( SST = SSR + SSE \).

• Df:

  Degrees of freedom associated with each term. For example:

  • hours: 1 degree of freedom because there’s one predictor variable.
  • Residuals: 3 degrees of freedom, calculated as \( n - k - 1 \), where \( n \) is the number of observations, and \( k \) is the number of predictors.
• F value:

  The F-statistic tests whether the predictor variable significantly contributes to explaining the variability in the response variable. A higher F-value indicates greater significance.

• Pr(>F):

  The p-value associated with the F-statistic. It indicates the probability of observing an F-value as extreme as the calculated value under the null hypothesis (i.e., the predictor variable has no effect). Small p-values (e.g., < 0.05) indicate statistical significance.

• Significance Codes:

  A shorthand interpretation of the p-value:

  • ***: Highly significant (p < 0.001)
  • **: Significant (p < 0.01)
  • *: Moderate significance (p < 0.05)
  • .: Weak significance (p < 0.1)
  • Blank: Not significant (p ≥ 0.1)

Together, these components provide a detailed view of how the predictor variable contributes to the variability in the response variable and the overall model fit.

Key Takeaways

• The car package simplifies regression analysis by providing detailed ANOVA tables.
• The relationship \( SST = SSR + SSE \) is verified using the ANOVA table output.
• The \( R^2 \) value confirms the proportion of variability explained by the model.
• This method is particularly useful for models with multiple terms, as it partitions the sum of squares by term.

Using modelr Package

The modelr package is part of the Tidyverse ecosystem and provides utilities for working with models in a pipeline-friendly way. It allows you to easily add predictions, residuals, and other calculations to your dataset for further analysis and visualization. This section demonstrates how to calculate SST, SSR, and SSE using modelr.

Step-by-Step Code

# If required, install the modelr package
if (!requireNamespace("modelr", quietly = TRUE)) {
  install.packages("modelr")
}

# Load required libraries
library(modelr)
library(tidyverse)

# Step 1: Create the dataset
df <- tibble(
  hours = c(2, 4, 6, 8, 10),
  scores = c(65, 75, 85, 90, 95)
)

# Step 2: Fit the regression model
model <- lm(scores ~ hours, data = df)

# Step 3: Add predictions and residuals to the dataset
df <- df %>%
  add_predictions(model) %>%    # Adds predicted values as a column
  add_residuals(model)          # Adds residuals (y - y_pred) as a column

# Step 4: Calculate SST, SSR, and SSE
sst <- df %>%
  summarise(sst = sum((scores - mean(scores))^2)) %>%
  pull(sst)

ssr <- df %>%
  summarise(ssr = sum((pred - mean(scores))^2)) %>%
  pull(ssr)

sse <- df %>%
  summarise(sse = sum(resid^2)) %>%
  pull(sse)

# Step 5: Compute R-squared
r_squared <- ssr / sst

# Step 6: Print results
cat("SST:", sst, "\n")
cat("SSR:", ssr, "\n")
cat("SSE:", sse, "\n")
cat("R-squared:", r_squared, "\n")

Detailed Explanation

Here’s a breakdown of each step in the workflow:

• Step 1: Create the dataset

  A tibble is created containing the independent variable (hours) and the dependent variable (scores).

• Step 2: Fit the regression model

  The lm() function fits a linear regression model, where scores is the response variable and hours is the predictor.

• Step 3: Add predictions and residuals

  The add_predictions() function appends a column to the dataset containing the predicted values (\( \hat{y}_i \)) for each observation. The add_residuals() function appends another column with the residuals (\( y_i - \hat{y}_i \)), which represent the errors.

• Step 4: Calculate SST, SSR, and SSE

  Using the updated dataset, the components are calculated as follows:

  • sst: Total sum of squares, calculated as the sum of squared differences between the observed values and their mean.
  • ssr: Regression sum of squares, calculated as the sum of squared differences between the predicted values and the mean of the observed values.
  • sse: Error sum of squares, calculated as the sum of squared residuals.
• Step 5: Compute R-squared

  The coefficient of determination (\( R^2 \)) is computed as the proportion of the total variability explained by the model: \( R^2 = \text{SSR} / \text{SST} \).

• Step 6: Print results

  The calculated values are displayed in the console using the cat() function.

Example Output

Key Takeaways

• The modelr package integrates seamlessly with the Tidyverse, making it easy to add predictions and residuals to datasets.
• The calculated components verify the relationship \( SST = SSR + SSE \), and the \( R^2 \) value confirms the proportion of variance explained by the model.
• The pipeline-friendly approach minimizes code duplication and enhances readability.
• This method can be easily extended to include more diagnostic calculations or visualizations.

Conclusion

R offers diverse and powerful methods to calculate and analyze the sum of squares components in regression analysis. From base R to Tidyverse and specialized packages like car and modelr, you can tailor your approach based on your workflow preferences and project requirements. These methods enable precise evaluation of model performance through SST, SSR, and SSE.

If you'd like to explore more about regression analysis or try out our interactive tools, visit the Further Reading section for additional resources and calculators.

Have fun and happy learning!