README for Assignment 1: Model Selection in R

Data Information

• Data File: jobs.txt
• Columns:
  • y: Job proficiency score (first column)
  • t1, t2, t3, t4: Scores on four aptitude tests
• Ensure to assign appropriate column headings after importing the data.

Assignment Tasks 1

1. Graphical Summaries: Generate scatterplots of job proficiency against each predictor and interpret what the plots suggest.
2. All Possible Regressions: Fit all 16 possible models with the four predictors and record model selection metrics: p, R2, R2a,p, PRESSp, AICp, BICp, Mallows Cp.
3. Best Model Selection: Identify the best model based on each criterion and determine which variable(s) can be excluded.
4. Model Fitting & Diagnostics: Fit the best model according to R2a,p and assess if model assumptions are met.
5. Best Model with Two Predictors: Using BICp and Mallows Cp, find the best model with two or fewer predictors.
6. Validation: Compare SSEp and PRESSp for Model 11.
7. Automated Search Procedures: Perform forward selection, backward elimination, and stepwise regression using the step() function and compare selected models.
8. Model Search with Conditions: Use the regsubsets() function from the leaps package to find the best models based on R2a,p, Mallows Cp, and BICp.

Package Requirements

• DAAG Package: For computing PRESSp
• Leaps Package: For regsubsets() function

Additional Notes

• Ensure all plots and results are properly labeled and interpreted.
• Check that your models meet statistical assumptions and provide diagnostics when needed.
• Maintain clear and concise code structure and comments for readability.

Assignment Tasks 2 Outlier Detection in Grocery Data using R

This project aims to detect influential outliers in grocery data using R. The dataset contains weekly activity data from a national grocery retailer, and the analysis focuses on identifying anomalies in the response and predictor variables.

Dataset Overview

• File: grocery.xlsx
• Variables:
  • labor: Total labor hours per week (response variable)
  • shipped: Number of cases shipped in a week
  • cost: Labor cost as a percentage of total costs
  • holiday: Binary indicator (1 if the week includes a holiday, 0 otherwise)

Analysis Steps

1. Residual Analysis

We fit a first-order linear regression model without interaction effects and analyze different types of residuals:

library(readxl)
data <- read_excel("grocery.xlsx")
result <- lm(labor ~ shipped + cost + holiday, data)

# Visualizing residuals
par(mfrow = c(1, 3))

# Plot of residuals vs fitted values
plot(result$fitted.values, result$residuals,
     main = "Residuals vs Fitted",
     xlab = "Fitted Values", ylab = "Residuals")

# Plot of standardized residuals vs fitted values
plot(result$fitted.values, rstandard(result),
     main = "Standardized Residuals vs Fitted",
     xlab = "Fitted Values", ylab = "Standardized Residuals")

# Plot of studentized residuals vs fitted values
plot(result$fitted.values, rstudent(result),
     main = "Studentized Residuals vs Fitted",
     xlab = "Fitted Values", ylab = "Studentized Residuals")

# Reset graphics layout
par(mfrow = c(1, 1))

2. Detecting Outliers in the Response Variable

We use studentized residuals and the Bonferroni correction to identify potential outliers:

# Calculate sample size and number of predictors
n <- nrow(data)
p <- length(coef(result)) - 1

# Compute studentized residuals
student.res <- rstudent(result)

# Calculate critical value using Bonferroni correction
alpha <- 0.05
crit <- qt(1 - alpha / (2 * n), df = n - p - 1)

# Plot studentized residuals with critical thresholds
plot(student.res,
     main = "Studentized Residuals with Critical Values",
     ylab = "Studentized Residuals")
abline(h = c(crit, -crit), col = "red", lty = 2)

# Identify outliers
outliers <- which(abs(student.res) > crit)
print("Indices of potential outliers:")
print(outliers)

How to Run the Analysis

1. Ensure that the grocery.xlsx file is in your working directory.
2. Open the R environment (e.g., RStudio, VSCode) and load the provided scripts.
3. Execute the code blocks in the order presented.

Results and Observations

• Visual analysis: Residual plots highlight any patterns or anomalies in the data.
• Outlier detection: Identified observations with studentized residuals beyond the Bonferroni-corrected thresholds.