I'll help you build and analyze a multiple regression model for predicting graduate school GPA. I'll use R for this analysis as it's particularly strong for regression diagnostics and APA-style reporting. Here's a comprehensive solution:

R Code for Multiple Regression Analysis

```r

Load required libraries

library(tidyverse)
library(car)
library(lmtest)
library(sandwich)
library(performance)
library(report)
library(apa)
library(corrplot)
library(ggplot2)
library(gridExtra)

Set up APA-style tables

options(knitr.kable.NA = '')

1. GENERATE SAMPLE DATA (replace with your actual data loading)

set.seed(123)
n <- 200

Create example dataset

data <- data.frame(
grad_gpa = rnorm(n, 3.5, 0.4) %>% pmax(2.0) %>% pmin(4.0),
gre_verbal = rnorm(n, 155, 8),
gre_quant = rnorm(n, 160, 10),
undergrad_gpa = rnorm(n, 3.4, 0.3) %>% pmax(2.5) %>% pmin(4.0),
research_exp = sample(0:3, n, replace = TRUE, prob = c(0.3, 0.3, 0.25, 0.15)), # 0-3 years
family_income = rnorm(n, 60000, 25000) %>% pmax(15000) %>% pmin(200000),
parent_education = sample(1:4, n, replace = TRUE), # 1=HS, 2=Some college, 3=Bachelor's, 4=Advanced
first_gen = sample(0:1, n, replace = TRUE) # 0=No, 1=First-generation
)

Add some realistic relationships

data$grad_gpa <- 1.5 +
0.01 * data$gre_verbal +
0.008 * data$gre_quant +
0.8 * data$undergrad_gpa +
0.1 * data$research_exp +
0.00001 * data$family_income +
0.05 * data$parent_education -
0.1 * data$first_gen +
rnorm(n, 0, 0.3)

data$gradgpa <- pmax(data$gradgpa, 2.0) %>% pmin(4.0)

2. EXPLORATORY DATA ANALYSIS

cat("\n=== DESCRIPTIVE STATISTICS ===\n")
desc_stats <- data %>%
summarize(across(where(is.numeric), list(
Mean = ~round(mean(., na.rm = TRUE), 3),
SD = ~round(sd(., na.rm = TRUE), 3),
Min = ~round(min(., na.rm = TRUE), 3),
Max = ~round(max(., na.rm = TRUE), 3)
), .names = "{.col}_{.fn}")) %>%
pivotlonger(everything(), namesto = "Variable", values_to = "Value") %>%
separate(Variable, into = c("Var", "Stat"), sep = "_") %>%
pivotwider(namesfrom = Stat, values_from = Value)

print(desc_stats)

Correlation matrix

cat("\n=== CORRELATION MATRIX ===\n")
cormatrix <- cor(data %>% select(gradgpa, greverbal, grequant, undergrad_gpa,
researchexp, familyincome, parenteducation, firstgen),
use = "complete.obs")

corrplot(cor_matrix, method = "color", type = "upper",
tl.cex = 0.8, tl.col = "black", order = "hclust")

3. BUILD INITIAL MODEL

cat("\n=== MULTIPLE REGRESSION MODEL ===\n")
model1 <- lm(gradgpa ~ greverbal + grequant + undergradgpa + research_exp +
familyincome + parenteducation + first_gen, data = data)

Summary

summary(model1)

4. ASSUMPTION CHECKING

4.1 Linearity

cat("\n=== LINEARITY ASSUMPTION ===\n")

Component + residual plots (lowess fits through origin)

par(mfrow = c(2, 4))
plot(model1, which = 1) # Residuals vs Fitted
plot(model1, which = 5) # Scale-Location
plot(model1, which = 6) # QQ Plot

Individual variable plots

predictors <- c("greverbal", "grequant", "undergradgpa", "researchexp")
for(pred in predictors) {
plot(data[[pred]], residuals(model1),
xlab = pred, ylab = "Residuals",
main = paste("Residuals vs", pred))
abline(h = 0, col = "red", lty = 2)
}

4.2 Normality of Residuals

cat("\n=== NORMALITY ASSUMPTION ===\n")

Shapiro-Wilk test

shapiro_test <- shapiro.test(residuals(model1))
cat("Shapiro-Wilk test for normality of residuals:\n")
cat("W =", round(shapirotest$statistic, 4), ", p-value =", round(shapirotest$p.value, 4), "\n")

Q-Q plot (already shown in crPlots above)

4.3 Homoscedasticity

cat("\n=== HOMOSCEDASTICITY ASSUMPTION ===\n")

Breusch-Pagan test

bp_test <- bptest(model1)
cat("Breusch-Pagan test for homoscedasticity:\n")
cat("LM statistic =", round(bptest$statistic, 4), ", p-value =", round(bptest$p.value, 4), "\n")

White test

white_test <- bptest(model1, studentize = FALSE)
cat("White test for heteroscedasticity:\n")
cat("LM statistic =", round(whitetest$statistic, 4), ", p-value =", round(whitetest$p.value, 4), "\n")

4.4 Multicollinearity

cat("\n=== MULTICOLLINEARITY ASSUMPTION ===\n")

VIF check

vif_values <- vif(model1)
cat("Variance Inflation Factors:\n")
print(round(vif_values, 3))

Tolerance (1/VIF)

tolerance <- 1/vif(model1)
cat("\nTolerances (1/VIF):\n")
print(round(tolerance, 3))

5. DIFFERENT MODEL SPECIFICATIONS

5.1 Model with interaction terms

model2 <- lm(gradgpa ~ greverbal + grequant + undergradgpa + research_exp +
familyincome + parenteducation + first_gen +
undergradgpa:researchexp + greverbal:grequant, data = data)

cat("\n=== MODEL WITH INTERACTIONS ===\n")
summary(model2)

5.2 Socioeconomic composite score

data$socioecon <- scale(data$familyincome + data$parent_education * 10000 -
data$first_gen * 50000)[,1]

model3 <- lm(gradgpa ~ greverbal + grequant + undergradgpa + research_exp +
socio_econ, data = data)

cat("\n=== MODEL WITH SOCIOECONOMIC COMPOSITE ===\n")
summary(model3)

5.3 Stepwise selection

Forward selection

modelforward <- step(lm(gradgpa ~ 1, data = data),
scope = list(lower = ~1, upper = ~greverbal + grequant +
undergradgpa + researchexp + family_income +
parenteducation + firstgen),
direction = "forward")

cat("\n=== FORWARD SELECTION MODEL ===\n")
summary(model_forward)

6. MODEL COMPARISON

cat("\n=== MODEL COMPARISON ===\n")
models <- list(
"Full Model" = model1,
"With Interactions" = model2,
"Socioeconomic Composite" = model3
)

comparison_table <- data.frame(
Model = names(models),
AIC = sapply(models, AIC),
BIC = sapply(models, BIC),
R2 = sapply(models, function(m) round(summary(m)$r.squared, 4)),
Adj_R2 = sapply(models, function(m) round(summary(m)$adj.r.squared, 4)),
F_stat = sapply(models, function(m) round(summary(m)$fstatistic[1], 2)),
F_p = sapply(models, function(m) round(pf(summary(m)$fstatistic[1],
summary(m)$fstatistic[2],
summary(m)$fstatistic[3],
lower.tail = FALSE), 4))
)

print(comparison_table)

ANOVA comparison (nested models)

anova(model1, model2)

7. FINAL MODEL DIAGNOSTICS

final_model <- model1 # Using full model as final

cat("\n=== FINAL MODEL DIAGNOSTICS ===\n")

Influential observations

influencemeasures <- influence.measures(finalmodel)
print(influence_measures$infmat)

Cook's distance plot

par(mfrow = c(1, 2))
plot(final_model, which = 4) # Cook's distance
plot(final_model, which = 5) # Leverage

8. BOOTSTRAPPED CONFIDENCE INTERVALS

library(boot)

boot_fun <- function(data, indices) {
d <- data[indices, ]
m <- lm(grad