brfss_subset_2023 <- readRDS("brfss_subset_2023.rds")

Introduction

Statistical modeling is a fundamental tool in epidemiology that allows us to:

Describe relationships between variables
Predict outcomes based on risk factors
Estimate associations while controlling for confounding

This lecture introduces key concepts in regression modeling using real-world data from the Behavioral Risk Factor Surveillance System (BRFSS) 2023.

Setup and Data Preparation

# Load required packages
library(tidyverse)
library(haven)
library(knitr)
library(kableExtra)
library(plotly)
library(broom)
library(car)
library(ggeffects)
library(gtsummary)
library(ggstats)

Descriptive Statistics

# Summary table by diabetes status
desc_table <- brfss_subset_2023 %>%
  group_by(diabetes) %>%
  summarise(
    N = n(),
    `Mean Age` = round(mean(age_cont), 1),
    `% Male` = round(100 * mean(sex == "Male"), 1),
    `% Obese` = round(100 * mean(bmi_cat == "Obese", na.rm = TRUE), 1),
    `% Physically Active` = round(100 * mean(phys_active), 1),
    `% Current Smoker` = round(100 * mean(current_smoker), 1),
    `% Hypertension` = round(100 * mean(hypertension), 1),
    `% High Cholesterol` = round(100 * mean(high_chol), 1)
  ) %>%
  mutate(diabetes = ifelse(diabetes == 1, "Diabetes", "No Diabetes"))

desc_table %>%
  kable(caption = "Descriptive Statistics by Diabetes Status",
        align = "lrrrrrrrr") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

Descriptive Statistics by Diabetes Status
diabetes	N	Mean Age	% Male	% Obese	% Physically Active	% Current Smoker	% Hypertension	% High Cholesterol
No Diabetes	1053	58.2	49.0	34.8	69.4	29.3	47.5	42.5
Diabetes	228	63.1	53.9	56.1	53.5	27.6	76.8	67.1

Part 1: Statistical Modeling Concepts

1. What is Statistical Modeling?

A statistical model is a mathematical representation of the relationship between:

An outcome variable (dependent variable, response)
One or more predictor variables (independent variables, exposures, covariates)

General Form of a Statistical Model

\[f(Y) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p + \epsilon\]

Where:

\(f(Y)\) is a function of the outcome (identity, log, logit, etc.)
\(\beta_0\) is the intercept (baseline value)
\(\beta_1, \beta_2, \ldots, \beta_p\) are coefficients (effect sizes)
\(X_1, X_2, \ldots, X_p\) are predictor variables
\(\epsilon\) is the error term (random variation)

2. Types of Regression Models

The choice of regression model depends on the type of outcome variable:

Common Regression Models in Epidemiology
Outcome Type	Regression Type	Link Function	Example
Continuous	Linear	Identity: Y	Blood pressure, BMI
Binary	Logistic	Logit: log(p/(1-p))	Disease status, mortality
Count	Poisson/Negative Binomial	Log: log(Y)	Number of infections
Time-to-event	Cox Proportional Hazards	Log: log(h(t))	Survival time

Simple vs. Multiple Regression

Simple regression: One predictor variable
Multiple regression: Two or more predictor variables (controls for confounding)

3. Linear Regression Example

Let’s model the relationship between age and diabetes prevalence.

Simple Linear Regression

# Simple linear regression: diabetes ~ age
model_linear_simple <- lm(diabetes ~ age_cont, data = brfss_subset_2023                         )

# Display results
tidy(model_linear_simple, conf.int = TRUE) %>%
  kable(caption = "Simple Linear Regression: Diabetes ~ Age",
        digits = 4,
        col.names = c("Term", "Estimate", "Std. Error", "t-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

Simple Linear Regression: Diabetes ~ Age
Term	Estimate	Std. Error	t-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	-0.0632	0.0481	-1.3125	0.1896	-0.1576	0.0312
age_cont	0.0041	0.0008	5.1368	0.0000	0.0025	0.0056

Interpretation:

Intercept (\(\beta_0\)): -0.0632 - Expected probability of diabetes at age 0 (not meaningful in this context)
Slope (\(\beta_1\)): 0.0041 - For each 1-year increase in age, the probability of diabetes increases by 0.41%

Visualization

With continuous age

# Create scatter plot with regression line
p1 <- ggplot(brfss_subset_2023, aes(x = age_cont, y = diabetes)) +
  geom_jitter(alpha = 0.2, width = 0.5, height = 0.02, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red", linewidth = 1.2) +
  labs(
    title = "Relationship Between Age and Diabetes",
    subtitle = "Simple Linear Regression",
    x = "Age (years)",
    y = "Probability of Diabetes"
  ) +
  theme_minimal(base_size = 12)

ggplotly(p1) %>%
  layout(hovermode = "closest")

Diabetes Prevalence by Age

4. Logistic Regression: The Preferred Model for Binary Outcomes

Problem with linear regression for binary outcomes:

Predicted probabilities can fall outside [0, 1]
Assumes constant variance (violated for binary data)

Solution: Logistic Regression

Uses the logit link function to ensure predicted probabilities stay between 0 and 1:

\[\text{logit}(p) = \log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X_1 + \cdots + \beta_p X_p\]

Simple Logistic Regression

# Simple logistic regression: diabetes ~ age
model_logistic_simple <- glm(diabetes ~ age_cont,
                              data = brfss_subset_2023,
                              family = binomial(link = "logit"))

# Display results with odds ratios
tidy(model_logistic_simple, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Simple Logistic Regression: Diabetes ~ Age (Odds Ratios)",
        digits = 3,
        col.names = c("Term", "Odds Ratio", "Std. Error", "z-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

Simple Logistic Regression: Diabetes ~ Age (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	0.029	0.423	-8.390	0	0.012	0.064
age_cont	1.034	0.007	4.978	0	1.021	1.048

Interpretation:

Odds Ratio (OR): 1.034
For each 1-year increase in age, the odds of diabetes increase by 3.4%
The relationship is highly statistically significant (p < 0.001)

Predicted Probabilities

# From ggeffects package
pp <- predict_response(model_logistic_simple, terms = "age_cont")
plot(pp)

Predicted Diabetes Probability by Age

# Generate predicted probabilities
pred_data <- data.frame(age_cont = seq(18, 80, by = 1))
pred_data$predicted_prob <- predict(model_logistic_simple,
                                    newdata = pred_data,
                                    type = "response")

# Plot
p2 <- ggplot(pred_data, aes(x = age_cont, y = predicted_prob)) +
  geom_line(color = "darkred", linewidth = 1.5) +
  geom_ribbon(aes(ymin = predicted_prob - 0.02,
                  ymax = predicted_prob + 0.02),
              alpha = 0.2, fill = "darkred") +
  labs(
    title = "Predicted Probability of Diabetes by Age",
    subtitle = "Simple Logistic Regression",
    x = "Age (years)",
    y = "Predicted Probability of Diabetes"
  ) +
  scale_y_continuous(labels = scales::percent_format(), limits = c(0, 0.6)) +
  theme_minimal(base_size = 12)

ggplotly(p2)

Predicted Diabetes Probability by Age

5. Multiple Regression: Controlling for Confounding

What is Confounding?

A confounder is a variable that:

Is associated with both the exposure and the outcome
Is not on the causal pathway between exposure and outcome
Distorts the true relationship between exposure and outcome

Example: The relationship between age and diabetes may be confounded by BMI, physical activity, and other factors.

Multiple Logistic Regression

# Multiple logistic regression with potential confounders
model_logistic_multiple <- glm(diabetes ~ age_cont + sex + bmi_cat +
                                phys_active + current_smoker + education,
                               data = brfss_subset_2023,
                               family = binomial(link = "logit"))

# Display results
tidy(model_logistic_multiple, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Multiple Logistic Regression: Diabetes ~ Age + Covariates (Odds Ratios)",
        digits = 3,
        col.names = c("Term", "Odds Ratio", "Std. Error", "z-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  scroll_box(height = "400px")

Multiple Logistic Regression: Diabetes ~ Age + Covariates (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	0.009	1.177	-4.001	0.000	0.000	0.065
age_cont	1.041	0.007	5.515	0.000	1.027	1.057
sexMale	1.191	0.154	1.133	0.257	0.880	1.613
bmi_catNormal	1.971	1.052	0.645	0.519	0.378	36.309
bmi_catOverweight	3.155	1.044	1.101	0.271	0.621	57.679
bmi_catObese	6.834	1.041	1.845	0.065	1.354	124.675
phys_active	0.589	0.157	-3.373	0.001	0.433	0.802
current_smoker	1.213	0.178	1.085	0.278	0.852	1.716
educationHigh school graduate	0.634	0.288	-1.579	0.114	0.364	1.131
educationSome college	0.542	0.294	-2.081	0.037	0.307	0.977
educationCollege graduate	0.584	0.305	-1.763	0.078	0.324	1.074

Interpretation:

Age (adjusted OR): 1.041
- After adjusting for sex, BMI, physical activity, smoking, and education, each 1-year increase in age is associated with a 4.1% increase in the odds of diabetes
Sex (Male vs Female): OR = 1.191
- Males have 19.1% higher odds of diabetes compared to females, adjusting for other variables

BMI (Obese vs Normal): OR = 6.834
- Obese individuals had 6.83 times higher odds of diabetes compared to normal-weight individuals.

6. Dummy Variables: Coding Categorical Predictors

Categorical variables with \(k\) levels are represented using \(k-1\) dummy variables (indicator variables).

Example: Education Level

Education has 4 levels: 1. < High school (reference category) 2. High school graduate 3. Some college 4. College graduate

R automatically creates 3 dummy variables:

# Extract dummy variable coding
dummy_table <- data.frame(
  Education = c("< High school", "High school graduate", "Some college", "College graduate"),
  `Dummy 1 (HS grad)` = c(0, 1, 0, 0),
  `Dummy 2 (Some college)` = c(0, 0, 1, 0),
  `Dummy 3 (College grad)` = c(0, 0, 0, 1),
  check.names = FALSE
)

dummy_table %>%
  kable(caption = "Dummy Variable Coding for Education",
        align = "lccc") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  row_spec(1, bold = TRUE, background = "#ffe6e6")  # Highlight reference category

Dummy Variable Coding for Education
Education	Dummy 1 (HS grad)	Dummy 2 (Some college)	Dummy 3 (College grad)
< High school	0	0	0
High school graduate	1	0	0
Some college	0	1	0
College graduate	0	0	1

Reference Category: The category with all zeros (< High school) is the reference group. All other categories are compared to this reference.

Visualizing Education Effects

# Extract education coefficients
educ_coefs <- tidy(model_logistic_multiple, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(str_detect(term, "education")) %>%
  mutate(
    education_level = str_remove(term, "education"),
    education_level = factor(education_level,
                             levels = c("High school graduate",
                                       "Some college",
                                       "College graduate"))
  )

# Add reference category
ref_row <- data.frame(
  term = "education< High school",
  estimate = 1.0,
  std.error = 0,
  statistic = NA,
  p.value = NA,
  conf.low = 1.0,
  conf.high = 1.0,
  education_level = factor("< High school (Ref)",
                          levels = c("< High school (Ref)",
                                    "High school graduate",
                                    "Some college",
                                    "College graduate"))
)

educ_coefs_full <- bind_rows(ref_row, educ_coefs) %>%
  mutate(education_level = factor(education_level,
                                 levels = c("< High school (Ref)",
                                           "High school graduate",
                                           "Some college",
                                           "College graduate")))

# Plot
p3 <- ggplot(educ_coefs_full, aes(x = education_level, y = estimate)) +
  geom_hline(yintercept = 1, linetype = "dashed", color = "gray50") +
  geom_pointrange(aes(ymin = conf.low, ymax = conf.high),
                  size = 0.8, color = "darkblue") +
  coord_flip() +
  labs(
    title = "Association Between Education and Diabetes",
    subtitle = "Adjusted Odds Ratios (reference: < High school)",
    x = "Education Level",
    y = "Odds Ratio (95% CI)"
  ) +
  theme_minimal(base_size = 12)

ggplotly(p3)

Odds Ratios for Education Levels

# Plot model coefficients with `ggcoef_model()`
ggcoef_model(model_logistic_multiple, exponentiate = TRUE,
  include = c("education"),
  variable_labels = c(
    education = "Education"),
  facet_labeller = ggplot2::label_wrap_gen(10)
)

7. Interactions (Effect Modification)

An interaction exists when the effect of one variable on the outcome differs across levels of another variable.

Epidemiologic term: Effect modification

Example: Age × Sex Interaction

Does the effect of age on diabetes differ between males and females?

# Model with interaction term
model_interaction <- glm(diabetes ~ age_cont * sex + bmi_cat + phys_active,
                         data = brfss_subset_2023 ,
                         family = binomial(link = "logit"))

# Display interaction results
tidy(model_interaction, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(str_detect(term, "age_cont")) %>%
  kable(caption = "Age × Sex Interaction Model (Odds Ratios)",
        digits = 3,
        col.names = c("Term", "Odds Ratio", "Std. Error", "z-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

Age × Sex Interaction Model (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
age_cont	1.031	0.009	3.178	0.001	1.012	1.051
age_cont:sexMale	1.015	0.014	1.084	0.278	0.988	1.044

Interpretation:

Main effect of age: OR among females (reference)
Interaction term (age:sexMale): Additional effect of age among males
If the interaction term is significant, the age-diabetes relationship differs by sex

Visualizing Interaction

# Generate predicted probabilities by sex
pred_interact <- ggpredict(model_interaction, terms = c("age_cont [18:80]", "sex"))

# Plot
p4 <- ggplot(pred_interact, aes(x = x, y = predicted, color = group, fill = group)) +
  geom_line(linewidth = 1.2) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.2, color = NA) +
  labs(
    title = "Predicted Probability of Diabetes by Age and Sex",
    subtitle = "Testing for Age × Sex Interaction",
    x = "Age (years)",
    y = "Predicted Probability of Diabetes",
    color = "Sex",
    fill = "Sex"
  ) +
  scale_y_continuous(labels = scales::percent_format()) +
  scale_color_manual(values = c("Female" = "#E64B35", "Male" = "#4DBBD5")) +
  scale_fill_manual(values = c("Female" = "#E64B35", "Male" = "#4DBBD5")) +
  theme_minimal(base_size = 12) +
  theme(legend.position = "bottom")

ggplotly(p4)

Age-Diabetes Relationship by Sex

8. Model Diagnostics

Every regression model makes assumptions about the data. If assumptions are violated, results may be invalid.

Key Assumptions for Logistic Regression

Linearity of log odds: Continuous predictors have a linear relationship with the log odds of the outcome
Independence of observations: Each observation is independent
No perfect multicollinearity: Predictors are not perfectly correlated
No influential outliers: Individual observations don’t overly influence the model

Checking for Multicollinearity

Variance Inflation Factor (VIF): Measures how much the variance of a coefficient is inflated due to correlation with other predictors.

VIF < 5: Generally acceptable
VIF > 10: Serious multicollinearity problem

# Calculate VIF
vif_values <- vif(model_logistic_multiple)

# Create VIF table
# For models with categorical variables, vif() returns GVIF (Generalized VIF)
if (is.matrix(vif_values)) {
  # If matrix (categorical variables present), extract GVIF^(1/(2*Df))
  vif_df <- data.frame(
    Variable = rownames(vif_values),
    VIF = vif_values[, "GVIF^(1/(2*Df))"]
  )
} else {
  # If vector (only continuous variables)
  vif_df <- data.frame(
    Variable = names(vif_values),
    VIF = as.numeric(vif_values)
  )
}

# Add interpretation
vif_df <- vif_df %>%
  arrange(desc(VIF)) %>%
  mutate(
    Interpretation = case_when(
      VIF < 5 ~ "Low (No concern)",
      VIF >= 5 & VIF < 10 ~ "Moderate (Monitor)",
      VIF >= 10 ~ "High (Problem)"
    )
  )

vif_df %>%
  kable(caption = "Variance Inflation Factors (VIF) for Multiple Regression Model",
        digits = 2,
        align = "lrc") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  row_spec(which(vif_df$VIF >= 10), bold = TRUE, color = "white", background = "#DC143C") %>%
  row_spec(which(vif_df$VIF >= 5 & vif_df$VIF < 10), background = "#FFA500") %>%
  row_spec(which(vif_df$VIF < 5), background = "#90EE90")

Variance Inflation Factors (VIF) for Multiple Regression Model
	Variable	VIF	Interpretation
age_cont	age_cont	1.05	Low (No concern)
current_smoker	current_smoker	1.05	Low (No concern)
phys_active	phys_active	1.02	Low (No concern)
sex	sex	1.01	Low (No concern)
education	education	1.01	Low (No concern)
bmi_cat	bmi_cat	1.01	Low (No concern)

Influential Observations

Cook’s Distance: Measures how much the model would change if an observation were removed.

Cook’s D > 1: Potentially influential observation

# Calculate Cook's distance
cooks_d <- cooks.distance(model_logistic_multiple)

# Create data frame
influence_df <- data.frame(
  observation = 1:length(cooks_d),
  cooks_d = cooks_d
) %>%
  mutate(influential = ifelse(cooks_d > 1, "Yes", "No"))

# Plot
p5 <- ggplot(influence_df, aes(x = observation, y = cooks_d, color = influential)) +
  geom_point(alpha = 0.6) +
  geom_hline(yintercept = 1, linetype = "dashed", color = "red") +
  labs(
    title = "Cook's Distance: Identifying Influential Observations",
    subtitle = "Values > 1 indicate potentially influential observations",
    x = "Observation Number",
    y = "Cook's Distance",
    color = "Influential?"
  ) +
  scale_color_manual(values = c("No" = "steelblue", "Yes" = "red")) +
  theme_minimal(base_size = 12)

ggplotly(p5)

Cook’s Distance for Influential Observations

# Count influential observations
n_influential <- sum(influence_df$influential == "Yes")
cat("Number of potentially influential observations:", n_influential, "\n")

## Number of potentially influential observations: 0

9. Model Comparison and Selection

Comparing Nested Models

Use Likelihood Ratio Test to compare nested models:

# Model 1: Age only
model1 <- glm(diabetes ~ age_cont,
              data = brfss_subset_2023,
              family = binomial)

# Model 2: Age + Sex
model2 <- glm(diabetes ~ age_cont + sex,
              data = brfss_subset_2023,
              family = binomial)

# Model 3: Full model
model3 <- model_logistic_multiple

# Likelihood ratio test
lrt_1_2 <- anova(model1, model2, test = "LRT")
lrt_2_3 <- anova(model2, model3, test = "LRT")

# Create comparison table
model_comp <- data.frame(
  Model = c("Model 1: Age only",
            "Model 2: Age + Sex",
            "Model 3: Full model"),
  AIC = c(AIC(model1), AIC(model2), AIC(model3)),
  BIC = c(BIC(model1), BIC(model2), BIC(model3)),
  `Deviance` = c(deviance(model1), deviance(model2), deviance(model3)),
  check.names = FALSE
)

model_comp %>%
  kable(caption = "Model Comparison: AIC, BIC, and Deviance",
        digits = 2,
        align = "lrrr") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  row_spec(which.min(model_comp$AIC), bold = TRUE, background = "#d4edda")

Model Comparison: AIC, BIC, and Deviance
Model	AIC	BIC	Deviance
Model 1: Age only	1175.08	1185.39	1171.08
Model 2: Age + Sex	1175.85	1191.32	1169.85
Model 3: Full model	1122.65	1179.36	1100.65

Interpretation:

Lower AIC/BIC indicates better model fit
Model 3 (full model) has the lowest AIC, suggesting it provides the best fit to the data

10. Error Term in Statistical Models

All statistical models include an error term (\(\epsilon\)) to account for:

Random variation in the outcome
Unmeasured variables not included in the model
Measurement error in variables

\[Y = \beta_0 + \beta_1 X_1 + \cdots + \beta_p X_p + \epsilon\]

Key points:

The model cannot perfectly predict every outcome
The difference between observed and predicted values is the error (residual)
We assume errors are normally distributed with mean 0 (for linear regression)

Part 2: Student Lab Activity

Lab Overview

In this lab, you will:

Build your own logistic regression model predicting hypertension (high blood pressure)
Create dummy variables for categorical predictors
Interpret regression coefficients
Test for confounding and interaction
Perform model diagnostics

Lab Instructions

Task 1: Explore the Outcome Variable

# YOUR CODE HERE: Create a frequency table of hypertension status

htn_freq <- brfss_subset_2023 %>%
  count(hypertension) %>%
  mutate(
    Percent = round(100 * n / sum(n), 1),
    hypertension = ifelse(hypertension == 1, 
                          "Hypertension", 
                          "No Hypertension")
  )

htn_freq %>%
  kable(caption = "Frequency of Hypertension Status",
        align = "lrr") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

Frequency of Hypertension Status
hypertension	n	Percent
No Hypertension	606	47.3
Hypertension	675	52.7

# YOUR CODE HERE: Calculate the prevalence of hypertension by age group
prev_age_table <- brfss_subset_2023 %>%
  group_by(age_group) %>%
  summarise(
    N = n(),
    Hypertension_Cases = sum(hypertension == 1, na.rm = TRUE),
    Prevalence_Percent = round(100 * mean(hypertension == 1, na.rm = TRUE), 1)
  )

prev_age_table %>%
  kable(caption = "Prevalence of Hypertension by Age Group",
        align = "lrrr") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

Prevalence of Hypertension by Age Group
age_group	N	Hypertension_Cases	Prevalence_Percent
18-24	12	1	8.3
25-34	77	15	19.5
35-44	138	42	30.4
45-54	161	61	37.9
55-64	266	137	51.5
65+	627	419	66.8

Questions:

What is the overall prevalence of hypertension in the dataset? The overall prevalence of hypertension in the dataset is 52.7%
How does hypertension prevalence vary by age group? Hypertension prevalence increases with age, starting from 8.3% among adults aged 18–24 to 66.8% among those aged 65 years and older. This pattern suggests a strong positive association between age and hypertension in this data.

Task 2: Build a Simple Logistic Regression Model

# YOUR CODE HERE: Fit a simple logistic regression model
# Outcome: hypertension
# Predictor: age_cont

model_logistic_simple <- glm(hypertension ~ age_cont,
                              data = brfss_subset_2023,
                              family = binomial(link = "logit"))

# YOUR CODE HERE: Display the results with odds ratios

tidy(model_logistic_simple, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Simple Logistic Regression: Hypertension ~ Age (Odds Ratios)",
        digits = 3,
        col.names = c("Term", "Odds Ratio", "Std. Error", "z-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

Simple Logistic Regression: Hypertension ~ Age (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	0.048	0.296	-10.293	0	0.026	0.084
age_cont	1.055	0.005	10.996	0	1.045	1.065

Questions:

What is the odds ratio for age? Interpret this value. The odds ratio for age is 1.055, meaning that each 1 year of age is associated with a 5.5% increase in the odds of hypertension.
Is the association statistically significant? he association is statistically significant (p < 0.001).
What is the 95% confidence interval for the odds ratio? The 95% confidence interval (1.045–1.065).

Task 3: Create a Multiple Regression Model

# YOUR CODE HERE: Fit a multiple logistic regression model
# Outcome: hypertension
# Predictors: age_cont, sex, bmi_cat, phys_active, current_smoker
# YOUR CODE HERE: Display the results

model_logistic_multiple <- glm(hypertension ~ age_cont + sex + bmi_cat +
                                phys_active + current_smoker + education,
                               data = brfss_subset_2023,
                               family = binomial(link = "logit"))

# Display results
tidy(model_logistic_multiple, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Multiple Logistic Regression: Hypertenstion ~ Age + Covariates (Odds Ratios)",
        digits = 3,
        col.names = c("Term", "Odds Ratio", "Std. Error", "z-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  scroll_box(height = "400px")

Multiple Logistic Regression: Hypertenstion ~ Age + Covariates (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	0.012	0.691	-6.435	0.000	0.003	0.043
age_cont	1.062	0.005	11.299	0.000	1.051	1.073
sexMale	1.245	0.124	1.772	0.076	0.977	1.586
bmi_catNormal	2.110	0.548	1.362	0.173	0.761	6.819
bmi_catOverweight	3.301	0.545	2.191	0.028	1.198	10.613
bmi_catObese	6.674	0.547	3.470	0.001	2.415	21.535
phys_active	0.930	0.132	-0.552	0.581	0.717	1.204
current_smoker	1.054	0.140	0.376	0.707	0.802	1.388
educationHigh school graduate	0.713	0.265	-1.275	0.202	0.421	1.194
educationSome college	0.554	0.267	-2.212	0.027	0.326	0.930
educationCollege graduate	0.651	0.274	-1.566	0.117	0.378	1.109

Questions:

How did the odds ratio for age change after adjusting for other variables? The odds ratio is 1.062 which means after adjusting for sex, BMI, physical activity, smoking, and education, each 1-year increase in age is associated with a 6.2 increase in the odds of hypertension.
What does this suggest about confounding? This suggests confounding may not convey the true relationship between the exposure and outcome.
Which variables are the strongest predictors of hypertension? Overall, the variables that are the strongest predictors of hypertension is age and higher BMI (either in the overweight or obese categories).

Task 4: Interpret Dummy Variables

# YOUR CODE HERE: Create a table showing the dummy variable coding for bmi_cat
dummy_table <- data.frame(
  bmi__cat = c("Underweight", "Normal", "Overweight", "Obese"),
  `Dummy 1 (Normal)` = c(0, 1, 0, 0),
  `Dummy 2 (Overweight)` = c(0, 0, 1, 0),
  `Dummy 3 (Obese)` = c(0, 0, 0, 1),
  check.names = FALSE
)

# YOUR CODE HERE: Extract and display the odds ratios for BMI categories
dummy_table %>%
  kable(caption = "Dummy Variable Coding for BMI",
        align = "lccc") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  row_spec(1, bold = TRUE, background = "#ffe6e6")  # Highlight reference category

Dummy Variable Coding for BMI
bmi__cat	Dummy 1 (Normal)	Dummy 2 (Overweight)	Dummy 3 (Obese)
Underweight	0	0	0
Normal	1	0	0
Overweight	0	1	0
Obese	0	0	1

Questions:

What is the reference category for BMI? The reference category is underweight.
Interpret the odds ratio for “Obese” compared to the reference category. The odds ratio for Obese is 6.674. This means that individuals classified as obese have approximately 6.67 times the odds of hypertension compared to individuals in the reference BMI category (Underweight).
How would you explain this to a non-statistician? People who are classified as obese are approximately 6 times more likely to have hypertension compared to people in the lowest weight category.

Task 5: Test for Interaction

# YOUR CODE HERE: Fit a model with Age × BMI interaction
model_interaction <- glm(hypertension ~ age_cont * bmi_cat,
                         data = brfss_subset_2023 ,
                         family = binomial(link = "logit"))

# Display interaction results
tidy(model_interaction, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(str_detect(term, "age_cont")) %>%
  kable(caption = "Age × BMI Interaction Model (Odds Ratios)",
        digits = 3,
        col.names = c("Term", "Odds Ratio", "Std. Error", "z-statistic", "p-value", "95% CI Lower", "95% CI Upper")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

Age × BMI Interaction Model (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
age_cont	1.004	0.042	0.102	0.918	0.929	1.108
age_cont:bmi_catNormal	1.058	0.043	1.306	0.192	0.957	1.147
age_cont:bmi_catOverweight	1.063	0.043	1.423	0.155	0.962	1.151
age_cont:bmi_catObese	1.054	0.042	1.232	0.218	0.954	1.140

# Fit the model without the interaction
model_no_interaction <- glm(
  hypertension ~ age_cont + bmi_cat,
  data = brfss_subset_2023,
  family = binomial(link = "logit")
)

# Generate predicted probabilities by BMI
pred_interact <- ggpredict(model_interaction, terms = c("age_cont [18:80]", "bmi_cat"))

# Plot
p4 <- ggplot(pred_interact, aes(x = x, y = predicted, color = group, fill = group)) +
  geom_line(linewidth = 1.2) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.15, color = NA) +
  labs(
    title = "Predicted Probability of Diabetes by Age and BMI",
    subtitle = "Testing for Age × BMI Interaction",
    x = "Age (years)",
    y = "Predicted Probability of Hypertension",
    color = "BMI",
    fill = "BMI"
  ) +
  scale_y_continuous(labels = scales::percent_format()) +
  scale_color_manual(values = c(
    "Underweight" = "#E64B35", 
    "Normal" = "#4DBBD5",
    "Overweight" = "#1B9E77", 
    "Obese" = "#E7298A")) +
  scale_fill_manual(values = c(
    "Underweight" = "#E64B35", 
    "Normal" = "#4DBBD5", 
    "Overweight" = "#1B9E77", 
    "Obese" = "#E7298A")) +
    
  guides(fill = "none") +
  theme_minimal(base_size = 10) +
  theme(legend.position = "bottom")

ggplotly(p4)

# YOUR CODE HERE: Perform a likelihood ratio test comparing models with and without interaction

# Likelihood ratio test
lrt_result <- anova(model_interaction, model_no_interaction, test = "LRT")

lrt_result

## Analysis of Deviance Table
## 
## Model 1: hypertension ~ age_cont * bmi_cat
## Model 2: hypertension ~ age_cont + bmi_cat
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1      1273     1566.1                     
## 2      1276     1568.1 -3  -1.9645   0.5798

Questions:

Is the interaction term statistically significant? The p-value is 0.5798, meaning it is not statistically significant
What does this mean in epidemiologic terms (effect modification)? Since the interaction is not significant, we can interpret that the effect of age on hypertension is roughly the same across all BMI categories.
Create a visualization showing predicted probabilities by age and BMI category

Task 6: Model Diagnostics

# YOUR CODE HERE: Calculate VIF for your multiple regression model

vif_values <- vif(model_logistic_multiple)

# Create VIF table
# For models with categorical variables, vif() returns GVIF (Generalized VIF)
if (is.matrix(vif_values)) {
  # If matrix (categorical variables present), extract GVIF^(1/(2*Df))
  vif_df <- data.frame(
    Variable = rownames(vif_values),
    VIF = vif_values[, "GVIF^(1/(2*Df))"]
  )
} else {
  # If vector (only continuous variables)
  vif_df <- data.frame(
    Variable = names(vif_values),
    VIF = as.numeric(vif_values)
  )
}

# Add interpretation
vif_df <- vif_df %>%
  arrange(desc(VIF)) %>%
  mutate(
    Interpretation = case_when(
      VIF < 5 ~ "Low (No concern)",
      VIF >= 5 & VIF < 10 ~ "Moderate (Monitor)",
      VIF >= 10 ~ "High (Problem)"
    )
  )

vif_df %>%
  kable(caption = "Variance Inflation Factors (VIF) for Multiple Regression Model",
        digits = 2,
        align = "lrc") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  row_spec(which(vif_df$VIF >= 10), bold = TRUE, color = "white", background = "#DC143C") %>%
  row_spec(which(vif_df$VIF >= 5 & vif_df$VIF < 10), background = "#FFA500") %>%
  row_spec(which(vif_df$VIF < 5), background = "#90EE90")

Variance Inflation Factors (VIF) for Multiple Regression Model
	Variable	VIF	Interpretation
age_cont	age_cont	1.07	Low (No concern)
current_smoker	current_smoker	1.04	Low (No concern)
phys_active	phys_active	1.02	Low (No concern)
bmi_cat	bmi_cat	1.02	Low (No concern)
sex	sex	1.01	Low (No concern)
education	education	1.01	Low (No concern)

# YOUR CODE HERE: Create a Cook's distance plot to identify influential observations

cooks_d <- cooks.distance(model_logistic_multiple)

# Create data frame
influence_df <- data.frame(
  observation = 1:length(cooks_d),
  cooks_d = cooks_d
) %>%
  mutate(influential = ifelse(cooks_d > 1, "Yes", "No"))

# Plot
p5 <- ggplot(influence_df, aes(x = observation, y = cooks_d, color = influential)) +
  geom_point(alpha = 0.6) +
  geom_hline(yintercept = 1, linetype = "dashed", color = "red") +
  labs(
    title = "Cook's Distance: Identifying Influential Observations",
    subtitle = "Values > 1 indicate potentially influential observations",
    x = "Observation Number",
    y = "Cook's Distance",
    color = "Influential?"
  ) +
  scale_color_manual(values = c("No" = "steelblue", "Yes" = "red")) +
  theme_minimal(base_size = 12)

ggplotly(p5)

# Count influential observations
n_influential <- sum(influence_df$influential == "Yes")
cat("Number of potentially influential observations:", n_influential, "\n")

## Number of potentially influential observations: 0

Questions:

Are there any concerns about multicollinearity? There are no concerns about multicollinearity.
Are there any influential observations that might affect your results? There are no influential observations that affect my results.
What would you do if you found serious violations? If I found serious violations, I would either adjust the model by removing problematic predictor or remove an observation.

Task 7: Model Comparison

# YOUR CODE HERE: Compare three models using AIC and BIC
# Model A: Age only
model1 <- glm(hypertension ~ age_cont,
              data = brfss_subset_2023,
              family = binomial)

# Model B: Age + sex + bmi_cat
model2 <- glm(hypertension ~ age_cont + sex + bmi_cat,
              data = brfss_subset_2023,
              family = binomial)

# Model C: Age + sex + bmi_cat + phys_active + current_smoker
model3 <- glm(hypertension ~ age_cont + sex +bmi_cat + current_smoker,
              data = brfss_subset_2023,
              family = binomial)

# YOUR CODE HERE: Create a comparison table
model_comp <- data.frame(
  Model = c("Model 1: Age only",
            "Model 2: Age + Sex + BMI",
            "Model 3: Age + Sex + BMI + Physical Activity + Current Smoker"),
  
  AIC = c(AIC(model1), AIC(model2), AIC(model3)),
  BIC = c(BIC(model1), BIC(model2), BIC(model3)),
  `Deviance` = c(deviance(model1), deviance(model2), deviance(model3)),
  check.names = FALSE
)

model_comp %>%
  kable(caption = "Model Comparison: AIC, BIC, and Deviance",
        digits = 2,
        align = "lrrr") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE) %>%
  row_spec(which.min(model_comp$AIC), bold = TRUE, background = "#d4edda")

Model Comparison: AIC, BIC, and Deviance
Model	AIC	BIC	Deviance
Model 1: Age only	1636.61	1646.92	1632.61
Model 2: Age + Sex + BMI	1576.49	1607.42	1564.49
Model 3: Age + Sex + BMI + Physical Activity + Current Smoker	1578.15	1614.24	1564.15

lrt_1_2 <- anova(model1, model2, test = "LRT")
lrt_2_3 <- anova(model2, model3, test = "LRT")

lrt_1_2

## Analysis of Deviance Table
## 
## Model 1: hypertension ~ age_cont
## Model 2: hypertension ~ age_cont + sex + bmi_cat
##   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
## 1      1279     1632.6                          
## 2      1275     1564.5  4   68.126 5.643e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Questions:

Which model has the best fit based on AIC? Model 2 has the best fit based on AIC.
Is the added complexity of the full model justified? No, the added complexity is not justified. Model 3 does not show a significant increase from Model 1.
Which model would you choose for your final analysis? Why? I would choose Model 2 because it has the lowest AIC, and by adding physical activity and current smoker, the AIC increases and deviance barely improves meaning these extra variables don’t meaningfully improve the model’s predictive ability.

Lab Report Guidelines

Write a brief report (1-2 pages) summarizing your findings:

Introduction: State your research question
Methods: Describe your analytic approach
Results: Present key findings with tables and figures
Interpretation: Explain what your results mean
Limitations: Discuss potential issues with your analysis

Submission: Submit your completed R Markdown file and knitted HTML report.

Research Question: How do different factors such as age, sex, BMI, physical activity, smoking status, and education affect hypertension (high blood pressure) in adults?

Hypertension (high blood pressure) is a public health concern and the goal of this analysis is to examine the associations between factors such as age, sex, BMI, physical activity, smoking status, and education and the likelihood of hypertension (high blood pressure) in adults using logistic regression and multiple logistic regression. Furthermore, we investigated the increase in the odds of hypertension and whether BMI modifies the effect of age on hypertension risk. I used a subset from the BRFSS 2023 dataset.

First, I calculated the overall prevalence of hypertension in the dataset and the prevalence of hypertension by age group. Second, I built both a simple logistic regression model and a multiple regression model to find the odd ratio and the adjusted odds ratio. Third, I created and interpreted dummy variables for BMI. Fourth, I tested for interaction between age and BMI and performing a likelihood ratio test to compare the models with and without interaction. Then I generated the predicted probabilities by BMI. These steps were for analyzing the effect modification of BMI modifies the effect of age on hypertension risk. Fifth, I ran model diagnostics - calculating VIF for your multiple regression model and creating a Cook’s distance plot to identify any influential observations that may affect my results. Finally, I did a model comparison, comparing three models using AIC and BIC - the models being age only, age + sex + BMI, and age + sex + BMI + physical activity + current smoker.

From my analysis, the following results are below:

Task 1 (Descriptive Statistics): Hypertension prevalence increased steadily across age groups, rising from 8.3% among adults aged 18-24 to 66.8% among those aged 65 years and older. This pattern indicates a strong positive association between age and hypertension. The overall prevalence of hypertension in the dataset is 52.7%.

Task 2 (Logistic Regression Results): In the unadjusted logistic regression model, the odds ratio (OR) for age was 1.055 (95% CI: 1.045-1.065, p < 0.001). This indicates that each additional year of age is associated with a 5.5% increase in the odds of hypertension, and this association is statistically significant.

Task 3 (Multiple Logistic Regression Model): After adjusting for sex, BMI, physical activity, smoking status, and education, the odds ratio for age increased slightly to 1.062, indicating that each additional year of age is associated with a 6.2% increase in the odds of hypertension. The small change in the age coefficient after adjustment suggests minimal confounding by the other variables included in the model or that confounding may not convey the true relationship between the exposure and outcome. The strongest predictors of hypertension were age and BMI category particularly in the overweight and obese categories.

Task 4 (Dummy Variables): Individuals classified as obese had an odds ratio of 6.674, meaning they had approximately 6.7 times the odds of hypertension compared to individuals in the reference category (underweight).

Task 5 (Interaction): The interaction between age and BMI was not statistically significant, the p-value = 0.5798. This indicates that the effect of age on hypertension does not significantly differ across BMI categories.

Task 6 (Model Diagnostics): Multicollinearity: VIF values ranged from 1.01 to 1.07, indicating no concerns. No influential data points were identified that substantially affected my results.

Task 7: Model fit was evaluated using AIC, BIC, and deviance: Model 1 (Age only): AIC = 1636.61, Model 2 (Age + Sex + BMI): AIC = 1576.49, and Model 3 (Full model): AIC = 1578.15. Model 2 had the lowest AIC, indicating the best fit model complexity. Although Model 3 included additional variables (physical activity and smoking status), its AIC increased slightly and deviance improved only minimally. Therefore, Model 2 was selected as the final model.

The results indicate that age and BMI are the primary determinants of hypertension risk in this dataset. Age demonstrated a strong and statistically significant association with hypertension. The steady increase in prevalence across age groups and the consistent odds ratio in both unadjusted and adjusted models suggest that aging is an independent and important risk factor. BMI, particularly obesity, showed a strong association with hypertension. Individuals classified as obese had substantially higher odds of hypertension compared to those in the reference category, highlighting excess body weight as a major modifiable risk factor. The absence of a significant age-by-BMI interaction suggests that the effect of age on hypertension is consistent across BMI categories. In epidemiologic terms, BMI increases overall risk but does not modify how age influences hypertension. Model comparison supported the selection of Model 2. Although additional behavioral variables were considered, they did not meaningfully improve model performance. Overall, these findings reinforce established evidence that aging and higher BMI (also body fat which BMI does not account for) are key contributors to hypertension risk and should remain central targets for prevention and public health interventions.

There are some limitations to account for in my analysis. One, the cross-sectional design prevents establishing causality between risk factors and hypertension. Second, many variables are self-reported, which may introduce recall or reporting bias. Residual confounding is also possible, as not all relevant factors (such as diet or family history) were included in the model. Finally, small sample sizes in certain BMI categories may reduce the precision of some estimates. Despite these limitations, the findings provide useful key predictors of hypertension in this dataset.

Summary

Key Concepts Covered

Statistical modeling describes relationships between variables
Regression types depend on the outcome variable type
Logistic regression is appropriate for binary outcomes
Multiple regression controls for confounding
Dummy variables represent categorical predictors
Interactions test for effect modification
Model diagnostics check assumptions and identify problems
Model comparison helps select the best model

Important Formulas

Logistic Regression:

\[\text{logit}(p) = \log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X_1 + \cdots + \beta_p X_p\]

Odds Ratio:

\[\text{OR} = e^{\beta_i}\]

Predicted Probability:

\[p = \frac{e^{\beta_0 + \beta_1 X_1 + \cdots + \beta_p X_p}}{1 + e^{\beta_0 + \beta_1 X_1 + \cdots + \beta_p X_p}}\]

References

Agresti, A. (2018). An Introduction to Categorical Data Analysis (3rd ed.). Wiley.
Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.). Wiley.
Vittinghoff, E., Glidden, D. V., Shiboski, S. C., & McCulloch, C. E. (2012). Regression Methods in Biostatistics (2nd ed.). Springer.
Centers for Disease Control and Prevention. (2023). Behavioral Risk Factor Surveillance System.

Session Info

sessionInfo()

## R version 4.5.1 (2025-06-13)
## Platform: aarch64-apple-darwin20
## Running under: macOS Sonoma 14.6
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## time zone: America/New_York
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] ggstats_0.12.0   gtsummary_2.5.0  ggeffects_2.3.2  car_3.1-3       
##  [5] carData_3.0-5    broom_1.0.11     plotly_4.12.0    kableExtra_1.4.0
##  [9] knitr_1.51       haven_2.5.5      lubridate_1.9.4  forcats_1.0.1   
## [13] stringr_1.6.0    dplyr_1.2.0      purrr_1.2.1      readr_2.1.5     
## [17] tidyr_1.3.2      tibble_3.3.0     ggplot2_4.0.1    tidyverse_2.0.0 
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.6         xfun_0.56            bslib_0.10.0        
##  [4] htmlwidgets_1.6.4    insight_1.4.5        lattice_0.22-7      
##  [7] tzdb_0.5.0           crosstalk_1.2.2      vctrs_0.7.1         
## [10] tools_4.5.1          generics_0.1.4       datawizard_1.3.0    
## [13] pkgconfig_2.0.3      Matrix_1.7-3         data.table_1.18.0   
## [16] RColorBrewer_1.1-3   S7_0.2.1             lifecycle_1.0.5     
## [19] compiler_4.5.1       farver_2.1.2         textshaping_1.0.4   
## [22] htmltools_0.5.9      sass_0.4.10          yaml_2.3.12         
## [25] lazyeval_0.2.2       Formula_1.2-5        pillar_1.11.0       
## [28] jquerylib_0.1.4      broom.helpers_1.22.0 cachem_1.1.0        
## [31] abind_1.4-8          nlme_3.1-168         tidyselect_1.2.1    
## [34] digest_0.6.39        stringi_1.8.7        labeling_0.4.3      
## [37] splines_4.5.1        labelled_2.16.0      fastmap_1.2.0       
## [40] grid_4.5.1           cli_3.6.5            magrittr_2.0.3      
## [43] cards_0.7.1          withr_3.0.2          scales_1.4.0        
## [46] backports_1.5.0      timechange_0.3.0     rmarkdown_2.30      
## [49] httr_1.4.7           otel_0.2.0           hms_1.1.3           
## [52] evaluate_1.0.5       viridisLite_0.4.2    mgcv_1.9-3          
## [55] rlang_1.1.7          glue_1.8.0           xml2_1.5.2          
## [58] svglite_2.2.2        rstudioapi_0.18.0    jsonlite_2.0.0      
## [61] R6_2.6.1             systemfonts_1.3.1

Statistical Modeling in Epidemiology

EPI 553 - Advanced Epidemiologic Methods

Lauren Martin

2-24-26

Introduction

Setup and Data Preparation

Descriptive Statistics

Part 1: Statistical Modeling Concepts

1. What is Statistical Modeling?

General Form of a Statistical Model

2. Types of Regression Models

Simple vs. Multiple Regression

3. Linear Regression Example

Simple Linear Regression

Visualization

With continuous age

4. Logistic Regression: The Preferred Model for Binary Outcomes

Simple Logistic Regression

Predicted Probabilities

5. Multiple Regression: Controlling for Confounding

What is Confounding?

Multiple Logistic Regression

6. Dummy Variables: Coding Categorical Predictors

Example: Education Level

Visualizing Education Effects

7. Interactions (Effect Modification)

Example: Age × Sex Interaction

Visualizing Interaction

8. Model Diagnostics

Key Assumptions for Logistic Regression

Checking for Multicollinearity

Influential Observations

9. Model Comparison and Selection

Comparing Nested Models

10. Error Term in Statistical Models

Part 2: Student Lab Activity

Lab Overview

Lab Instructions

Task 1: Explore the Outcome Variable

Task 2: Build a Simple Logistic Regression Model

Task 3: Create a Multiple Regression Model

Task 4: Interpret Dummy Variables

Task 5: Test for Interaction

Task 6: Model Diagnostics

Task 7: Model Comparison

Lab Report Guidelines

Summary

Key Concepts Covered

Important Formulas

References

`2-24-26`