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)

Loading BRFSS 2023 Data

The BRFSS is a large-scale telephone survey that collects data on health-related risk behaviors, chronic health conditions, and use of preventive services from U.S. residents.

# Load the full BRFSS 2023 dataset
brfss_full <- read_xpt("/Users/mm992584/Library/CloudStorage/OneDrive-UniversityatAlbany-SUNY/Spring 2026/Epi 553/Lectures Notes/Modeling/LLCP2023.XPT") %>%
janitor::clean_names()

# Load the full BRFSS 2023 dataset
brfss_clean <- readRDS("~/Downloads/brfss_subset_2023.rds")
library(janitor)


# Display dataset dimensions
names(brfss_clean)

##  [1] "diabetes"       "age_group"      "age_cont"       "sex"           
##  [5] "race"           "education"      "income"         "bmi_cat"       
##  [9] "phys_active"    "current_smoker" "gen_health"     "hypertension"  
## [13] "high_chol"

Creating a Working Subset

For computational efficiency and teaching purposes, we’ll create a subset with relevant variables and complete cases.

# Display dataset dimensions
names(brfss_full)

# Select variables of interest and create analytic dataset
set.seed(553)  # For reproducibility

brfss_subset <- brfss_full %>%
  select(
    # Outcome: Diabetes status
    diabete4,
    # Demographics
    age_g,      # Age category
    sex,         # Sex
    race,       # Race/ethnicity
    educag,     # Education level
    incomg1,    # Income category
    # Health behaviors
    bmi5cat,    # BMI category
    exerany2,     # Physical activity
    smokday2,     # Smoking frequency
    # Health status
    genhlth,      # General health
    rfhype6,    # High blood pressure
    rfchol3     # High cholesterol
  ) %>%
  # Filter to complete cases only
  drop_na() %>%
  # Sample 2000 observations for manageable analysis
  slice_sample(n = 2000)

# Display subset dimensions
cat("Working subset dimensions:",
    nrow(brfss_subset), "observations,",
    ncol(brfss_subset), "variables\n")

Data Recoding and Cleaning

Descriptive Statistics

# Summary table by diabetes status
desc_table <- brfss_clean %>%
  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_clean)

# 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_clean, 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_clean,
                              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_clean,
                               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_clean,
                         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_clean,
              family = binomial)

# Model 2: Age + Sex
model2 <- glm(diabetes ~ age_cont + sex,
              data = brfss_clean,
              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

tab_htn <- table(brfss_clean$hypertension)
tab_htn

## 
##   0   1 
## 606 675

prop.table(tab_htn)

## 
##         0         1 
## 0.4730679 0.5269321

# YOUR CODE HERE: Calculate the prevalence of hypertension by age group

# Overall prevalence
prev_htn <- mean(brfss_clean$hypertension == 1)
cat("Overall hypertension prevalence:", round(100 * prev_htn, 1), "%\n")

## Overall hypertension prevalence: 52.7 %

# Prevalence by age group
prev_by_age <- brfss_clean %>%
  group_by(age_group) %>%
  summarise(
    N = n(),
    prev_htn = mean(hypertension == 1),
    prev_htn_pct = round(100 * prev_htn, 1),
    .groups = "drop"
  )

prev_by_age %>%
  kable(caption = "Hypertension Prevalence by Age Group",
        digits = 3,
        align = "lrrr",
        col.names = c("Age group", "N", "Prevalence (proportion)", "Prevalence (%)")) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

Hypertension Prevalence by Age Group
Age group	N	Prevalence (proportion)	Prevalence (%)
18-24	12	0.083	8.3
25-34	77	0.195	19.5
35-44	138	0.304	30.4
45-54	161	0.379	37.9
55-64	266	0.515	51.5
65+	627	0.668	66.8

Questions:

What is the overall prevalence of hypertension in the dataset? The overall prevalence of in the dataset was 52.7%, showiing that slightly more than half of the study population had hypertension.
How does hypertension prevalence vary by age group? Young adults have low prevalence, while older adults have much higher prevalence. The hypertension prevalence increases with age. ————————————————————————

Task 2: Build a Simple Logistic Regression Model

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

# Simple logistic regression: hypertension ~ age_cont
model_htn_simple <- glm(hypertension ~ age_cont,
                        data = brfss_clean,
                        family = binomial(link = "logit"))



# YOUR CODE HERE: Display the results with odds ratios

tidy(model_htn_simple, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Simple Logistic Regression: Hypertension and Age With 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 and Age With 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

# Store key values for inline interpretation
or_age_simple <- exp(coef(model_htn_simple)["age_cont"])
ci_age_simple <- exp(confint(model_htn_simple)["age_cont", ])
p_age_simple <- tidy(model_htn_simple) %>% filter(term == "age_cont") %>% pull(p.value)

Questions:

What is the odds ratio for age? Interpret this value. Odds ratio for age is 1.055 meaninf that for each 1-year increase in age, the odds of hypertension increase by about 5.5%.
Is the association statistically significant? Yes, because the p-value is less than .001.
What is the 95% confidence interval for the odds ratio? The 95% confidence interval for the odds ratio was 1.045 to 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

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

# YOUR CODE HERE: Display the results

tidy(model_htn_mult, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Multiple Logistic Regression: Hypertension ~ Age + Covariates in 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 = "350px")

Multiple Logistic Regression: Hypertension ~ Age + Covariates in Odds Ratios
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	0.008	0.653	-7.355	0.000	0.002	0.028
age_cont	1.061	0.005	11.234	0.000	1.050	1.073
sexMale	1.270	0.123	1.950	0.051	0.999	1.616
bmi_catNormal	2.097	0.546	1.356	0.175	0.759	6.756
bmi_catOverweight	3.241	0.543	2.166	0.030	1.183	10.385
bmi_catObese	6.585	0.545	3.459	0.001	2.394	21.176
phys_active	0.900	0.130	-0.808	0.419	0.697	1.162
current_smoker	1.071	0.139	0.495	0.621	0.817	1.407

# Values for confounding 
or_age_adj <- exp(coef(model_htn_mult)["age_cont"])
pct_change_age_or <- 100 * (or_age_adj - or_age_simple) / or_age_simple

# Identify strongest predictors by distance from OR=1 
strongest <- tidy(model_htn_mult, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(term != "(Intercept)") %>%
  mutate(dist_from_1 = abs(estimate - 1)) %>%
  arrange(desc(dist_from_1)) %>%
  slice(1:5) %>%
  select(term, estimate, conf.low, conf.high, p.value)
strongest

## # A tibble: 5 × 5
##   term              estimate conf.low conf.high  p.value
##   <chr>                <dbl>    <dbl>     <dbl>    <dbl>
## 1 bmi_catObese         6.59     2.39      21.2  0.000542
## 2 bmi_catOverweight    3.24     1.18      10.4  0.0303  
## 3 bmi_catNormal        2.10     0.759      6.76 0.175   
## 4 sexMale              1.27     0.999      1.62 0.0511  
## 5 phys_active          0.900    0.697      1.16 0.419

Questions:

How did the odds ratio for age change after adjusting for other variables? After adjusting for sex, BMI, physical activity, and smoking, the odds ratio for age increased slightly from 1.055 to 1.061, showing a small strength of the association.
What does this suggest about confounding? The change in the age odds ratio shows limited confounding, proviing that age is a predictor of hypertension even after controlling for other variables.
Which variables are the strongest predictors of hypertension? The strongest predictors were BMI and age. People who were obese had over six times higher odds of hypertension compared with the other group, and overweight individuals had more than three times higher odds. ————————————————————————

Task 4: Interpret Dummy Variables

# YOUR CODE HERE: Create a table showing the dummy variable coding for bmi_cat

bmi_mm <- model.matrix(~ bmi_cat, data = brfss_clean)
head(bmi_mm)

##   (Intercept) bmi_catNormal bmi_catOverweight bmi_catObese
## 1           1             0                 0            1
## 2           1             0                 0            1
## 3           1             1                 0            0
## 4           1             1                 0            0
## 5           1             0                 1            0
## 6           1             1                 0            0

# YOUR CODE HERE: Extract and display the odds ratios for BMI categories

bmi_or <- tidy(model_htn_mult, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(str_detect(term, "^bmi_cat")) %>%
  mutate(
    BMI_level = str_remove(term, "bmi_cat")
  ) %>%
  select(BMI_level, estimate, conf.low, conf.high, p.value)

bmi_or %>%
  kable(caption = "BMI Category Odds Ratios, Multiple Model",
        digits = 3,
        col.names = c("BMI Level", "OR", "95% CI Low", "95% CI High", "p-value")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

BMI Category Odds Ratios, Multiple Model
BMI Level	OR	95% CI Low	95% CI High	p-value
Normal	2.097	0.759	6.756	0.175
Overweight	3.241	1.183	10.385	0.030
Obese	6.585	2.394	21.176	0.001

levels(brfss_clean$bmi_cat)

## [1] "Underweight" "Normal"      "Overweight"  "Obese"

Questions:

What is the reference category for BMI? Underweight is the reference category.
Interpret the odds ratio for “Obese” compared to the reference category. People who are obese have about 6.6 times higher odds of hypertension compared to underweight people, after adjusting for other variables.
How would you explain this to a non-statistician? People who are obese are more likely to have high blood pressure than people who are underweight. The results shown suggest a strong relationship between higher body weight and hypertension risk.

Task 5: Test for Interaction

# YOUR CODE HERE: Fit a model with Age × BMI interaction
# Test if the effect of age on hypertension differs by BMI category

# Without interaction
m_no_int <- glm(hypertension ~ age_cont + bmi_cat + sex + phys_active + current_smoker,
                data = brfss_clean, family = binomial)

# With interaction
m_int <- glm(hypertension ~ age_cont * bmi_cat + sex + phys_active + current_smoker,
             data = brfss_clean, family = binomial)


# YOUR CODE HERE: Perform a likelihood ratio test comparing models with and without interaction
lrt_int <- anova(m_no_int, m_int, test = "LRT")
lrt_int

## Analysis of Deviance Table
## 
## Model 1: hypertension ~ age_cont + bmi_cat + sex + phys_active + current_smoker
## Model 2: hypertension ~ age_cont * bmi_cat + sex + phys_active + current_smoker
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1      1273     1563.5                     
## 2      1270     1561.3  3   2.2363   0.5248

# Visualize predicted probabilities
pred_int <- ggpredict(m_int, terms = c("age_cont [18:80]", "bmi_cat"))
p_int <- ggplot(pred_int, aes(x = x, y = predicted, color = group, fill = group)) +
  geom_line(linewidth = 1.1) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.15, color = NA) +
  scale_y_continuous(labels = scales::percent_format()) +
  labs(
    title = "Predicted Probability of Hypertension by Age and BMI",
    subtitle = "Testing Age × BMI Interaction (Effect Modification)",
    x = "Age in years",
    y = "Predicted Probability of Hypertension",
    color = "BMI category",
    fill = "BMI category"
  ) +
  theme_minimal()

ggplotly(p_int)

Questions:

Is the interaction term statistically significant? No, age and BMI aren’t statistically significant because p > 0.05
What does this mean in epidemiologic terms (effect modification)? The relationship between age and hypertension are similar across BMI categories, meaning there’s no evidence of effect modification.
Create a visualization showing predicted probabilities by age and BMI category In the figure, individuals with obesity consistently have the highest predicted probability, followed by overweight and normal BMI groups. ————————————————————————

Task 6: Model Diagnostics

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

v <- vif(model_htn_mult)

if (is.matrix(v)) {
  vif_df <- data.frame(
    Variable = rownames(v),
    VIF = v[, "GVIF^(1/(2*Df))"],
    row.names = NULL
  )
} else {
  vif_df <- data.frame(
    Variable = names(v),
    VIF = as.numeric(v),
    row.names = NULL
  )
}

vif_df %>%
  arrange(desc(VIF)) %>%
  kable(caption = "VIF (or adjusted GVIF) for Multiple Logistic Model",
        digits = 2,
        align = "lr") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

VIF (or adjusted GVIF) for Multiple Logistic Model
Variable	VIF
age_cont	1.06
current_smoker	1.04
bmi_cat	1.02
phys_active	1.01
sex	1.01

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

cd <- cooks.distance(model_htn_mult)
influence_df <- tibble(
  observation = seq_along(cd),
  cooks_d = cd,
  influential = ifelse(cd > 1, "Yes", "No")
)

p_cd <- ggplot(influence_df, aes(x = observation, y = cooks_d, color = influential)) +
  geom_point(alpha = 0.6) +
  geom_hline(yintercept = 1, linetype = "dashed") +
  labs(
    title = "Cook's Distance: Multiple Model",
    subtitle = "Cook's D > 1 indicates potentially influential observations",
    x = "Observation",
    y = "Cook's Distance",
    color = "Influential?"
  ) +
  theme_minimal()

ggplotly(p_cd)

cat("Number of observations with Cook's D > 1:", sum(cd > 1), "\n")

## Number of observations with Cook's D > 1: 0

Questions:

Are there any concerns about multicollinearity? There is no evidence of multicollinearity, since all VIF values are close to 1.
Are there any influential observations that might affect your results? No, because all Cook’s distance values were below 1.
What would you do if you found serious violations? If there were serious violations, I would examine the data for errors and consider removing or combining correlated variables. I would also create sensitivity analyses to assess how influential observations affect model results.

Task 7: Model Comparison

# YOUR CODE HERE: Compare three models using AIC and BIC
# Model 1: Age
# Model 2: Age + sex + bmi_cat
# Model 3: Age + sex + bmi_cat + phys_active + current_smoker

model1 <- glm(hypertension ~ age_cont,
          data = brfss_clean, family = binomial)

model2 <- glm(hypertension ~ age_cont + sex + bmi_cat,
          data = brfss_clean, family = binomial)

model3 <- glm(hypertension ~ age_cont + sex + bmi_cat + phys_active + current_smoker,
          data = brfss_clean, family = binomial)

# YOUR CODE HERE: Create a comparison table

model_comp <- data.frame(
  Model = c("1: Age",
            "2: Age + Sex + BMI",
            "3: + Physical activity + Smoking"),
  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 for Hypertension",
        digits = 2,
        align = "lrrr") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE)

Model Comparison for Hypertension
Model	AIC	BIC	Deviance
1: Age	1636.61	1646.92	1632.61
2: Age + Sex + BMI	1576.49	1607.42	1564.49
3: + Physical activity + Smoking	1579.50	1620.74	1563.50

Questions:

Which model has the best fit based on AIC? Model 2 has the best fit because it has the lowest AIC, based on the table.
Is the added complexity of the full model justified?
Which model would you choose for your final analysis? Why? The added complexity of the full model is not justified because it doesn’t improve AIC or deviance. ————————————————————————

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

The goal of this analysis was to examine factors associated with hypertension using the BRFSS 2023 dataset. The main research question was whether age, sex, BMI, physical activity, and smoking are related to the likelihood of having hypertension. We also explored whether the relationship between age and hypertension differs across BMI categories. We used a cleaned subset of the BRFSS dataset containing demographic and health behavior variables. Hypertension was the outcome variable. First, descriptive statistics were calculated to understand the prevalence of hypertension overall and by age group. Then, logistic regression models were used to estimate associations between predictors and hypertension. We built a simple model with age only, multiple model including age, sex, BMI, physical activity, and smoking, and a interaction model to test whether BMI modifies the effect of age. The model diagnostics such as VIF and Cook’s distance were used to check assumptions. Model fit was compared using AIC and BIC. The overall prevalence of hypertension in the dataset was about 53%, and prevalence increased steadily with age. Older age groups showed much higher hypertension rates compared to younger groups. In the simple logistic regression, age was significantly associated with hypertension. Each one-year increase in age increased the odds of hypertension by about 5–6%. In the multiple model, age remained a strong predictor after adjustment. BMI was also an important predictor. Individuals with obesity had about 6.6 times higher odds of hypertension compared to the reference BMI group, and overweight individuals also had higher odds. Sex showed a borderline association, while physical activity and smoking were not significant predictors. The interaction between age and BMI was not statistically significant, suggesting that the effect of age on hypertension is similar across BMI categories. Model diagnostics showed no multicollinearity and no influential observations. Model comparison indicated that the model including age, sex, and BMI had the best fit. These findings show that age and BMI are major factors related to hypertension. As people get older, their risk of hypertension increases. Higher BMI, especially obesity, is strongly associated with hypertension. Although lifestyle behaviors like smoking and physical activity are important for health, they did not significantly improve prediction in this model. The lack of interaction indicates that BMI does not change how age affects hypertension, meaning age increases risk in a similar way across BMI groups. This analysis has a few limitations. The data are cross-sectional, so we cannot determine cause and effect. Also, many variables are self-reported, which may introduce measurement error. The subset sample size is smaller than the full BRFSS dataset, which may limit precision. Important factors that can alter values such as diet, medication use, and genetics were not included, which could affect the results. Submission: Submit your completed R Markdown file and knitted HTML report.

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 4.5.2 (2025-10-31)
## Platform: aarch64-apple-darwin20
## Running under: macOS Sequoia 15.6
## 
## Matrix products: default
## BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.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_2.1.6     
## [17] tidyr_1.3.2      tibble_3.3.1     ggplot2_4.0.1    tidyverse_2.0.0 
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.6         xfun_0.56            bslib_0.9.0         
##  [4] htmlwidgets_1.6.4    insight_1.4.6        lattice_0.22-7      
##  [7] tzdb_0.5.0           crosstalk_1.2.2      vctrs_0.7.1         
## [10] tools_4.5.2          generics_0.1.4       datawizard_1.3.0    
## [13] pkgconfig_2.0.3      Matrix_1.7-4         data.table_1.18.0   
## [16] RColorBrewer_1.1-3   S7_0.2.1             lifecycle_1.0.5     
## [19] compiler_4.5.2       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_1.11.1       
## [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_4.5.2        labelled_2.16.0      fastmap_1.2.0       
## [40] grid_4.5.2           cli_3.6.5            magrittr_2.0.4      
## [43] cards_0.7.1          utf8_1.2.6           withr_3.0.2         
## [46] scales_1.4.0         backports_1.5.0      timechange_0.3.0    
## [49] rmarkdown_2.30       httr_1.4.7           otel_0.2.0          
## [52] hms_1.1.4            evaluate_1.0.5       viridisLite_0.4.2   
## [55] mgcv_1.9-3           rlang_1.1.7          glue_1.8.0          
## [58] xml2_1.5.2           svglite_2.2.2        rstudioapi_0.18.0   
## [61] jsonlite_2.0.0       R6_2.6.1             systemfonts_1.3.1

Statistical Modeling in Epidemiology

EPI 553 - Advanced Epidemiologic Methods

Muntasir Masum

2026-02-25

Introduction

Setup and Data Preparation

Loading BRFSS 2023 Data

Creating a Working Subset

Data Recoding and Cleaning

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