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_clean <- read_rds("brfss_subset_2023.rds")

# Display dataset dimensions
## 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

Creating a Working Subset

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

# Create analytic dataset from cleaned BRFSS subset
set.seed(553)  # For reproducibility

brfss_subset <- brfss_clean %>%
  select(any_of(c(
    "diabetes",
    "age_cont",
    "sex",
    "race",
    "educag",
    "incomg1",
    "bmi_cat",
    "phys_active",
    "current_smoker",
    "hypertension",
    "high_chol"
  ))) %>%
  drop_na() %>%
  slice_sample(n = 2000)

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

## Working subset dimensions: 1281 observations, 9 variables

Data Recoding and Cleaning

library(dplyr)
library(readr)

brfss_clean2 <- brfss_subset %>%
  mutate(
    # Make sure outcome is 0/1 (if it’s already 0/1, this won’t change it)
    diabetes = case_when(
      diabetes == 1 ~ 1,
      diabetes == 0 ~ 0,
      TRUE ~ NA_real_
    ),

    # Create age groups from age_cont (since you have age_cont, not age_g)
    age_group = cut(
      age_cont,
      breaks = c(18, 25, 35, 45, 55, 65, Inf),
      right = FALSE,
      labels = c("18-24", "25-34", "35-44", "45-54", "55-64", "65+")
    ),

    # Ensure factors look right
    sex = factor(sex),
    bmi_cat = factor(bmi_cat, levels = c("Underweight", "Normal", "Overweight", "Obese"))
  ) %>%
  select(diabetes, age_group, age_cont, sex, bmi_cat,
         phys_active, current_smoker, hypertension, high_chol) %>%
  drop_na()

cat("Clean dataset created with", nrow(brfss_clean2), "complete observations\n")

## Clean dataset created with 1281 complete observations

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

# Frequency table of hypertension
table(brfss_clean2$hypertension)

## 
##   0   1 
## 606 675

# Proportions (overall prevalence)
prop.table(table(brfss_clean2$hypertension))

## 
##         0         1 
## 0.4730679 0.5269321

# Prevalence of hypertension by age group
brfss_clean2 %>%
  group_by(age_group) %>%
  summarise(
    N = n(),
    Hypertension_Cases = sum(hypertension == 1),
    Prevalence = round(mean(hypertension == 1) * 100, 1)
  )

## # A tibble: 6 × 4
##   age_group     N Hypertension_Cases Prevalence
##   <fct>     <int>              <int>      <dbl>
## 1 18-24        12                  1        8.3
## 2 25-34        77                 15       19.5
## 3 35-44       138                 42       30.4
## 4 45-54       161                 61       37.9
## 5 55-64       266                137       51.5
## 6 65+         627                419       66.8

Questions:

What is the overall prevalence of hypertension in the dataset?

675 individuals with hypertension

606 without hypertension

Prevalence = 675/ 1281 = 0.527

The analytic sample included 1,281 adults, and the overall prevalence of hypertension was 52.7%.
How does hypertension prevalence vary by age group?

Hypertension prevalence increases steadily with age. It is lowest among adults aged 18–24 (8.3%) and rises progressively across age categories, reaching its highest level among adults aged 65+ (66.8%). This pattern suggests a strong positive association between age and hypertension prevalence.

Task 2: Build a Simple Logistic Regression Model

# Fit simple logistic regression model
model1 <- glm(hypertension ~ age_cont,
              data = brfss_clean2,
              family = binomial)

# View standard model output
summary(model1)

## 
## Call:
## glm(formula = hypertension ~ age_cont, family = binomial, data = brfss_clean2)
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.042577   0.295584  -10.29   <2e-16 ***
## age_cont     0.053119   0.004831   11.00   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1772.1  on 1280  degrees of freedom
## Residual deviance: 1632.6  on 1279  degrees of freedom
## AIC: 1636.6
## 
## Number of Fisher Scoring iterations: 4

# Display odds ratios with 95% CI
exp(cbind(OR = coef(model1), confint(model1)))

##                     OR      2.5 %     97.5 %
## (Intercept) 0.04771176 0.02644276 0.08431815
## age_cont    1.05455475 1.04476526 1.06475213

Questions:

What is the odds ratio for age? Interpret this value.

OR = 1.055. For each one-year increase in age, the odds of hypertension increase by approximately 5.5%, holding all else constant. We get 5.5% from: (1.055 − 1) × 100.
Is the association statistically significant?

Yes, the association between age and hypertension is statistically significant, meaning there is strong evidence that age is related to hypertension risk.
What is the 95% confidence interval for the odds ratio?

The 95% confidence interval for the odds ratio is 1.045–1.065, suggesting a precise and positive association between age and hypertension. Meaning, we are 95% confident that each additional year of age increases the odds of hypertension by between 4.5% and 6.5%.

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

Questions:

How did the odds ratio for age change after adjusting for other variables?
What does this suggest about confounding?
Which variables are the strongest predictors of hypertension?

Task 4: Interpret Dummy Variables

# Set reference categories (run BEFORE model)
brfss_clean2 <- brfss_clean2 %>%
  mutate(
    sex = relevel(factor(sex), ref = "Female"),
    bmi_cat = relevel(factor(bmi_cat), ref = "Normal")
  )

# Fit multiple logistic regression model
model2 <- glm(hypertension ~ age_cont + sex + bmi_cat + phys_active + current_smoker,
              data = brfss_clean2,
              family = binomial)

# Show model summary
summary(model2)

## 
## Call:
## glm(formula = hypertension ~ age_cont + sex + bmi_cat + phys_active + 
##     current_smoker, family = binomial, data = brfss_clean2)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)        -4.065489   0.385202 -10.554  < 2e-16 ***
## age_cont            0.059453   0.005292  11.234  < 2e-16 ***
## sexMale             0.239129   0.122612   1.950  0.05114 .  
## bmi_catUnderweight -0.740579   0.546292  -1.356  0.17521    
## bmi_catOverweight   0.435353   0.158073   2.754  0.00588 ** 
## bmi_catObese        1.144248   0.162600   7.037 1.96e-12 ***
## phys_active        -0.105371   0.130457  -0.808  0.41926    
## current_smoker      0.068533   0.138515   0.495  0.62076    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1772.1  on 1280  degrees of freedom
## Residual deviance: 1563.5  on 1273  degrees of freedom
## AIC: 1579.5
## 
## Number of Fisher Scoring iterations: 4

# Show odds ratios + 95% CI
exp(cbind(OR = coef(model2), confint(model2)))

##                           OR       2.5 %     97.5 %
## (Intercept)        0.0171546 0.007943934 0.03599689
## age_cont           1.0612558 1.050496837 1.07253490
## sexMale            1.2701421 0.998922794 1.61567286
## bmi_catUnderweight 0.4768376 0.148012712 1.31683702
## bmi_catOverweight  1.5455093 1.134572337 2.10902614
## bmi_catObese       3.1400805 2.288478035 4.33033241
## phys_active        0.8999907 0.696650987 1.16203458
## current_smoker     1.0709359 0.816955023 1.40654285

Questions:

What is the reference category for BMI?

The reference category is “Normal BMI.”
Interpret the odds ratio for “Obese” compared to the reference category.

OR = 3.14

95% CI: 2.29 – 4.33

p < 0.001

Individuals classified as obese have 3.14 times the odds of hypertension compared to individuals with normal BMI, after adjusting for age, sex, physical activity, and smoking.
How would you explain this to a non-statistician?

People who are obese are much more likely to have high blood pressure than people with a normal weight. In fact, their risk is more than three times higher, even after accounting for age, sex, physical activity, and smoking.

Task 5: Test for Interaction

# Model WITHOUT interaction (main effects only)
model_no_int <- glm(hypertension ~ age_cont + sex + bmi_cat + phys_active + current_smoker,
                    data = brfss_clean2,
                    family = binomial)

# Model WITH Age × BMI interaction
model_int <- glm(hypertension ~ age_cont * bmi_cat + sex + phys_active + current_smoker,
                 data = brfss_clean2,
                 family = binomial)

# Compare models using Likelihood Ratio Test (LRT)
anova(model_no_int, model_int, test = "Chisq")

## Analysis of Deviance Table
## 
## Model 1: hypertension ~ age_cont + sex + bmi_cat + 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

# ---- Visualization: predicted probabilities by age and BMI ----
library(ggplot2)

# Create a grid of values for prediction
newdata <- expand.grid(
  age_cont = seq(min(brfss_clean2$age_cont),
                 max(brfss_clean2$age_cont),
                 length.out = 100),
  bmi_cat = levels(brfss_clean2$bmi_cat),
  sex = "Female",
  phys_active = 0,
  current_smoker = 0
)

# Predicted probabilities from the interaction model
newdata$pred <- predict(model_int, newdata = newdata, type = "response")

# Plot predicted probabilities
ggplot(newdata, aes(x = age_cont, y = pred, color = bmi_cat)) +
  geom_line(linewidth = 1.2) +
  labs(
    x = "Age",
    y = "Predicted Probability of Hypertension",
    color = "BMI Category",
    title = "Predicted Probability of Hypertension by Age and BMI Category"
  ) +
  theme_minimal()

Questions:

Is the interaction term statistically significant?

p = 0.5248

Decision rule:

p < 0.05 → significant interaction

p ≥ 0.05 → not significant

No, the interaction between age and BMI is not statistically significant (p = 0.525).
What does this mean in epidemiologic terms (effect modification)?

There is no evidence that BMI modifies the relationship between age and hypertension. The effect of age on hypertension appears to be similar across BMI categories. This indicates that BMI does not modify the association between age and hypertension. Therefore, the effect of age on hypertension is consistent across BMI categories.
Create a visualization showing predicted probabilities by age and BMI category

Task 6: Model Diagnostics

# ---- VIF (multicollinearity) ----
library(car)

vif(model2)

##                    GVIF Df GVIF^(1/(2*Df))
## age_cont       1.126628  1        1.061428
## sex            1.016509  1        1.008221
## bmi_cat        1.103045  3        1.016480
## phys_active    1.024820  1        1.012334
## current_smoker 1.073574  1        1.036134

# ---- Cook's distance (influential observations) ----
cooksD <- cooks.distance(model2)

# Basic Cook's D plot
plot(cooksD, type = "h",
     main = "Cook's Distance for Multiple Logistic Regression Model",
     ylab = "Cook's Distance", xlab = "Observation Index")

# Common cutoff line: 4/n
abline(h = 4/length(cooksD), lty = 2)

# Identify which observations exceed cutoff
which(cooksD > 4/length(cooksD))

##   31  168  228  272  280  366  426  440  496  552  629  646  783  814  910 1047 
##   31  168  228  272  280  366  426  440  496  552  629  646  783  814  910 1047 
## 1061 1081 1137 1140 1158 1186 
## 1061 1081 1137 1140 1158 1186

Questions:

Are there any concerns about multicollinearity?

There is no evidence of problematic multicollinearity. The predictors represent distinct constructs (demographic, behavioral, and clinical factors), and variance inflation factors were within acceptable limits.
Are there any influential observations that might affect your results?

A small number of observations exceed the Cook’s distance cutoff, indicating they may be influential. However, the majority of observations fall well below the threshold, suggesting that influential points are limited and unlikely to substantially affect model results.
What would you do if you found serious violations?

If serious multicollinearity were present, I would examine correlations among predictors and consider removing or combining highly correlated variables. If influential observations were identified, I would inspect those cases for errors or unusual values and evaluate whether model results change when they are excluded.

Task 7: Model Comparison

library(dplyr)

# Model A: Age only
modelA <- glm(hypertension ~ age_cont,
              data = brfss_clean2,
              family = binomial)

# Model B: Age + sex + bmi_cat
modelB <- glm(hypertension ~ age_cont + sex + bmi_cat,
              data = brfss_clean2,
              family = binomial)

# Model C: Age + sex + bmi_cat + phys_active + current_smoker
modelC <- glm(hypertension ~ age_cont + sex + bmi_cat + phys_active + current_smoker,
              data = brfss_clean2,
              family = binomial)

# Create comparison table
model_comp <- tibble(
  Model = c("A: Age only",
            "B: Age + sex + BMI",
            "C: Age + sex + BMI + phys_active + current_smoker"),
  AIC = c(AIC(modelA), AIC(modelB), AIC(modelC)),
  BIC = c(BIC(modelA), BIC(modelB), BIC(modelC))
) %>%
  mutate(
    AIC = round(AIC, 1),
    BIC = round(BIC, 1)
  )

model_comp

## # A tibble: 3 × 3
##   Model                                               AIC   BIC
##   <chr>                                             <dbl> <dbl>
## 1 A: Age only                                       1637. 1647.
## 2 B: Age + sex + BMI                                1576. 1607.
## 3 C: Age + sex + BMI + phys_active + current_smoker 1580. 1621.

Questions:

Which model has the best fit based on AIC?

Model B (Age + sex + BMI) has the lowest AIC (1576.5).
Is the added complexity of the full model justified?

The added complexity of the full model is not justified because it results in a higher AIC and BIC compared to Model B.
Which model would you choose for your final analysis? Why?

Model B (Age + sex + BMI) has the lowest AIC (1576.5), indicating the best model fit. The full model (Model C) has a higher AIC and BIC, suggesting that the additional predictors (physical activity and smoking) do not meaningfully improve model performance. Therefore, Model B would be selected for the final analysis because it provides the best balance between model fit and parsimony.

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.

Lab Report:

Introduction

Hypertension is a major public health concern and a leading risk factor for cardiovascular disease. Identifying predictors of hypertension can help inform prevention strategies and clinical screening efforts. The objective of this analysis was to evaluate demographic and behavioral predictors of hypertension and determine which statistical model best predicts hypertension risk (Schmidt et al., 2020).

Methods

We conducted a cross-sectional analysis using a cleaned subset of BRFSS 2023 data (n = 1,281). The outcome variable was hypertension status (yes/no). Predictor variables included age (continuous), sex, BMI category, physical activity, and smoking status. Logistic regression models were used to estimate odds ratios (ORs) and 95% confidence intervals (CIs). Effect modification was assessed using an Age × BMI interaction term and likelihood ratio testing. Model diagnostics included variance inflation factors (VIF) and Cook’s distance. Model fit was compared using AIC and BIC.

Results

The analytic sample included 1,281 adults, and the overall prevalence of hypertension was 52.7%. Hypertension prevalence increased steadily with age.

In adjusted logistic regression models, age and BMI were the strongest predictors of hypertension. Each additional year of age increased the odds of hypertension by approximately 6%. Compared with individuals with normal BMI, overweight individuals had higher odds of hypertension, and obese individuals had more than three times the odds. Sex, smoking, and physical activity were not statistically significant predictors after adjustment.

Model comparison showed that the model including age, sex, and BMI had the lowest AIC and BIC, indicating the best fit.

Interpretation

Age and BMI were the strongest predictors of hypertension in this sample. Each additional year of age increased the odds of hypertension by approximately 6%, and obesity was associated with more than a threefold increase in odds compared with normal BMI. These findings are consistent with established epidemiologic evidence linking aging and adiposity to cardiovascular risk.

The lack of significant associations for smoking and physical activity in adjusted models suggests their effects may be weaker or mediated through other factors in this dataset.

The absence of interaction between age and BMI indicates that the effect of age on hypertension risk is similar across BMI groups.

Limitations

Several limitations should be considered. First, the analysis was cross-sectional, so causal relationships cannot be inferred. Second, BRFSS data rely on self-report, which may introduce misclassification or recall bias. Third, residual confounding may exist due to unmeasured factors such as diet, medication use, or genetic predisposition. Finally, some age groups had small sample sizes, which may reduce estimate precision.

Conclusion

This analysis demonstrates that age and BMI are strong predictors of hypertension, with obesity representing a particularly important risk factor. The most parsimonious and best-fitting model included age, sex, and BMI. These findings reinforce the importance of weight management and age-related screening strategies in hypertension prevention efforts.

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.
Centers for Disease Control and Prevention. (2023). Behavioral Risk Factor Surveillance System.
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.
Schmidt, B. M., Durao, S., Toews, I., Bavuma, C. M., Hohlfeld, A., Nury, E., Meerpohl, J. J., & Kredo, T. (2020). Screening strategies for hypertension. The Cochrane database of systematic reviews, 5(5), CD013212. https://doi.org/10.1002/14651858.CD013212.pub2

Session Info

sessionInfo()

## R version 4.5.2 (2025-10-31 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 11 x64 (build 26200)
## 
## Matrix products: default
##   LAPACK version 3.12.1
## 
## locale:
## [1] LC_COLLATE=English_United States.utf8 
## [2] LC_CTYPE=English_United States.utf8   
## [3] LC_MONETARY=English_United States.utf8
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.utf8    
## 
## 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_1.4.4        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] codetools_0.2-20     htmltools_0.5.9      sass_0.4.10         
## [25] yaml_2.3.12          lazyeval_0.2.2       Formula_1.2-5       
## [28] pillar_1.11.1        jquerylib_0.1.4      broom.helpers_1.22.0
## [31] cachem_1.1.0         abind_1.4-8          nlme_3.1-168        
## [34] tidyselect_1.2.1     digest_0.6.39        stringi_1.8.7       
## [37] labeling_0.4.3       splines_4.5.2        labelled_2.16.0     
## [40] fastmap_1.2.0        grid_4.5.2           cli_3.6.5           
## [43] magrittr_2.0.4       cards_0.7.1          utf8_1.2.6          
## [46] withr_3.0.2          scales_1.4.0         backports_1.5.0     
## [49] timechange_0.3.0     rmarkdown_2.30       httr_1.4.7          
## [52] otel_0.2.0           hms_1.1.4            evaluate_1.0.5      
## [55] viridisLite_0.4.2    mgcv_1.9-3           rlang_1.1.7         
## [58] glue_1.8.0           xml2_1.5.1           svglite_2.2.2       
## [61] rstudioapi_0.18.0    jsonlite_2.0.0       R6_2.6.1            
## [64] systemfonts_1.3.1

Statistical Modeling in Epidemiology

EPI 553 - Advanced Epidemiologic Methods

Jessica Okoroji

2026-02-24

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

Lab Report:

Summary

Key Concepts Covered

Important Formulas

References