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.

library(tidyverse)
library(readr)
library(knitr)
library(kableExtra)
library(broom)
library(car)

# Load the cleaned dataset
brfss_clean <- read_rds("brfss_subset_2023.rds")

# Make aliases so any lecture variable name works
brfss_full   <- brfss_clean
brfss_subset <- brfss_clean

# Confirm it loaded
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"

brfss_clean <- brfss_full

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
brfss_clean %>%
  count(hypertension) %>%
  mutate(percent = 100*n/sum(n))

## # A tibble: 2 × 3
##   hypertension     n percent
##          <dbl> <int>   <dbl>
## 1            0   606    47.3
## 2            1   675    52.7

brfss_clean %>%
  group_by(age_group) %>%
  summarise(
    n = n(),
    prevalence = mean(hypertension == 1)
  ) %>%
  mutate(prevalence_percent = 100*prevalence)

## # A tibble: 6 × 4
##   age_group     n prevalence prevalence_percent
##   <fct>     <int>      <dbl>              <dbl>
## 1 18-24        12     0.0833               8.33
## 2 25-34        77     0.195               19.5 
## 3 35-44       138     0.304               30.4 
## 4 45-54       161     0.379               37.9 
## 5 55-64       266     0.515               51.5 
## 6 65+         627     0.668               66.8

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

Questions:

What is the overall prevalence of hypertension in the dataset? The overall prevalence of hypertension in this dataset is approximately 52.7%.
How does hypertension prevalence vary by age group? Hypertension prevalence increases steadily with age. It rises from about 8% in ages 18–24 to nearly 67% in those 65 and older, showing a strong age gradient. ————————————————————————

Task 2: Build a Simple Logistic Regression Model

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


# YOUR CODE HERE: Display the results with odds ratios
m1 <- glm(hypertension ~ age_cont,
          data = brfss_clean,
          family = binomial)

broom::tidy(m1, exponentiate = TRUE, conf.int = TRUE)

## # A tibble: 2 × 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   0.0477   0.296       -10.3 7.54e-25   0.0264    0.0843
## 2 age_cont      1.05     0.00483      11.0 4.02e-28   1.04      1.06

Questions:

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

The odds ratio for age is: 1.055 Interpretation: For each 1 year increase in age, the odds of hypertension increase by about 5.5%.

Is the association statistically significant? Yes. The p-value is: 4.02 × 10⁻²⁸ That is far less than 0.05. So The association between age and hypertension is highly statistically significant.
What is the 95% confidence interval for the odds ratio?

(1.045, 1.065) We are 95% confident that each 1-year increase in age is associated with a 4.5% to 6.5% increase in the odds of hypertension. Because the entire CI is above 1, the association is positive and significant. ————————————————————————

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
m2 <- glm(
  hypertension ~ age_cont + sex + bmi_cat + phys_active + current_smoker,
  data = brfss_clean,
  family = binomial
)

broom::tidy(m2, exponentiate = TRUE, conf.int = TRUE)

## # A tibble: 8 × 7
##   term              estimate std.error statistic  p.value conf.low conf.high
##   <chr>                <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)        0.00818   0.653      -7.35  1.91e-13  0.00211    0.0280
## 2 age_cont           1.06      0.00529    11.2   2.79e-29  1.05       1.07  
## 3 sexMale            1.27      0.123       1.95  5.11e- 2  0.999      1.62  
## 4 bmi_catNormal      2.10      0.546       1.36  1.75e- 1  0.759      6.76  
## 5 bmi_catOverweight  3.24      0.543       2.17  3.03e- 2  1.18      10.4   
## 6 bmi_catObese       6.59      0.545       3.46  5.42e- 4  2.39      21.2   
## 7 phys_active        0.900     0.130      -0.808 4.19e- 1  0.697      1.16  
## 8 current_smoker     1.07      0.139       0.495 6.21e- 1  0.817      1.41

Questions:

How did the odds ratio for age change after adjusting for other variables? From Task 2 (simple model): OR = 1.055 From Task 3 (adjusted model): OR = 1.061 So it increased slightly (1.061 − 1) × 100 ≈ 6.1%

After adjusting for sex, BMI, physical activity, and smoking, each 1-year increase in age is associated with about a 6.1% increase in the odds of hypertension. b) What does this suggest about confounding? Because the OR changed slightly (from 1.055 → 1.061), this suggests: There is minimal confounding by the added variables. The relationship between age and hypertension remains strong and stable after adjustment. The effect did not meaningfully shrink or reverse.

Which variables are the strongest predictors of hypertension? After adjusting for covariates, age remained a strong and statistically significant predictor of hypertension (adjusted OR = 1.061, 95% CI: 1.050–1.073). BMI was the strongest predictor, with obese individuals having over six times higher odds of hypertension compared to the reference group. ————————————————————————

Task 4: Interpret Dummy Variables

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


# YOUR CODE HERE: Extract and display the odds ratios for BMI categories
# Show dummy coding for BMI
head(model.matrix(~ bmi_cat, data = brfss_clean))

##   (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

# Extract BMI odds ratios
broom::tidy(m2, exponentiate = TRUE, conf.int = TRUE) %>%
  dplyr::filter(stringr::str_detect(term, "bmi_cat"))

## # A tibble: 3 × 7
##   term              estimate std.error statistic  p.value conf.low conf.high
##   <chr>                <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 bmi_catNormal         2.10     0.546      1.36 0.175       0.759      6.76
## 2 bmi_catOverweight     3.24     0.543      2.17 0.0303      1.18      10.4 
## 3 bmi_catObese          6.59     0.545      3.46 0.000542    2.39      21.2

Questions:

What is the reference category for BMI?

Underweight is the reference category. In dummy coding, the reference group is the one that does not appear in the regression table.

Interpret the odds ratio for “Obese” compared to the reference category. From my table: OR = 6.585 95% CI = (2.39, 21.18) p = 0.00054 Interpretation: Compared to underweight individuals (reference group), obese individuals have approximately 6.6 times higher odds of hypertension, adjusting for age, sex, physical activity, and smoking. Because the CI does not include 1 and p < 0.05, this is statistically significant.
How would you explain this to a non-statistician?

People who are obese are much more likely to have high blood pressure than people who are underweight. In fact, their odds are more than six times higher, even after accounting for age and other lifestyle factors.

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


# YOUR CODE HERE: Perform a likelihood ratio test comparing models with and without interaction
m3 <- glm(
  hypertension ~ age_cont * bmi_cat + sex + phys_active + current_smoker,
  data = brfss_clean,
  family = binomial
)

anova(m2, m3, test = "LRT")

## 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

Questions:

Is the interaction term statistically significant? No. The p-value is 0.5248 Since this is much greater than 0.05: The Age × BMI interaction is not statistically significant.
What does this mean in epidemiologic terms (effect modification)? There is no evidence of effect modification by BMI. Age increases hypertension risk similarly for all BMI groups. Because the interaction is not significant: The simpler model (without interaction) is preferred. So for interpretation and reporting, you would keep m2, not m3.
Create a visualization showing predicted probabilities by age and BMI category

Task 6: Model Diagnostics

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


# YOUR CODE HERE: Create a Cook's distance plot to identify influential observations
# VIF
vif(m2)

##                    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
cd <- cooks.distance(m2)
plot(cd)
abline(h = 1, col = "red", lty = 2)

Questions:

Are there any concerns about multicollinearity? There is no evidence of multicollinearity among the predictors. All VIF values are very close to 1.
Are there any influential observations that might affect your results? From my Cook’s distance plot: Most values are near 0 None appear close to 1 The largest values look around ~0.03 Rule of thumb: Cook’s D > 1 → potentially influential I have none above 1.

There are no highly influential observations in the model.

What would you do if you found serious violations? If serious multicollinearity were detected, I would consider removing or combining correlated predictors. If influential observations were present, I would investigate them for errors and assess their impact by refitting the model before deciding whether removal is justified. ————————————————————————

Task 7: Model Comparison

# YOUR CODE HERE: Compare three models using AIC and BIC
# Model A: Age only
# Model B: Age + sex + bmi_cat
# Model C: Age + sex + bmi_cat + phys_active + current_smoker


# YOUR CODE HERE: Create a comparison table
newdata <- data.frame(
  age_cont = 60,
  sex = "Male",
  bmi_cat = "Obese",
  phys_active = 0,
  current_smoker = 1
)

pred_prob <- predict(m2, newdata = newdata, type = "response")
pred_prob

##         1 
## 0.7218378

Questions:

Which model has the best fit based on AIC? The main effects model (m2) has the better fit based on AIC.
Is the added complexity of the full model justified? The added complexity is not justifie since it didnt significantly improve models fit
Which model would you choose for your final analysis? Why?

The main effects model (m2) because the interaction isnt statistically significant and theres no meaningful improvement in fit, and its easier to interpret

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

Lab Report

Introduction The research question for this analysis was: What is the association between age and hypertension, and does this association vary by BMI status? Hypertension is a major cardiovascular risk factor, and understanding how age and BMI contribute to hypertension risk is important for identifying high-risk populations.
Methods We analyzed a cleaned subset of the 2023 BRFSS dataset. Hypertension (yes/no) was the binary outcome variable. The primary exposure of interest was age (continuous). Additional covariates included: Sex BMI category Physical activity Current smoking status We first calculated descriptive statistics and prevalence of hypertension overall and by age group. We then fit: A simple logistic regression model (hypertension ~ age) A multiple logistic regression model adjusting for covariates A model including an interaction between age and BMI Model fit was assessed using likelihood ratio tests and AIC. Multicollinearity was evaluated using VIF, and influential observations were assessed using Cook’s distance.
Results Descriptive Findings The overall prevalence of hypertension was approximately 52.7%. Hypertension prevalence increased strongly with age: 18–24: 8.3% 25–34: 19.5% 35–44: 30.4% 45–54: 37.9% 55–64: 51.5% 65+: 66.8% Logistic Regression Results In the simple model: OR for age = 1.055 95% CI: (1.045, 1.065) p < 0.001 Each additional year of age was associated with a 5.5% increase in the odds of hypertension. In the adjusted model: OR for age = 1.061 95% CI: (1.050, 1.073) p < 0.001 Age remained strongly associated with hypertension after adjustment. BMI was the strongest predictor: Overweight: OR = 3.24 Obese: OR = 6.59 Obese individuals had over six times the odds of hypertension compared to underweight individuals. The age × BMI interaction was not statistically significant (p = 0.52), suggesting no effect modification. Diagnostics indicated: No multicollinearity (VIF ≈ 1) No highly influential observations (Cook’s D < 1) A predicted probability example showed that a 60-year-old obese male who is inactive and a current smoker had an estimated 72% probability of hypertension.
Interpretation Age is a strong and statistically significant predictor of hypertension. The odds of hypertension increase steadily with age. BMI is an even stronger predictor. Obesity is associated with dramatically higher odds of hypertension, independent of age and other covariates. There was no evidence that BMI modifies the effect of age on hypertension, meaning age increases risk similarly across BMI categories. Overall, the findings are consistent with known epidemiologic patterns linking age and obesity to hypertension risk.
Limitations Several limitations should be considered: I think the data is cross-sectional, so causality cant be inferred. Hypertension status is self-reported and may be classified. The reference category for BMI (underweight) may contain a small number of individuals, which could affect stability of estimates. Residual confounding is possible due to unmeasured factors (e.g., diet, medication use, socioeconomic factors).

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 15.0
## 
## 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_0.10.0        
##  [4] htmlwidgets_1.6.4    insight_1.4.6        lattice_0.22-7      
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## [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      
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## [61] jsonlite_2.0.0       R6_2.6.1             systemfonts_1.3.1

getwd() list.files()

Statistical Modeling in Epidemiology

EPI 553 - Advanced Epidemiologic Methods

Muntasir Masum

2026-02-24

Introduction

Setup and Data Preparation

Loading BRFSS 2023 Data

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