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)

###Load brfss_subset_2023.rd

brfss_subset_2023 <- read_rds ("~/Downloads/HEPI553/brfss_subset_2023.rds")

Introduction

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

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

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

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

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

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

Part 1: Statistical Modeling Concepts

1. What is Statistical Modeling?

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

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

General Form of a Statistical Model

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

Where:

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

2. Types of Regression Models

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

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

Simple vs. Multiple Regression

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

3. Linear Regression Example

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

Simple Linear Regression

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

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

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

Interpretation:

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

Visualization

With continuous age

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

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

Diabetes Prevalence by Age

4. Logistic Regression: The Preferred Model for Binary Outcomes

Problem with linear regression for binary outcomes:

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

Solution: Logistic Regression

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

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

Simple Logistic Regression

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

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

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

Interpretation:

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

Predicted Probabilities

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

Predicted Diabetes Probability by Age

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

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

ggplotly(p2)

Predicted Diabetes Probability by Age

5. Multiple Regression: Controlling for Confounding

What is Confounding?

A confounder is a variable that:

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

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

Multiple Logistic Regression

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

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

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

Interpretation:

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

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

6. Dummy Variables: Coding Categorical Predictors

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

Example: Education Level

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

R automatically creates 3 dummy variables:

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

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

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

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

Visualizing Education Effects

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

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

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

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

ggplotly(p3)

Odds Ratios for Education Levels

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

***

7. Interactions (Effect Modification)

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

Epidemiologic term: Effect modification

Example: Age × Sex Interaction

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

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

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

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

Interpretation:

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

Visualizing Interaction

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

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

ggplotly(p4)

Age-Diabetes Relationship by Sex

8. Model Diagnostics

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

Key Assumptions for Logistic Regression

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

Checking for Multicollinearity

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

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

# Calculate VIF
vif_values <- vif(model_logistic_multiple)

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

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

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

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

Influential Observations

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

Cook’s D > 1: Potentially influential observation

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

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

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

ggplotly(p5)

Cook’s Distance for Influential Observations

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

## Number of potentially influential observations: 0

9. Model Comparison and Selection

Comparing Nested Models

Use Likelihood Ratio Test to compare nested models:

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

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

# Model 3: Full model
model3 <- model_logistic_multiple

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

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

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

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

Interpretation:

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

10. Error Term in Statistical Models

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

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

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

Key points:

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

Part 2: Student Lab Activity

Lab Overview

In this lab, you will:

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

Lab Instructions

Task 1: Explore the Outcome Variable

# YOUR CODE HERE: Create a frequency table of hypertension status
tab <- table (brfss_subset_2023$hypertension)
cbind(
  Frequency = tab,
  Percentage = round(prop.table(tab) * 100, 2)
)

##   Frequency Percentage
## 0       606      47.31
## 1       675      52.69

# YOUR CODE HERE: Calculate the prevalence of hypertension by age group
brfss_subset_2023 %>%
  dplyr::group_by(age_group) %>%
  dplyr::summarise(
    total = dplyr::n(),
    cases = sum(hypertension == 1, na.rm = TRUE),
    prevalence = round((cases / total) * 100, 2)
  )

## # A tibble: 6 × 4
##   age_group total cases prevalence
##   <fct>     <int> <int>      <dbl>
## 1 18-24        12     1       8.33
## 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

# Total prevalence as a percentage
overall_prev <- mean(brfss_subset_2023$hypertension == 1, na.rm = TRUE) * 100
overall_prev

## [1] 52.69321

Questions:

What is the overall prevalence of hypertension in the dataset? The overall prevalence of hypertension in the dataset is 52.7.
How does hypertension prevalence vary by age group? As shown above, the prevalence increases as the age increases. This shows that the likelihood of a person to get hypertension is higher as they get older compared to someone who is younger.

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

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

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

Questions:

What is the odds ratio for age? Interpret this value. The odds ratio for age is 1.055. This means that for each 1-year increase in age, the odds of hypertension increase by 5.5%
Is the association statistically significant? The relationship is highly statistically significant (p < 0.001).
What is the 95% confidence interval for the odds ratio? The 95% confidence interval for the odds ratio is 1.045-1.065. ***

### Task 3: Create a Multiple Regression Model
# YOUR CODE HERE: Fit a multiple logistic regression model
# Outcome: hypertension
# Predictors: age_cont, sex, bmi_cat, phys_active, current_smoker


# YOUR CODE HERE: Display the results
# Multiple logistic regression with potential confounders
model_logistic_multiple <- glm(hypertension ~ age_cont + sex + bmi_cat +
                                phys_active + current_smoker,
                               data = brfss_subset_2023,
                               family = binomial(link = "logit"))

# Display results
tidy(model_logistic_multiple, exponentiate = TRUE, conf.int = TRUE) %>%
  kable(caption = "Multiple Logistic Regression: Hypertension ~ 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: Hypertension ~ Age + Covariates (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

Questions:

How did the odds ratio for age change after adjusting for other variables? The odds ratio for age changed after adjusting for other variables because when considering other factors for hypertension, the odds ratio will increase or decrease due to other health predictors such as smoking status, physical activity and BMI.
What does this suggest about confounding? This suggest that confounding may not convey the true relationship between an exposure (variables) and outcome (hypertension).
Which variables are the strongest predictors of hypertension? The variables with the strongest predictors of hypertension are overweight and obese. This shows that the higher in BMI a person is, the higher the chances of them experiencing hypertension.

Task 4: Interpret Dummy Variables

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

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

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

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

tidy(model_logistic_multiple, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(grepl("bmi_cat", term)) %>%
  kable(caption = "Multiple Logistic Regression: Hypertension ~ BMI (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: Hypertension ~ BMI (Odds Ratios)
Term	Odds Ratio	Std. Error	z-statistic	p-value	95% CI Lower	95% CI Upper
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

Questions: a) What is the reference category for BMI? The reference category for BMI is “Underweight” since the category contains all zeros. b) Interpret the odds ratio for “Obese” compared to the reference category. Individuals have a 6.585 times the odds of the outcome (hypertension) compared to those who are underweight. c) How would you explain this to a non-statistician? People who are obese have a 6.585 times as likely to experience hypertension compared to those who are underweight.

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

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

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

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

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

# Display the result
lrt_result

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

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

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

ggplotly(p4)

Questions:

Is the interaction term statistically significant? The p-value = 0.5798, therefore we fail to reject the null and it is not statistically significant.
What does this mean in epidemiologic terms (effect modification)? This means in epidemiologic terms that an interaction exists when the effect of age on the outcome of hypertension is same across levels of the other variables.
Create a visualization showing predicted probabilities by age and BMI category Visualization shown above. ***

Task 6: Model Diagnostics

# YOUR CODE HERE: Calculate VIF for your multiple regression model
# 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	1.06	Low (No concern)
current_smoker	current_smoker	1.04	Low (No concern)
bmi_cat	bmi_cat	1.02	Low (No concern)
phys_active	phys_active	1.01	Low (No concern)
sex	sex	1.01	Low (No concern)

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

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

## Number of potentially influential observations: 0

Questions:

Are there any concerns about multicollinearity? There are no multicollinearity issues since each interpretation is low (no concern).
Are there any influential observations that might affect your results? There are no influential observations that affect my results.
What would you do if you found serious violations? I might have to remove a predictor if there’s a multicollinearity issue or I might have to remove an observation if there’s an influential point.

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

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

# Model 3: Age + sex + bmi_cat + phys_active + current_smoker
model3 <- glm(hypertension ~ age_cont + sex + bmi_cat +phys_active + current_smoker,
              data = brfss_subset_2023,
              family = binomial)


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

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

Model Comparison: AIC, BIC, and Deviance
Model	AIC	BIC	Deviance
Model 1: Age only	1636.61	1646.92	1632.61
Model 2: Age + Sex + bmi_cat	1576.49	1607.42	1564.49
Model 3: Age + sex + bmi_cat + phys_active + current_smoker	1579.50	1620.74	1563.50

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

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

Questions:

Which model has the best fit based on AIC? Model 2 is the best fit since it shows that from model 1 to model 2, it shows significant results.
Is the added complexity of the full model justified? No, because model 3 did not show a significant increase from model 1.
Which model would you choose for your final analysis? Why? I would choose model 2 since it shows a balance of complexity and fit.

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.

Introduction: State your research question My research question was to see if the levels of blood pressure (hypertension) increased based on a series of variables such as age, sex, BMI, physical activity, and smoking status for adults. In the lab, I needed to conduct a series of statistical methods, such as determining the outcome variable, performing an odds ratio for age, performing a simple and multiple logistic regression models, interpreting the dummy variables, testing for interactions, analyzing model diagnostics, and comparing models based on AIC/BIC.

Methods: Describe your analytic approach First, I calculated the frequency table and prevalence of hypertension by age group. I then needed to perform simple and multiple logistic regression models to determine the significance of the odds ratio for age, the 95% confidence interval, and also identify how confounding plays a role when adjusting for other variables. Conducting both methods allowed me to see the relationship between age and the significance of the variable when comparing it to the other variables. This can show the strongest predictors of hypertension. I then had to interpret the dummy variables for the BMI_cat and the odds ratio for the BMI category. This allowed me to identify the reference category for BMI while also interpreting the odds ratio for Obese compared to the reference category. By testing the interaction between model 1, model 2, and model 3, and testing for the likelihood ratio test, I was able to identify which interaction term was statistically significant, and I was able to predict the probabilities by age and bmi_cat. When performing a diagnostic test, I was able to look at the VIF and interpretation, since each showed low, I acknowledge that there were no multicollinearity issues or influential observation issues. Lastly, when performing the model comparison for model A, model B, and Model C, and using the likelihood ratio test, I was able to identify which model was the best fit on AIC and why its best fit for my final analysis.

Results: Present key findings with tables and figures Task 1: A tibble shows the age variable with prevalence Task 2: Simple logistic regression for hypertension and age Task 3: Multiple logistic regression for hypertension, age, and covariates Task 4: Dummy variable coding for BMI Task 5: Analysis of Deviance Table, predicted probability of hypertension by age and BMI, and age x bmi interaction model Task 6: Variance inflation factors (VIF) for multiple regression model, Cook’s Distance: identifying influential observations Task 7: Model comparison: AIC, BIC, and Deviance

Interpretation: Explain what your results mean Task 1: As the prevalence increases, so does the age variable. Which shows that hypertension is significant among older individuals compared to younger individuals. Task 2: For the age variable, the odds ratio is 1.055, which means for each 1-year increase in age, the odds of hypertension increase by 5.5%. These results show it is highly statistically significant, and the 95% confidence interval for the odds ratio is 1.045-1.065. Task 3: I adjusted for other variables when considering the age variable. It shows that the odds ratio for age changed. This shows that hypertension can increase or decrease when considering other predictors. Task 4: The reference category for hypertension is underweight since the category contains all zeroes. This also shows that people who are obese are 6.585 times as likely to experience hypertension compared to those who are underweight. Task 5: Since the p-value = 0.5655, it is statistically significant, and there is an interaction that exists when the effect of age on the outcome of hypertension differs across levels of the other variables, bmi_cat and phys-active. Task 6: With both the VIF and interaction being low, there is no indication of multicollinearity issues or influential observations that would affect my results. Task 7: Model 2 shows the best fit and the most significant results because it conveys a balance of complexity and fit.

Limitations: Discuss potential issues with your analysis A potential issue with my analysis is that when working with logistic regression models, it’s crucial not to put many input variables since it can distort the true associations and lead to large standard errors (Ranganathan, 2017). As the example of hypertension was used with multiple variables, it’s important not to add more than what was given because it can affect the associations and tamper with the accuracy of the data.

Reference: Ranganathan, P., Pramesh, C. S., & Aggarwal, R. (2017, Summer 8). Common pitfalls in statistical analysis: Logistic regression. Perspectives in clinical research. https://pmc.ncbi.nlm.nih.gov/articles/PMC5543767/ ***

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 Tahoe 26.3
## 
## 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

June McCarthy

2026-02-24

Setup and Data Preparation

Introduction

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 4: Interpret Dummy Variables

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

Task 5: Test for Interaction

Task 6: Model Diagnostics

Task 7: Model Comparison

Lab Report Guidelines

Summary

Key Concepts Covered

Important Formulas

References