Introduction

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

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

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

Setup and Data Preparation

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

Loading BRFSS 2023 Data

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

# Load the full BRFSS 2023 dataset
brfss_full <- read_xpt("C:/Users/tahia/OneDrive/Desktop/UAlbany PhD/Epi 553/LLCP2023XPT/LLCP2023.XPT") %>%
  janitor::clean_names()

# Display dataset dimensions
names(brfss_full)

##   [1] "state"    "fmonth"   "idate"    "imonth"   "iday"     "iyear"   
##   [7] "dispcode" "seqno"    "psu"      "ctelenm1" "pvtresd1" "colghous"
##  [13] "statere1" "celphon1" "ladult1"  "numadult" "respslc1" "landsex2"
##  [19] "lndsxbrt" "safetime" "ctelnum1" "cellfon5" "cadult1"  "cellsex2"
##  [25] "celsxbrt" "pvtresd3" "cclghous" "cstate1"  "landline" "hhadult" 
##  [31] "sexvar"   "genhlth"  "physhlth" "menthlth" "poorhlth" "primins1"
##  [37] "persdoc3" "medcost1" "checkup1" "exerany2" "exract12" "exeroft1"
##  [43] "exerhmm1" "exract22" "exeroft2" "exerhmm2" "strength" "bphigh6" 
##  [49] "bpmeds1"  "cholchk3" "toldhi3"  "cholmed3" "cvdinfr4" "cvdcrhd4"
##  [55] "cvdstrk3" "asthma3"  "asthnow"  "chcscnc1" "chcocnc1" "chccopd3"
##  [61] "addepev3" "chckdny2" "havarth4" "diabete4" "diabage4" "marital" 
##  [67] "educa"    "renthom1" "numhhol4" "numphon4" "cpdemo1c" "veteran3"
##  [73] "employ1"  "children" "income3"  "pregnant" "weight2"  "height3" 
##  [79] "deaf"     "blind"    "decide"   "diffwalk" "diffdres" "diffalon"
##  [85] "fall12mn" "fallinj5" "smoke100" "smokday2" "usenow3"  "ecignow2"
##  [91] "alcday4"  "avedrnk3" "drnk3ge5" "maxdrnks" "flushot7" "flshtmy3"
##  [97] "pneuvac4" "shingle2" "hivtst7"  "hivtstd3" "seatbelt" "drnkdri2"
## [103] "covidpo1" "covidsm1" "covidact" "pdiabts1" "prediab2" "diabtype"
## [109] "insulin1" "chkhemo3" "eyeexam1" "diabeye1" "diabedu1" "feetsore"
## [115] "arthexer" "arthedu"  "lmtjoin3" "arthdis2" "joinpai2" "lcsfirst"
## [121] "lcslast"  "lcsnumcg" "lcsctsc1" "lcsscncr" "lcsctwhn" "hadmam"  
## [127] "howlong"  "cervscrn" "crvclcnc" "crvclpap" "crvclhpv" "hadhyst2"
## [133] "psatest1" "psatime1" "pcpsars2" "psasugs1" "pcstalk2" "hadsigm4"
## [139] "colnsigm" "colntes1" "sigmtes1" "lastsig4" "colncncr" "vircolo1"
## [145] "vclntes2" "smalstol" "stoltest" "stooldn2" "bldstfit" "sdnates1"
## [151] "cncrdiff" "cncrage"  "cncrtyp2" "csrvtrt3" "csrvdoc1" "csrvsum" 
## [157] "csrvrtrn" "csrvinst" "csrvinsr" "csrvdein" "csrvclin" "csrvpain"
## [163] "csrvctl2" "indortan" "numburn3" "sunprtct" "wkdayout" "wkendout"
## [169] "cimemlo1" "cdworry"  "cddiscu1" "cdhous1"  "cdsocia1" "caregiv1"
## [175] "crgvrel4" "crgvlng1" "crgvhrs1" "crgvprb3" "crgvalzd" "crgvper1"
## [181] "crgvhou1" "crgvexpt" "lastsmk2" "stopsmk2" "mentcigs" "mentecig"
## [187] "heattbco" "firearm5" "gunload"  "loadulk2" "hasymp1"  "hasymp2" 
## [193] "hasymp3"  "hasymp4"  "hasymp5"  "hasymp6"  "strsymp1" "strsymp2"
## [199] "strsymp3" "strsymp4" "strsymp5" "strsymp6" "firstaid" "aspirin" 
## [205] "birthsex" "somale"   "sofemale" "trnsgndr" "marijan1" "marjsmok"
## [211] "marjeat"  "marjvape" "marjdab"  "marjothr" "usemrjn4" "acedeprs"
## [217] "acedrink" "acedrugs" "aceprisn" "acedivrc" "acepunch" "acehurt1"
## [223] "aceswear" "acetouch" "acetthem" "acehvsex" "aceadsaf" "aceadned"
## [229] "imfvpla4" "hpvadvc4" "hpvadsht" "tetanus1" "covidva1" "covacge1"
## [235] "covidnu2" "lsatisfy" "emtsuprt" "sdlonely" "sdhemply" "foodstmp"
## [241] "sdhfood1" "sdhbills" "sdhutils" "sdhtrnsp" "sdhstre1" "rrclass3"
## [247] "rrcognt2" "rrtreat"  "rratwrk2" "rrhcare4" "rrphysm2" "rcsgend1"
## [253] "rcsxbrth" "rcsrltn2" "casthdx2" "casthno2" "qstver"   "qstlang" 
## [259] "metstat"  "urbstat"  "mscode"   "ststr"    "strwt"    "rawrake" 
## [265] "wt2rake"  "imprace"  "chispnc"  "crace1"   "cageg"    "cllcpwt" 
## [271] "dualuse"  "dualcor"  "llcpwt2"  "llcpwt"   "rfhlth"   "phys14d" 
## [277] "ment14d"  "hlthpl1"  "hcvu653"  "totinda"  "metvl12"  "metvl22" 
## [283] "maxvo21"  "fc601"    "actin13"  "actin23"  "padur1"   "padur2"  
## [289] "pafreq1"  "pafreq2"  "minac12"  "minac22"  "strfreq"  "pamiss3" 
## [295] "pamin13"  "pamin23"  "pa3min"   "pavig13"  "pavig23"  "pa3vigm" 
## [301] "pacat3"   "paindx3"  "pa150r4"  "pa300r4"  "pa30023"  "pastrng" 
## [307] "parec3"   "pastae3"  "rfhype6"  "cholch3"  "rfchol3"  "michd"   
## [313] "ltasth1"  "casthm1"  "asthms1"  "drdxar2"  "mrace1"   "hispanc" 
## [319] "race"     "raceg21"  "racegr3"  "raceprv"  "sex"      "ageg5yr" 
## [325] "age65yr"  "age80"    "age_g"    "htin4"    "htm4"     "wtkg3"   
## [331] "bmi5"     "bmi5cat"  "rfbmi5"   "chldcnt"  "educag"   "incomg1" 
## [337] "smoker3"  "rfsmok3"  "cureci2"  "drnkany6" "drocdy4"  "rfbing6" 
## [343] "drnkwk2"  "rfdrhv8"  "flshot7"  "pneumo3"  "aidtst4"  "rfseat2" 
## [349] "rfseat3"  "drnkdrv"

Creating a Working Subset

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

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

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

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

## Working subset dimensions: 2000 observations, 12 variables

Data Recoding and Cleaning

# Create clean dataset with recoded variables
brfss_clean <- brfss_subset %>%
  mutate(
    # Outcome: Diabetes (binary)
    diabetes = case_when(
      diabete4 == 1 ~ 1,  # Yes
      diabete4 %in% c(2, 3, 4) ~ 0,  # No, pre-diabetes, or gestational only
      TRUE ~ NA_real_
    ),

    # Age groups
    age_group = factor(case_when(
      age_g == 1 ~ "18-24",
      age_g == 2 ~ "25-34",
      age_g == 3 ~ "35-44",
      age_g == 4 ~ "45-54",
      age_g == 5 ~ "55-64",
      age_g == 6 ~ "65+"
    ), levels = c("18-24", "25-34", "35-44", "45-54", "55-64", "65+")),

    # Age continuous (midpoint of category)
    age_cont = case_when(
      age_g == 1 ~ 21,
      age_g == 2 ~ 29.5,
      age_g == 3 ~ 39.5,
      age_g == 4 ~ 49.5,
      age_g == 5 ~ 59.5,
      age_g == 6 ~ 70
    ),

    # Sex
    sex = factor(ifelse(sex == 1, "Male", "Female")),

    # Race/ethnicity
    race = factor(case_when(
      race == 1 ~ "White",
      race == 2 ~ "Black",
      race == 3 ~ "Native American",
      race == 4 ~ "Asian",
      race == 5 ~ "Native Hawaiian/PI",
      race == 6 ~ "Other",
      race == 7 ~ "Multiracial",
      race == 8 ~ "Hispanic"
    )),

    # Education (simplified)
    education = factor(case_when(
      educag == 1 ~ "< High school",
      educag == 2 ~ "High school graduate",
      educag == 3 ~ "Some college",
      educag == 4 ~ "College graduate"
    ), levels = c("< High school", "High school graduate", "Some college", "College graduate")),

    # Income (simplified)
    income = factor(case_when(
      incomg1 == 1 ~ "< $25,000",
      incomg1 == 2 ~ "$25,000-$49,999",
      incomg1 == 3 ~ "$50,000-$74,999",
      incomg1 == 4 ~ "$75,000+",
      incomg1 == 5 ~ "Unknown"
    ), levels = c("< $25,000", "$25,000-$49,999", "$50,000-$74,999", "$75,000+", "Unknown")),

    # BMI category
    bmi_cat = factor(case_when(
      bmi5cat == 1 ~ "Underweight",
      bmi5cat == 2 ~ "Normal",
      bmi5cat == 3 ~ "Overweight",
      bmi5cat == 4 ~ "Obese"
    ), levels = c("Underweight", "Normal", "Overweight", "Obese")),

    # Physical activity (binary)
    phys_active = ifelse(exerany2 == 1, 1, 0),

    # Current smoking
    current_smoker = case_when(
      smokday2 == 1 ~ 1,  # Every day
      smokday2 == 2 ~ 1,  # Some days
      smokday2 == 3 ~ 0,  # Not at all
      TRUE ~ 0
    ),

    # General health (simplified)
    gen_health = factor(case_when(
      genhlth %in% c(1, 2) ~ "Excellent/Very good",
      genhlth == 3 ~ "Good",
      genhlth %in% c(4, 5) ~ "Fair/Poor"
    ), levels = c("Excellent/Very good", "Good", "Fair/Poor")),

    # Hypertension
    hypertension = ifelse(rfhype6 == 2, 1, 0),

    # High cholesterol
    high_chol = ifelse(rfchol3 == 2, 1, 0)
  ) %>%
  # Select only the clean variables
  select(diabetes, age_group, age_cont, sex, race, education, income,
         bmi_cat, phys_active, current_smoker, gen_health,
         hypertension, high_chol) %>%
  # Remove any remaining missing values
  drop_na()

# Save the cleaned subset for future use
write_rds(brfss_clean,
          "C:/Users/tahia/OneDrive/Desktop/UAlbany PhD/Epi 553/Lab 4/brfss_subset_2023.rds")

cat("Clean dataset saved with", nrow(brfss_clean), "complete observations\n")

## Clean dataset saved with 1281 complete observations

brfss_clean <- read_rds("C:/Users/tahia/OneDrive/Desktop/UAlbany PhD/Epi 553/Lab 4/brfss_subset_2023.rds")

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
# Summary table by hypertension status
table(brfss_clean$hypertension)

## 
##   0   1 
## 606 675

prev_ht <- mean(brfss_clean$hypertension == 1, na.rm = TRUE) * 100
prev_ht

## [1] 52.69321

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

prev_by_age <- brfss_clean %>%
  group_by(age_group) %>%
  summarise(
    n = n(),
    ht_cases = sum(hypertension == 1, na.rm = TRUE),
    prevalence = mean(hypertension == 1, na.rm = TRUE) * 100,
    .groups = "drop"
  )

prev_by_age

## # A tibble: 6 × 4
##   age_group     n ht_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

Questions:

What is the overall prevalence of hypertension in the dataset?

Ans: The overall prevalence of hypertension is 52.69%.

How does hypertension prevalence vary by age group?

Ans: As per the prevalence of hypertension by age group, it can be seen that the prevalence of hypertension increases by the age group. the youngest age group has the lowest prevalence rate compared to the older age group.

Task 2: Build a Simple Logistic Regression Model

# YOUR CODE HERE: Fit a simple logistic regression model
# Outcome: hypertension
# Predictor: age_cont
model_logistic_simple1 <- glm(hypertension ~ age_cont,
                              data = brfss_clean,
                              family = binomial(link = "logit"))

# Display results with odds ratios
tidy(model_logistic_simple1, 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

# Create scatter plot with regression line
p1 <- ggplot(model_logistic_simple1, aes(x = age_cont, y = hypertension)) +
  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 Hypertension",
    subtitle = "Simple Linear Regression",
    x = "Age (years)",
    y = "Odds Ratio of Hypertension"
  ) +
  theme_minimal(base_size = 12)

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

Hypertension Prevalence by Age

Questions:

What is the odds ratio for age? Interpret this value. Ans: The odds ratio of 1.034 indicates that for each one-year increase in age, the odds of hypertension increase by approximately 3.4%, holding sex, BMI, physical activity, and smoking constant.
Is the association statistically significant?

Ans:Yes. The association is statistically significant because the 95% confidence interval does not include 1.0, and the p-value is less than 0.05.

What is the 95% confidence interval for the odds ratio?

Ans: 1.021-1.048, This means we are 95% confident that the true increase in odds of hypertension per one-year increase in age lies between 2.1% and 4.8%.

Task 3: Create a Multiple Regression Model

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

model_logistic_multiple1 <- glm(hypertension ~ age_cont + sex + bmi_cat +
                                phys_active + current_smoker,
                               data = brfss_clean,
                               family = binomial(link = "logit"))

# Display results
tidy(model_logistic_multiple1, 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?

Ans: with out adjusting the confounders, the odds ratio for age was 1.03 and after adjusting the confounders, it is 1.06, indicating not much effect of the counfounders are impacting the association.

What does this suggest about confounding?

Ans:After adjusting the confounder, there is not difference in the odds ratio of age categories. so it can be said that these confounders does nt have much impact of the association of age with hypertension.

Which variables are the strongest predictors of hypertension?

And: From the multiple regression moedl, the BMI has the most high number as a predistor of hypertention.

Task 4: Interpret Dummy Variables

# YOUR CODE HERE: Create a table showing the dummy variable coding for bmi_cat
# Extract dummy variable coding
dummy_table1 <- data.frame(
  BMI = c("Normal", "Overweight", "Obese"),
  `Dummy 1 (Overweight)` = c(0, 1, 0),
  `Dummy 2 (Obese)` = c(0, 0, 1),
  check.names = FALSE
)

dummy_table1 %>%
  kable(caption = "Dummy Variable Coding for BMI", align = "lcc") %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
  row_spec(1, bold = TRUE, background = "#ffe6e6")

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

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

# Extract BMI odds ratios
bmi_coefs <- tidy(model_logistic_multiple1, exponentiate = TRUE, conf.int = TRUE) %>%
  filter(str_detect(term, "^bmi_cat")) %>%
  mutate(
    bmi_cat = str_remove(term, "^bmi_cat_?"),
    bmi_cat = factor(bmi_cat, levels = c("Overweight", "Obese"))
  )

# Add reference category
ref_row1 <- data.frame(
  term = "bmi_catNormal",
  estimate = 1.0,
  std.error = NA,
  statistic = NA,
  p.value = NA,
  conf.low = 1.0,
  conf.high = 1.0,
  bmi_cat = factor("Normal (Ref)", levels = c("Normal (Ref)", "Overweight", "Obese"))
)

bmi_coefs_full <- bind_rows(ref_row1, bmi_coefs) %>%
  mutate(bmi_cat = factor(bmi_cat, levels = c("Normal (Ref)", "Overweight", "Obese")))

# Plot
p3 <- ggplot(bmi_coefs_full, aes(x = bmi_cat, y = estimate)) +
  geom_hline(yintercept = 1, linetype = "dashed") +
  geom_pointrange(aes(ymin = conf.low, ymax = conf.high)) +
  coord_flip() +
  labs(
    title = "Association Between BMI and Hypertension",
    subtitle = "Adjusted Odds Ratios (reference: Normal)",
    x = "BMI",
    y = "Odds Ratio (95% CI)"
  ) +
  theme_minimal(base_size = 12)

ggplotly(p3)

Questions:

What is the reference category for BMI?

Ans: Normal

Interpret the odds ratio for “Obese” compared to the reference category.

Ans: Compared to individuals with normal BMI, obese individuals have higher odds of hypertension. Specifically, the odds ratio indicates that obese individuals have approximately 6.58 times the odds of having hypertension relative to those with normal BMI, after adjusting for age, sex, physical activity, and smoking status.

How would you explain this to a non-statistician? Ans: People who are classified as obese are much more likely to have high blood pressure than people with a normal weight, even after accounting for age, sex, exercise habits, and smoking. In simple terms, carrying excess body weight substantially increases the likelihood of developing hypertension.

Task 5: Test for Interaction

# YOUR CODE HERE: Fit a model with Age × BMI interaction

model_interaction1 <- glm(diabetes ~ age_cont * bmi_cat,
                         data = brfss_clean,
                         family = binomial(link = "logit"))

# Display interaction results
tidy(model_interaction1, 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	3.530	40.609	0.031	0.975	853637484	84628801
age_cont:bmi_catNormal	0.291	40.609	-0.030	0.976	0	0
age_cont:bmi_catOverweight	0.296	40.609	-0.030	0.976	0	0
age_cont:bmi_catObese	0.294	40.609	-0.030	0.976	0	0

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

# Reduced model (without interaction)
model_reduced <- glm(
  hypertension ~ age_cont + bmi_cat + sex + phys_active + current_smoker,
  data = brfss_clean,
  family = binomial(link = "logit")
)

# Full model (with Age × BMI interaction)
model_full <- glm(
  hypertension ~ age_cont * bmi_cat + sex + phys_active + current_smoker,
  data = brfss_clean,
  family = binomial(link = "logit")
)

# Perform Likelihood Ratio Test
lrt_result <- anova(model_reduced, model_full, test = "LRT")

# Display clean table
broom::tidy(lrt_result) %>%
  knitr::kable(
    caption = "Likelihood Ratio Test Comparing Models With and Without Age × BMI Interaction",
    digits = 4
  ) %>%
  kableExtra::kable_styling(
    bootstrap_options = c("striped", "hover"),
    full_width = FALSE
  )

Likelihood Ratio Test Comparing Models With and Without Age × BMI Interaction
term	df.residual	residual.deviance	df	deviance	p.value
hypertension ~ age_cont + bmi_cat + sex + phys_active + current_smoker	1273	1563.496	NA	NA	NA
hypertension ~ age_cont * bmi_cat + sex + phys_active + current_smoker	1270	1561.260	3	2.2363	0.5248

# Test if the effect of age on hypertension differs by BMI category
pred_interact <- ggpredict(
  model_full,
  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 Category",
    subtitle = "Age × BMI Interaction Model",
    x = "Age (years)",
    y = "Predicted Probability of Hypertension",
    color = "BMI Category",
    fill = "BMI Category"
  ) +
  scale_y_continuous(labels = scales::percent_format()) +
  theme_minimal(base_size = 12) +
  theme(legend.position = "bottom")

p4

Questions:

Is the interaction term statistically significant?

Ans: No, The likelihood ratio test comparing the model with and without the Age × BMI interaction yielded a p-value of 0.5248, which is greater than 0.05. Therefore, the interaction term is not statistically significant.

What does this mean in epidemiologic terms (effect modification)?

Ans: In epidemiologic terms, this indicates no evidence of effect modification by BMI category.

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
# Calculate VIF
vif_values <- car::vif(model_logistic_multiple1)

# Handle categorical variables (GVIF adjustment)
if (is.matrix(vif_values)) {
  
  vif_df <- data.frame(
    Variable = rownames(vif_values),
    GVIF = vif_values[, "GVIF"],
    Df = vif_values[, "Df"],
    Adjusted_VIF = vif_values[, "GVIF^(1/(2*Df))"]
  )
  
} else {
  
  vif_df <- data.frame(
    Variable = names(vif_values),
    Adjusted_VIF = as.numeric(vif_values)
  )
}

# Display table
knitr::kable(
  vif_df,
  caption = "Variance Inflation Factors (VIF) for Hypertension Logistic Model",
  digits = 3
) %>%
  kableExtra::kable_styling(
    bootstrap_options = c("striped", "hover"),
    full_width = FALSE
  )

Variance Inflation Factors (VIF) for Hypertension Logistic Model
	Variable	GVIF	Df	Adjusted_VIF
age_cont	age_cont	1.127	1	1.061
sex	sex	1.017	1	1.008
bmi_cat	bmi_cat	1.103	3	1.016
phys_active	phys_active	1.025	1	1.012
current_smoker	current_smoker	1.074	1	1.036

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


# Calculate Cook's distance
cooks_d <- cooks.distance(model_logistic_multiple1)

# Create data frame
influence_df <- data.frame(
  observation = 1:length(cooks_d),
  cooks_d = cooks_d
)

# Common threshold
threshold <- 4 / nrow(brfss_clean)

influence_df <- influence_df %>%
  dplyr::mutate(
    influential = ifelse(cooks_d > threshold, "Yes", "No")
  )

# Plot
p_cook <- ggplot(influence_df, aes(x = observation, y = cooks_d)) +
  geom_point(aes(color = influential), alpha = 0.6) +
  geom_hline(yintercept = threshold, linetype = "dashed", color = "red") +
  labs(
    title = "Cook's Distance Plot",
    subtitle = paste("Threshold =", round(threshold, 4)),
    x = "Observation Number",
    y = "Cook's Distance",
    color = "Influential?"
  ) +
  scale_color_manual(values = c("No" = "steelblue", "Yes" = "red")) +
  theme_minimal(base_size = 12)

p_cook

# Count influential observations
sum(influence_df$influential == "Yes")

## [1] 22

Questions:

Are there any concerns about multicollinearity?

Ans: No. All adjusted VIF values are close to 1 and well below the common threshold of 5, indicating no evidence of problematic multicollinearity among the predictors.

Are there any influential observations that might affect your results?

Ans: A small number of observations exceed the Cook’s distance threshold (4/n), suggesting potential mild influence.

What would you do if you found serious violations?

Ans: If serious multicollinearity were present, I would consider removing or combining correlated variables.

Task 7: Model Comparison

# YOUR CODE HERE: Compare three models using AIC and BIC

# Model A: Age only
model_A <- glm(
  hypertension ~ age_cont,
  data = brfss_clean,
  family = binomial(link = "logit")
)

# Model B: Age + sex + bmi_cat
model_B <- glm(
  hypertension ~ age_cont + sex + bmi_cat,
  data = brfss_clean,
  family = binomial(link = "logit")
)

# Model C: Age + sex + bmi_cat + phys_active + current_smoker
model_C <- glm(
  hypertension ~ age_cont + sex + bmi_cat + phys_active + current_smoker,
  data = brfss_clean,
  family = binomial(link = "logit")
)

# Create comparison table
model_comp <- tibble::tibble(
  Model = c("Model A: Age only",
            "Model B: Age + sex + BMI",
            "Model C: Age + sex + BMI + activity + smoking"),
  AIC = c(AIC(model_A), AIC(model_B), AIC(model_C)),
  BIC = c(BIC(model_A), BIC(model_B), BIC(model_C)),
  Deviance = c(deviance(model_A), deviance(model_B), deviance(model_C))
) %>%
  dplyr::arrange(AIC)

# Display table
model_comp %>%
  knitr::kable(
    caption = "Model Comparison for Hypertension (Lower AIC/BIC = Better Fit)",
    digits = 2
  ) %>%
  kableExtra::kable_styling(
    bootstrap_options = c("striped", "hover"),
    full_width = FALSE
  )

Model Comparison for Hypertension (Lower AIC/BIC = Better Fit)
Model	AIC	BIC	Deviance
Model B: Age + sex + BMI	1576.49	1607.42	1564.49
Model C: Age + sex + BMI + activity + smoking	1579.50	1620.74	1563.50
Model A: Age only	1636.61	1646.92	1632.61

Questions:

Which model has the best fit based on AIC?

Ans: Model C (Age + Sex + BMI + Physical Activity + Smoking) has the lowest AIC and BIC, indicating the best balance between model fit and model complexity.

Is the added complexity of the full model justified?

Ans:There is no evidence of problematic multicollinearity among predictors in the multiple logistic regression model.

Which model would you choose for your final analysis? Why?

Ans: I would choose Model C for the final analysis because it provides the best model fit based on AIC and BIC while also adjusting for important confounders such as BMI, physical activity, and smoking.

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

The goal of this analysis was to examine factors associated with hypertension among adults using BRFSS 2023 data. Specifically, I wanted to assess whether age, sex, BMI, physical activity, and smoking are related to hypertension. I also tested whether the effect of age on hypertension differs across BMI categories (i.e., effect modification).

Methods

Hypertension (yes/no) was the binary outcome. I used logistic regression to estimate odds ratios (ORs) and 95% confidence intervals (CIs).

Three models were compared:

Model A: Age only

Model B: Age + Sex + BMI

Model C: Age + Sex + BMI + Physical Activity + Smoking

Model fit was evaluated using AIC and BIC. I tested effect modification by including an Age × BMI interaction term and performing a likelihood ratio test (LRT). Model diagnostics included variance inflation factors (VIF) for multicollinearity and Cook’s distance for influential observations.

Results

The overall prevalence of hypertension in the sample was about 53%.

Age was significantly associated with hypertension (OR = 1.034, 95% CI: 1.021–1.048), meaning each additional year of age increased the odds of hypertension by about 3.4%. BMI showed a strong association, with obese individuals having substantially higher odds of hypertension compared to those with normal BMI. Physical activity appeared protective. Smoking showed weaker associations after adjustment.

The Age × BMI interaction was not statistically significant (LRT p = 0.525), indicating no evidence that the age–hypertension relationship differs by BMI category.

Model C had the lowest AIC and BIC, suggesting the best overall fit.

Diagnostic checks showed no multicollinearity concerns (all VIFs close to 1). Although a few observations exceeded the Cook’s distance threshold, none appeared highly influential.

Interpretation

Age and BMI are the strongest predictors of hypertension in this analysis. The risk of hypertension increases steadily with age and is much higher among overweight and obese individuals. The lack of a significant interaction suggests that the age-related increase in hypertension is fairly consistent across BMI categories.

Limitations

This analysis is cross-sectional, so causality cannot be established. Hypertension and other variables are self-reported, which may introduce misclassification. In addition, survey weights were not applied, which may limit generalizability to the U.S. population.

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 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] janitor_2.2.1        codetools_0.2-20     snakecase_0.11.1    
## [25] htmltools_0.5.9      sass_0.4.10          yaml_2.3.12         
## [28] lazyeval_0.2.2       Formula_1.2-5        pillar_1.11.1       
## [31] jquerylib_0.1.4      broom.helpers_1.22.0 cachem_1.1.0        
## [34] abind_1.4-8          nlme_3.1-168         tidyselect_1.2.1    
## [37] digest_0.6.39        stringi_1.8.7        labeling_0.4.3      
## [40] splines_4.5.2        labelled_2.16.0      fastmap_1.2.0       
## [43] grid_4.5.2           cli_3.6.5            magrittr_2.0.4      
## [46] cards_0.7.1          utf8_1.2.6           withr_3.0.2         
## [49] scales_1.4.0         backports_1.5.0      timechange_0.3.0    
## [52] rmarkdown_2.30       httr_1.4.7           otel_0.2.0          
## [55] hms_1.1.4            evaluate_1.0.5       viridisLite_0.4.2   
## [58] mgcv_1.9-3           rlang_1.1.7          glue_1.8.0          
## [61] xml2_1.5.2           svglite_2.2.2        rstudioapi_0.18.0   
## [64] jsonlite_2.0.0       R6_2.6.1             systemfonts_1.3.1

Statistical Modeling in Epidemiology

EPI 553 - Advanced Epidemiologic Methods

Muntasir Masum

2026-02-24

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

Introduction

Methods

Results

Interpretation

Limitations

Summary

Key Concepts Covered

Important Formulas

References