Introduction

In Part 1, we introduced the logistic model, the logit transformation, and the connection between logistic regression coefficients and odds ratios. We fit simple logistic regression models with single predictors and previewed multiple logistic regression.

In Part 2, we go deeper:

  1. Multiple logistic regression with adjusted odds ratios for confounder control
  2. Maximum likelihood estimation and the Wald and likelihood ratio tests
  3. Confidence intervals for adjusted odds ratios
  4. Interaction terms in logistic regression and effect modification
  5. Goodness-of-fit: deviance, Hosmer-Lemeshow test, pseudo-R²
  6. Model discrimination: ROC curves and the area under the curve (AUC)
  7. Diagnostics: residuals, influential observations, and linearity in the logit

Textbook reference: Kleinbaum et al., Chapter 22 (Sections 22.4 and 22.5)

Setup and Data

library(tidyverse)
library(knitr)
library(kableExtra)
library(broom)
library(gtsummary)
library(car)
library(ggeffects)
library(ResourceSelection)  # for Hosmer-Lemeshow
library(pROC)               # for ROC/AUC
library(performance)        # for model performance metrics

options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))
brfss_logistic <- readRDS(
  "E:/College WORK MS in EPI/SEM 2 MS EPI/Epi553 STATS 2/brfss_logistic_2020.rds"
)

dim(brfss_logistic)
## [1] 5000   10

Outcome: fmd — Frequent Mental Distress (1 = 14+ mentally unhealthy days in past 30, 0 = otherwise).

Part 1: Multiple Logistic Regression

The Adjusted Model

We extend the simple model to include several predictors:

\[\text{logit}[\Pr(Y = 1)] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_k X_k\]

Each coefficient \(\beta_j\) represents the change in log-odds for a one-unit increase in \(X_j\), holding all other predictors constant. Exponentiating gives the adjusted odds ratio:

\[\text{aOR}_j = e^{\beta_j}\]

mod_full <- glm(
  fmd ~ exercise + smoker + age + sex + sleep_hrs + income_cat + bmi + physhlth_days,
  data = brfss_logistic,
  family = binomial(link = "logit")
)

mod_full |>
  tbl_regression(
    exponentiate = TRUE,
    label = list(
      exercise      ~ "Exercise (past 30 days)",
      smoker        ~ "Smoking status",
      age           ~ "Age (per year)",
      sex           ~ "Sex",
      sleep_hrs     ~ "Sleep hours",
      income_cat    ~ "Income category (per unit)",
      bmi           ~ "BMI",
      physhlth_days ~ "Physically unhealthy days"
    )
  ) |>
  bold_labels() |>
  bold_p()
Characteristic OR 95% CI p-value
Exercise (past 30 days)


    No — —
    Yes 0.82 0.68, 0.99 0.041
Smoking status


    Former/Never — —
    Current 1.30 1.09, 1.56 0.004
Age (per year) 0.97 0.96, 0.97 <0.001
Sex


    Male — —
    Female 1.68 1.41, 1.98 <0.001
Sleep hours 0.86 0.81, 0.90 <0.001
Income category (per unit) 0.91 0.87, 0.95 <0.001
BMI 1.01 0.99, 1.02 0.4
Physically unhealthy days 1.06 1.06, 1.07 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

Interpretation: Each row gives the adjusted odds ratio (aOR) and 95% CI for one predictor, controlling for all others. For example, the aOR for current smoking compares the odds of frequent mental distress for current smokers vs. former/never smokers, after adjusting for age, sex, sleep, income, BMI, exercise, and physical health. An aOR > 1 indicates a risk factor; an aOR < 1 indicates a protective factor.

Scaling Continuous Predictors

A 1-year change in age or a 1-unit change in BMI is rarely the most clinically meaningful comparison. We can rescale to improve interpretation:

mod_scaled <- glm(
  fmd ~ exercise + smoker + I(age/10) + sex + sleep_hrs +
        income_cat + I(bmi/5) + physhlth_days,
  data = brfss_logistic,
  family = binomial
)

mod_scaled |>
  tbl_regression(
    exponentiate = TRUE,
    label = list(
      "I(age/10)"  ~ "Age (per 10 years)",
      "I(bmi/5)"   ~ "BMI (per 5 units)"
    )
  ) |>
  bold_labels()
Characteristic OR 95% CI p-value
exercise


    No — —
    Yes 0.82 0.68, 0.99 0.041
smoker


    Former/Never — —
    Current 1.30 1.09, 1.56 0.004
Age (per 10 years) 0.73 0.69, 0.77 <0.001
sex


    Male — —
    Female 1.68 1.41, 1.98 <0.001
sleep_hrs 0.86 0.81, 0.90 <0.001
income_cat 0.91 0.87, 0.95 <0.001
BMI (per 5 units) 1.03 0.97, 1.10 0.4
physhlth_days 1.06 1.06, 1.07 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

Interpretation: Now the aOR for age compares two individuals 10 years apart, and the aOR for BMI compares two individuals 5 BMI units apart, both more clinically interpretable.

Part 2: Maximum Likelihood Estimation and Hypothesis Tests

Maximum Likelihood Estimation (ML)

Unlike linear regression, which uses ordinary least squares, logistic regression coefficients are estimated by maximum likelihood. The algorithm finds the values of \(\beta_0, \beta_1, \ldots, \beta_k\) that maximize the likelihood of observing the data.

The likelihood function for \(n\) independent binary observations is:

\[L(\boldsymbol{\beta}) = \prod_{i=1}^{n} p_i^{y_i}(1 - p_i)^{1 - y_i}\]

where \(p_i = \Pr(Y_i = 1 \mid X_i)\) is the predicted probability for observation \(i\). Taking the log gives the log-likelihood:

\[\ln L(\boldsymbol{\beta}) = \sum_{i=1}^{n} \left[y_i \ln p_i + (1 - y_i) \ln(1 - p_i)\right]\]

The ML estimates \(\hat{\beta}\) are obtained iteratively (typically by Newton-Raphson). We never compute these by hand, but it is important to know that R’s glm() reports Deviance \(= -2 \ln \hat{L}\), which is the foundation for hypothesis testing.

Wald Test (z-test for individual coefficients)

The Wald test is the default test reported by summary() and tidy(). For each coefficient \(\beta_j\):

\[z = \frac{\hat{\beta}_j}{\text{SE}(\hat{\beta}_j)} \sim N(0, 1) \text{ under } H_0: \beta_j = 0\]

The p-value tests whether the coefficient is significantly different from zero, equivalently whether the OR is significantly different from 1.

tidy(mod_full, conf.int = TRUE, exponentiate = TRUE) |>
  mutate(across(c(estimate, std.error, statistic, conf.low, conf.high),
                \(x) round(x, 3)),
         p.value = format.pval(p.value, digits = 3)) |>
  kable(caption = "Wald Tests for Each Coefficient (Exponentiated)") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Wald Tests for Each Coefficient (Exponentiated)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 2.198 0.356 2.215 0.02675 1.095 4.414
exerciseYes 0.820 0.097 -2.044 0.04095 0.679 0.993
smokerCurrent 1.301 0.091 2.890 0.00386 1.088 1.555
age 0.969 0.003 -11.225 < 2e-16 0.963 0.974
sexFemale 1.675 0.086 5.975 2.30e-09 1.415 1.985
sleep_hrs 0.857 0.027 -5.643 1.67e-08 0.812 0.904
income_cat 0.909 0.020 -4.703 2.56e-06 0.873 0.946
bmi 1.006 0.006 0.932 0.35113 0.993 1.018
physhlth_days 1.065 0.004 15.634 < 2e-16 1.057 1.073

Caveat: The Wald test can be unreliable when sample sizes are small or when coefficients are large. The likelihood ratio test is generally preferred for these situations.

Likelihood Ratio Test (LR test)

The likelihood ratio test compares two nested models: a “full” model and a “reduced” model that drops one or more predictors. The test statistic is:

\[\text{LR} = -2(\ln \hat{L}_{\text{reduced}} - \ln \hat{L}_{\text{full}}) = D_{\text{reduced}} - D_{\text{full}}\]

Under \(H_0\) that the dropped predictors have no effect, LR follows a \(\chi^2\) distribution with degrees of freedom equal to the number of parameters dropped.

Example: Test the joint contribution of smoking and exercise

mod_reduced <- glm(
  fmd ~ age + sex + sleep_hrs + income_cat + bmi + physhlth_days,
  data = brfss_logistic,
  family = binomial
)

anova(mod_reduced, mod_full, test = "LRT") |>
  kable(digits = 3,
        caption = "LR Test: Does adding exercise + smoker improve the model?") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
LR Test: Does adding exercise + smoker improve the model?
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
4993 3641.913 NA NA NA
4991 3628.474 2 13.439 0.001

Interpretation: The LR test gives a \(\chi^2\) statistic on 3 degrees of freedom (1 for exercise, 2 for smoker — but smoker has 2 levels so 1 dummy variable is created, making df = 2 here actually). A small p-value means the dropped variables jointly contribute to model fit.

Wald vs. LR test in practice

Aspect Wald test LR test
What it tests Single coefficient or vector Nested model comparison
Computational cost Very fast Requires fitting two models
Reliability with small samples Less reliable Generally preferred
Reported by R summary(model) anova(m1, m2, test = "LRT")

In large samples (like ours with n = 5,000), the two tests usually agree. In smaller samples or with extreme estimates, prefer the LR test.

Part 3: Confidence Intervals for Adjusted Odds Ratios

For the OR of a single coefficient, the 95% CI is computed on the log-odds scale and then exponentiated:

\[95\% \text{ CI for } e^{\beta_j} = \exp\left(\hat{\beta}_j \pm 1.96 \cdot \text{SE}(\hat{\beta}_j)\right)\]

This is the default approach used by confint() and tidy(..., conf.int = TRUE).

ci_table <- tidy(mod_full, conf.int = TRUE, exponentiate = TRUE) |>
  filter(term != "(Intercept)") |>
  dplyr::select(term, estimate, conf.low, conf.high) |>
  mutate(across(c(estimate, conf.low, conf.high), \(x) round(x, 3)))

ci_table |>
  kable(col.names = c("Predictor", "aOR", "95% CI Lower", "95% CI Upper"),
        caption = "Adjusted Odds Ratios with 95% CIs") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Adjusted Odds Ratios with 95% CIs
Predictor aOR 95% CI Lower 95% CI Upper
exerciseYes 0.820 0.679 0.993
smokerCurrent 1.301 1.088 1.555
age 0.969 0.963 0.974
sexFemale 1.675 1.415 1.985
sleep_hrs 0.857 0.812 0.904
income_cat 0.909 0.873 0.946
bmi 1.006 0.993 1.018
physhlth_days 1.065 1.057 1.073

Forest Plot of Adjusted ORs

A forest plot is the standard way to visualize multiple ORs and their CIs:

ci_table |>
  ggplot(aes(x = estimate, y = reorder(term, estimate))) +
  geom_vline(xintercept = 1, linetype = "dashed", color = "red") +
  geom_point(size = 3, color = "steelblue") +
  geom_errorbarh(aes(xmin = conf.low, xmax = conf.high), height = 0.2,
                 color = "steelblue") +
  scale_x_log10() +
  labs(title = "Forest Plot of Adjusted Odds Ratios for Frequent Mental Distress",
       subtitle = "Reference line at OR = 1; log-scale x-axis",
       x = "Adjusted Odds Ratio (95% CI)", y = NULL) +
  theme_minimal()

Interpretation: Predictors whose CIs do not cross the dashed line at OR = 1 are statistically significantly associated with FMD at the 0.05 level. The log-scale x-axis ensures that ORs of 0.5 and 2.0 (which represent equally strong associations in opposite directions) appear equidistant from 1.

Part 4: Interaction (Effect Modification)

What Is Interaction in Logistic Regression?

Interaction (effect modification) occurs when the effect of one predictor on the outcome depends on the value of another predictor. In logistic regression, interaction is modeled by including a product term:

\[\text{logit}[\Pr(Y=1)] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 (X_1 \cdot X_2)\]

If \(\beta_3 \neq 0\), the OR for \(X_1\) depends on the value of \(X_2\).

Example: Does the effect of exercise differ by sex?

mod_interact <- glm(
  fmd ~ exercise * sex + age + smoker + sleep_hrs + income_cat,
  data = brfss_logistic,
  family = binomial
)

mod_interact |>
  tbl_regression(exponentiate = TRUE) |>
  bold_labels() |>
  bold_p()
Characteristic OR 95% CI p-value
exercise


    No — —
    Yes 0.61 0.47, 0.79 <0.001
sex


    Male — —
    Female 1.66 1.26, 2.21 <0.001
IMPUTED AGE VALUE COLLAPSED ABOVE 80 0.97 0.97, 0.98 <0.001
smoker


    Former/Never — —
    Current 1.25 1.05, 1.48 0.012
sleep_hrs 0.81 0.77, 0.86 <0.001
income_cat 0.85 0.82, 0.89 <0.001
exercise * sex


    Yes * Female 1.00 0.71, 1.41 >0.9
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

Testing the Interaction

mod_no_interact <- glm(
  fmd ~ exercise + sex + age + smoker + sleep_hrs + income_cat,
  data = brfss_logistic,
  family = binomial
)

anova(mod_no_interact, mod_interact, test = "LRT") |>
  kable(digits = 3,
        caption = "LR Test for Exercise × Sex Interaction") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
LR Test for Exercise × Sex Interaction
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
4993 3870.197 NA NA NA
4992 3870.197 1 0 0.987

Interpretation: If the p-value for the LR test is small (< 0.05), the interaction is statistically significant: the effect of exercise differs by sex. If not, we can drop the interaction term and use the simpler main-effects model.

Visualizing the Interaction

ggpredict(mod_interact, terms = c("exercise", "sex")) |>
  plot() +
  labs(title = "Predicted Probability of FMD by Exercise and Sex",
       x = "Exercise", y = "Predicted Probability of FMD",
       color = "Sex") +
  theme_minimal()

Interpretation: If the lines are parallel, there is no interaction. If they cross or diverge, the effect of exercise differs across sex.

Stratified Odds Ratios

When an interaction is present, we report stratum-specific odds ratios rather than a single overall OR:

# Stratum-specific ORs from the interaction model
ggpredict(mod_interact, terms = c("exercise", "sex")) |>
  as_tibble() |>
  pivot_wider(id_cols = group, names_from = x, values_from = predicted) |>
  mutate(OR_yes_vs_no = (Yes / (1 - Yes)) / (No / (1 - No))) |>
  dplyr::select(Sex = group, OR_yes_vs_no) |>
  kable(digits = 3,
        col.names = c("Sex", "OR (Exercise: Yes vs. No)"),
        caption = "Stratum-Specific Odds Ratios for Exercise") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Stratum-Specific Odds Ratios for Exercise
Sex OR (Exercise: Yes vs. No)
Male 0.612
Female 0.614

Part 5: Goodness-of-Fit

Deviance

The deviance of a logistic model is:

\[D = -2 \ln \hat{L}\]

It is analogous to the residual sum of squares in linear regression: smaller is better. By itself, the deviance is hard to interpret, but differences in deviance between nested models follow a \(\chi^2\) distribution and form the basis of the LR test.

glance(mod_full) |>
  dplyr::select(null.deviance, df.null, deviance, df.residual, AIC, BIC) |>
  kable(digits = 1, caption = "Model Fit Statistics") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Model Fit Statistics
null.deviance df.null deviance df.residual AIC BIC
4251.3 4999 3628.5 4991 3646.5 3705.1

Quick check: The difference between null.deviance and deviance represents the improvement from adding all predictors to an intercept-only model. We can test this with an LR test on df.null - df.residual degrees of freedom.

Pseudo-R²

There is no exact analog of \(R^2\) for logistic regression, but several “pseudo-R²” measures exist. The most common is McFadden’s R²:

\[R^2_{\text{McFadden}} = 1 - \frac{\ln \hat{L}_{\text{full}}}{\ln \hat{L}_{\text{null}}}\]

Values between 0.2 and 0.4 are considered excellent fit.

performance::r2_mcfadden(mod_full)
## # R2 for Generalized Linear Regression
##        R2: 0.147
##   adj. R2: 0.146

Interpretation: McFadden’s R² should not be interpreted on the same scale as linear regression R². Values are typically much smaller (e.g., 0.1 may indicate a reasonable fit).

Hosmer-Lemeshow Test

The Hosmer-Lemeshow test assesses the agreement between predicted and observed event rates within deciles of predicted probability. A non-significant p-value indicates adequate fit.

hl_test <- hoslem.test(
  x = as.numeric(brfss_logistic$fmd) - 1,
  y = fitted(mod_full),
  g = 10
)

hl_test
## 
##  Hosmer and Lemeshow goodness of fit (GOF) test
## 
## data:  as.numeric(brfss_logistic$fmd) - 1, fitted(mod_full)
## X-squared = 8.9639, df = 8, p-value = 0.3453

Interpretation: A small p-value (< 0.05) suggests that the model does not fit well in some regions of predicted probability. With large samples (like ours), the Hosmer-Lemeshow test can be over-powered and detect trivial misfit. Always pair it with a calibration plot.

Calibration Plot

brfss_logistic |>
  mutate(pred_prob = fitted(mod_full),
         obs_outcome = as.numeric(fmd) - 1,
         decile = ntile(pred_prob, 10)) |>
  group_by(decile) |>
  summarise(
    mean_pred = mean(pred_prob),
    mean_obs  = mean(obs_outcome),
    .groups = "drop"
  ) |>
  ggplot(aes(x = mean_pred, y = mean_obs)) +
  geom_abline(slope = 1, intercept = 0, color = "red", linetype = "dashed") +
  geom_point(size = 3, color = "steelblue") +
  geom_line(color = "steelblue") +
  labs(title = "Calibration Plot: Observed vs. Predicted Probability of FMD",
       subtitle = "Points should fall on the dashed line for perfect calibration",
       x = "Mean Predicted Probability (within decile)",
       y = "Observed Proportion (within decile)") +
  theme_minimal()

Interpretation: A well-calibrated model has points lying close to the 45-degree line. Systematic departures suggest miscalibration: points above the line indicate the model under-predicts; points below indicate over-prediction.

Part 6: Discrimination — ROC Curve and AUC

While calibration assesses how well predicted probabilities match observed rates, discrimination assesses how well the model separates events from non-events.

The ROC curve plots sensitivity (true positive rate) against 1 − specificity (false positive rate) across all possible probability cutoffs.

The AUC (area under the ROC curve) summarizes discrimination:

AUC Discrimination
0.5 No discrimination (chance)
0.6-0.7 Poor
0.7-0.8 Acceptable
0.8-0.9 Excellent
> 0.9 Outstanding 
roc_obj <- roc(
  response = brfss_logistic$fmd,
  predictor = fitted(mod_full),
  levels = c("No", "Yes"),
  direction = "<"
)

auc_value <- auc(roc_obj)

ggroc(roc_obj, color = "steelblue", linewidth = 1.2) +
  geom_abline(slope = 1, intercept = 1, linetype = "dashed", color = "red") +
  labs(title = "ROC Curve for Frequent Mental Distress Model",
       subtitle = paste0("AUC = ", round(auc_value, 3)),
       x = "Specificity", y = "Sensitivity") +
  theme_minimal()

Interpretation: An AUC of approximately 0.75-0.80 indicates acceptable to excellent discrimination, meaning the model can distinguish between individuals with and without FMD reasonably well. Note that calibration and discrimination are distinct concepts: a model can have good discrimination but poor calibration, or vice versa.

Part 7: Diagnostics for Logistic Regression

Linearity in the Logit (for continuous predictors)

For continuous predictors, logistic regression assumes a linear relationship between the predictor and the log-odds. We can check this with a smoothed plot of the logit against the predictor.

brfss_logistic |>
  mutate(logit_pred = predict(mod_full, type = "link")) |>
  ggplot(aes(x = age, y = logit_pred)) +
  geom_point(alpha = 0.2, color = "steelblue") +
  geom_smooth(method = "loess", se = FALSE, color = "darkorange") +
  labs(title = "Linearity in the Logit: Age",
       x = "Age (years)", y = "Predicted Log-Odds (logit)") +
  theme_minimal()

Interpretation: A roughly linear loess curve supports the linearity assumption. A clearly curved pattern suggests we should add a quadratic term or use a spline.

Influential Observations

Cook’s distance and standardized residuals from logistic regression can be examined the same way as in linear regression:

brfss_logistic |>
  mutate(cooks_d = cooks.distance(mod_full),
         row_id = row_number()) |>
  ggplot(aes(x = row_id, y = cooks_d)) +
  geom_point(alpha = 0.4, color = "steelblue") +
  geom_hline(yintercept = 4 / nrow(brfss_logistic),
             linetype = "dashed", color = "red") +
  labs(title = "Cook's Distance for Logistic Regression Model",
       subtitle = "Red line: 4/n threshold",
       x = "Observation Index", y = "Cook's Distance") +
  theme_minimal()

Multicollinearity

VIFs work the same way as in linear regression:

vif(mod_full) |>
  as.data.frame() |>
  rownames_to_column("Predictor") |>
  kable(digits = 2, caption = "Variance Inflation Factors") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Variance Inflation Factors
Predictor vif(mod_full)
exercise 1.13
smoker 1.12
age 1.17
sex 1.01
sleep_hrs 1.02
income_cat 1.14
bmi 1.03
physhlth_days 1.20

Rule of thumb: VIF > 5 (or 10) indicates problematic multicollinearity.

Part 8: Reporting Logistic Regression Results

A publication-quality logistic regression table should include:

  1. Sample size and number of events
  2. Adjusted odds ratios with 95% CIs
  3. p-values for each coefficient
  4. Model fit statistics (AIC or pseudo-R²)
  5. Discrimination metric (AUC)
mod_full |>
  tbl_regression(
    exponentiate = TRUE,
    label = list(
      exercise      ~ "Exercise (past 30 days)",
      smoker        ~ "Smoking status",
      age           ~ "Age",
      sex           ~ "Sex",
      sleep_hrs     ~ "Sleep hours",
      income_cat    ~ "Income category",
      bmi           ~ "BMI",
      physhlth_days ~ "Physically unhealthy days"
    )
  ) |>
  add_glance_source_note(
    include = c(nobs, AIC, BIC),
    label = list(nobs ~ "N", AIC ~ "AIC", BIC ~ "BIC")
  ) |>
  bold_labels() |>
  bold_p() |>
  modify_caption("**Adjusted Odds Ratios for Frequent Mental Distress, BRFSS 2020**")
Adjusted Odds Ratios for Frequent Mental Distress, BRFSS 2020
Characteristic OR 95% CI p-value
Exercise (past 30 days)


    No — —
    Yes 0.82 0.68, 0.99 0.041
Smoking status


    Former/Never — —
    Current 1.30 1.09, 1.56 0.004
Age 0.97 0.96, 0.97 <0.001
Sex


    Male — —
    Female 1.68 1.41, 1.98 <0.001
Sleep hours 0.86 0.81, 0.90 <0.001
Income category 0.91 0.87, 0.95 <0.001
BMI 1.01 0.99, 1.02 0.4
Physically unhealthy days 1.06 1.06, 1.07 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio
N = 5,000; AIC = 3,646; BIC = 3,705

Summary

Concept Tool / R function
Multiple logistic regression glm(y ~ x1 + x2 + ..., family = binomial)
Adjusted odds ratios tidy(model, exponentiate = TRUE)
Wald test Default in summary() and tidy()
Likelihood ratio test anova(reduced, full, test = "LRT")
Confidence intervals confint(model) (profile CI) or tidy(..., conf.int = TRUE)
Interaction glm(y ~ x1 * x2, ...); test with LR test
Pseudo-R² performance::r2_mcfadden()
Hosmer-Lemeshow ResourceSelection::hoslem.test()
Calibration plot Decile-based observed vs. predicted
ROC curve / AUC pROC::roc() and pROC::auc()
Diagnostics cooks.distance(), vif(), linearity plots
Publication table gtsummary::tbl_regression()

Next Lecture (April 16)

  • Polytomous logistic regression: outcomes with > 2 nominal categories
  • Ordinal logistic regression: outcomes with > 2 ordered categories
  • Proportional odds assumption and how to test it

References

  • Kleinbaum, D. G., Kupper, L. L., Nizam, A., & Rosenberg, E. S. (2013). Applied Regression Analysis and Other Multivariable Methods (5th ed.), Chapter 22.
  • Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.). Wiley.
  • Steyerberg, E. W. (2019). Clinical Prediction Models (2nd ed.). Springer.

Part 9: In-Class Lab Activity

EPI 553 — Logistic Regression Part 2 Lab Due: End of class, April 14, 2026

Instructions

In this lab, you will build a multiple logistic regression model, conduct hypothesis tests, examine an interaction, and assess goodness-of-fit and discrimination. Use the same BRFSS 2020 logistic dataset from Part 1. Work through each task systematically. You may discuss concepts with classmates, but your written answers and R code must be your own.

Submission: Knit your .Rmd to HTML and upload to Brightspace by end of class.

Data for the Lab

Variable Description Type
fmd Frequent mental distress (No/Yes) Factor (outcome)
menthlth_days Mentally unhealthy days (0–30) Numeric
physhlth_days Physically unhealthy days (0–30) Numeric
sleep_hrs Sleep hours per night (1–14) Numeric
age Age in years (capped at 80) Numeric
sex Sex (Male/Female) Factor
bmi Body mass index Numeric
exercise Exercised in past 30 days (No/Yes) Factor
income_cat Household income category (1–8) Numeric
smoker Former/Never vs. Current Factor 
library(tidyverse)
library(broom)
library(knitr)
library(kableExtra)
library(gtsummary)
library(car)
library(ggeffects)
library(ResourceSelection)
library(pROC)
library(performance)

brfss_logistic <- readRDS(
  "E:/College WORK MS in EPI/SEM 2 MS EPI/Epi553 STATS 2/brfss_logistic_2020.rds"
)

Task 1: Build a Multiple Logistic Regression Model (15 points)

1a. (5 pts) Fit a multiple logistic regression model predicting fmd from at least 5 predictors of your choice.

mod_mlr_full <- glm(
  fmd ~ sleep_hrs + exercise + smoker + sex + income_cat + physhlth_days,
  data   = brfss_logistic,
  family = binomial(link = "logit")
)

tidy(mod_full, conf.int = TRUE, exponentiate = FALSE) |>
  mutate(across(where(is.numeric), ~ round(.x, 3))) |>
  kable(caption = "Multiple Logistic Regression: FMD ~ 6 Predictors (Log-Odds Scale)") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Multiple Logistic Regression: FMD ~ 6 Predictors (Log-Odds Scale)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 0.788 0.356 2.215 0.027 0.091 1.485
exerciseYes -0.198 0.097 -2.044 0.041 -0.387 -0.007
smokerCurrent 0.263 0.091 2.890 0.004 0.084 0.442
age -0.032 0.003 -11.225 0.000 -0.038 -0.026
sexFemale 0.516 0.086 5.975 0.000 0.347 0.686
sleep_hrs -0.154 0.027 -5.643 0.000 -0.208 -0.101
income_cat -0.096 0.020 -4.703 0.000 -0.135 -0.056
bmi 0.006 0.006 0.932 0.351 -0.007 0.018
physhlth_days 0.063 0.004 15.634 0.000 0.055 0.071

1b. (5 pts) Report the adjusted ORs with 95% CIs in a publication-quality table using tbl_regression().

mod_mlr_full |>
  tbl_regression(
    exponentiate = TRUE,
    label = list(
      sleep_hrs     ~ "Sleep hours (per 1 hr)",
      exercise      ~ "Exercise (past 30 days)",
      smoker        ~ "Smoking status",
      sex           ~ "Sex",
      income_cat    ~ "Income category (per unit)",
      physhlth_days ~ "Physically unhealthy days"
    )
  ) |>
  bold_labels() |>
  bold_p() |>
  modify_caption("**Table 1. Adjusted Odds Ratios for Frequent Mental Distress, BRFSS 2020**")
Table 1. Adjusted Odds Ratios for Frequent Mental Distress, BRFSS 2020
Characteristic OR 95% CI p-value
Sleep hours (per 1 hr) 0.82 0.78, 0.87 <0.001
Exercise (past 30 days)


    No — —
    Yes 0.91 0.76, 1.10 0.3
Smoking status


    Former/Never — —
    Current 1.66 1.40, 1.96 <0.001
Sex


    Male — —
    Female 1.62 1.38, 1.92 <0.001
Income category (per unit) 0.92 0.88, 0.95 <0.001
Physically unhealthy days 1.06 1.05, 1.07 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

1c. (5 pts) Interpret the adjusted OR for two predictors of your choice in 1-2 sentences each. Make sure to mention what the OR represents (per unit change for continuous; reference category for categorical).

Sleep hours: For each additional hour of sleep per night, the adjusted odds of frequent mental distress decrease by 18% (OR = 0.82). This means that more sleep is associated with lower odds of reporting poor mental health, holding all other variables constant.

Exercise Compared with adults who did not exercise in the past 30 days, those who did exercise have slightly lower adjusted odds of frequent mental distress (OR = 0.91), although this association is not statistically significant (p = 0.30). This means exercise does not show a meaningful independent association with mental distress after adjusting for other variables.

Task 2: Wald and Likelihood Ratio Tests (15 points)

2a. (5 pts) Identify the Wald p-value for each predictor in your model from the tidy() or summary() output.

tidy(mod_mlr_full, conf.int = TRUE, exponentiate = TRUE) |>
  filter(term != "(Intercept)") |>
  mutate(
    across(c(estimate, std.error, statistic, conf.low, conf.high), ~ round(.x, 3)),
    p.value = format.pval(p.value, digits = 3, eps = 0.001)
  ) |>
  kable(
    col.names = c("Term", "aOR", "SE", "z (Wald)", "95% CI Lower", "95% CI Upper", "Wald p-value"),
    caption   = "Wald Tests for Each Coefficient in the Full Model"
  ) |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Wald Tests for Each Coefficient in the Full Model
Term aOR SE z (Wald) 95% CI Lower 95% CI Upper Wald p-value
sleep_hrs 0.823 0.027 -7.206 <0.001 0.781 0.868
exerciseYes 0.913 0.094 -0.967 0.334 0.761 1.099
smokerCurrent 1.656 0.087 5.801 <0.001 1.396 1.964
sexFemale 1.625 0.085 5.717 <0.001 1.376 1.920
income_cat 0.916 0.020 -4.386 <0.001 0.881 0.953
physhlth_days 1.057 0.004 14.483 <0.001 1.049 1.065

Interpretation: Sleep hours (aOR = 0.823, p < 0.001) Each additional hour of sleep is associated with 17.7% lower odds of frequent mental distress. The Wald test shows this effect is highly significant.

Exercise (aOR = 0.913, p = 0.334) People who exercised in the past 30 days have slightly lower odds of frequent mental distress, but the Wald test shows this effect is not statistically significant.

Current smoking (aOR = 1.656, p < 0.001) Current smokers have 66% higher odds of frequent mental distress compared with former/never smokers. The Wald test strongly supports this as a significant independent predictor.

Female sex (aOR = 1.625, p < 0.001) Women have 62.5% higher odds of frequent mental distress compared with men. The Wald test confirms this is a significant effect.

Income category (aOR = 0.916, p < 0.001) Each step up in income category reduces the odds of frequent mental distress by 8.4%. The Wald test shows this is a significant protective factor.

Physical health days (aOR = 1.057, p < 0.001) Each additional physically unhealthy day increases the odds of frequent mental distress by 5.7%. This is one of the strongest predictors, with a very large Wald statistic.

2b. (5 pts) Fit a reduced model that drops one predictor of your choice. Perform a likelihood ratio test comparing the full and reduced models using anova(reduced, full, test = "LRT").

We will drop exercise category from the full model to test whether it jointly improves model fit.

mod_mlr_reduced <- glm(
  fmd ~ sleep_hrs + smoker + sex + income_cat + physhlth_days,
  data   = brfss_logistic,
  family = binomial
)

anova(mod_mlr_reduced, mod_mlr_full, test = "LRT") |>
  kable(
    digits  = 3,
    caption = "LR Test: Does dropping exercise improve the model?"
  ) |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
LR Test: Does dropping exercise improve the model?
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
4994 3762.947 NA NA NA
4993 3762.018 1 0.928 0.335

2c. (5 pts) Compare the conclusions from the Wald test (for the dropped predictor) and the LR test. Do they agree? In 2-3 sentences, explain when the two tests might disagree.

The Wald test for exerciseYes showed a non‑significant effect (p = 0.334), and the LR test also showed no significant improvement when exercise was removed (Deviance = 0.928, df = 1, p = 0.335). Both tests conclude that exercise does not independently improve model fit. The Wald test can disagree with the LR test when the coefficient estimate is unstable, has a large standard error, or lies near the boundary of the parameter space. This is because the Wald test relies on normal approximation, which can perform poorly in small samples. The LR test is generally more robust because it compares the full and reduced models directly, so disagreement usually happens when the Wald test underestimates significance while the LR test detects a real effect. —

Task 3: Test an Interaction (20 points)

3a. (5 pts) Fit a model that includes an interaction between two predictors of your choice.

We will test whether the effect of sleeping hours on FMD differs by sex.

mod_mlr_interact <- glm(
  fmd ~ sleep_hrs* sex + smoker + exercise + income_cat + physhlth_days,
  data   = brfss_logistic,
  family = binomial
)

mod_mlr_interact |>
  tbl_regression(exponentiate = TRUE) |>
  bold_labels() |>
  bold_p() |>
  modify_caption("**Model with sleeping hours and Sex Interaction**")
Model with sleeping hours and Sex Interaction
Characteristic OR 95% CI p-value
sleep_hrs 0.80 0.74, 0.87 <0.001
sex


    Male — —
    Female 1.15 0.56, 2.38 0.7
smoker


    Former/Never — —
    Current 1.66 1.40, 1.96 <0.001
exercise


    No — —
    Yes 0.91 0.76, 1.10 0.3
income_cat 0.92 0.88, 0.95 <0.001
physhlth_days 1.06 1.05, 1.06 <0.001
sleep_hrs * sex


    sleep_hrs * Female 1.05 0.95, 1.17 0.3
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

Interpretation: The OR for sleep is 0.80, meaning each additional hour of sleep reduces the odds of FMD by 20% for the reference group (men). The OR for female sex is 1.15, but not significant (p = 0.70), meaning women do not differ from men in baseline odds of FMD after adjustment. —

3b. (5 pts) Perform a likelihood ratio test comparing the model with the interaction to the model without it.

mod_mlr_no_interact <- glm(
  fmd ~ sleep_hrs + sex + smoker + exercise + income_cat + physhlth_days,
  data   = brfss_logistic,
  family = binomial
)

anova(mod_mlr_no_interact, mod_mlr_interact, test = "LRT") |>
  kable(
    digits  = 3,
    caption = "LR Test for sleep hours and Sex Interaction"
  ) |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
LR Test for sleep hours and Sex Interaction
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
4993 3762.018 NA NA NA
4992 3761.115 1 0.903 0.342

Interpretation: Because the LR test p‑value is 0.342, > α = 0.05, we fail to reject the null hypothesis that the sleep and sex interaction coefficient equals zero. This means there is no statistically significant evidence that the effect of sleep hours on frequent mental distress differs between men and women.

3c. (5 pts) Create a visualization of the interaction using ggpredict() and plot().

ggpredict(mod_mlr_interact, terms = c("sleep_hrs", "sex")) |>
  plot() +
  labs(
    title    = "Predicted Probability of FMD by Sleep Hours and Sex",
    subtitle = "Interaction model; ribbons = 95% confidence intervals",
    x        = "Sleeping Hours",
    y        = "Predicted Probability of FMD",
    color    = "Sex"
  ) +
  scale_y_continuous(labels = scales::percent_format()) +
  theme_minimal()

Interpretation: More sleep is associated with a lower predicted probability of frequent mental distress (FMD) for both men and women. Women have somewhat higher predicted probabilities overall but the slopes are nearly parallel.

3d. (5 pts) In 3-4 sentences, interpret the interaction. Does the effect of one predictor differ across levels of the other? If statistically significant, report the stratum-specific odds ratios.

Since I want to look at sleep, I am comparing two sleep amounts Short Sleep (5 hours) vs. Healthy Sleep (8 hours).

ggpredict(mod_mlr_interact, terms = c("sleep_hrs [5, 8]", "sex")) |>
  as_tibble() |>
  pivot_wider(id_cols = group, names_from = x, values_from = predicted) |>
  mutate(OR_5vs8 = (`5` / (1 - `5`)) / (`8` / (1 - `8`))) |> 
  dplyr::select(Sex = group, OR_5vs8) |> 
  kable(
    digits   = 3,
     col.names = c("Sex", "OR (5 hrs vs. 8 hrs Sleep)"), 
    caption = "Odds of Outcome: Short Sleep (5h) vs. Adequate Sleep (8h) Stratified by Sex"
  ) |> 
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Odds of Outcome: Short Sleep (5h) vs. Adequate Sleep (8h) Stratified by Sex
Sex OR (5 hrs vs. 8 hrs Sleep)
Male 1.949
Female 1.672

Interpretation: Men (OR = 1.949) Among men, sleeping 5 hours instead of 8 hours is associated with approximately 95% higher odds of frequent mental distress. In other words, short‑sleeping men have nearly double the odds of FMD compared with men who sleep adequately.

Women (OR = 1.672) Among women, sleeping 5 hours instead of 8 hours is associated with about 67% higher odds of frequent mental distress. Women sleeping lower hours show elevated risk, but the increase is smaller than in men.

Task 4: Goodness-of-Fit and Discrimination (25 points)

4a. (5 pts) Compute McFadden’s pseudo-R² for your full model using performance::r2_mcfadden().

performance::r2_mcfadden(mod_mlr_full)
## # R2 for Generalized Linear Regression
##        R2: 0.115
##   adj. R2: 0.115

Interpretation: An R² of 0.115 means that full model explains about 11.5% of the variation in the likelihood of frequent mental distress. In logistic regression, R² values are typically much lower than in linear regression, so an R² around 0.10–0.20 is common even for well‑performing models. This shows that while the predictors (sleep, smoking, sex, income, physical health days, etc.) cover a meaningful risk, most of the variability in mental distress is driven by factors not included in the model, which is expected for complex behavioral health outcomes.

4b. (5 pts) Perform the Hosmer-Lemeshow goodness-of-fit test using ResourceSelection::hoslem.test(). Report the test statistic and p-value. Comment on the interpretation given your sample size.

hl_lab <- hoslem.test(
  x = as.numeric(brfss_logistic$fmd) - 1,
  y = fitted(mod_mlr_full),
  g = 10
)

hl_lab
## 
##  Hosmer and Lemeshow goodness of fit (GOF) test
## 
## data:  as.numeric(brfss_logistic$fmd) - 1, fitted(mod_mlr_full)
## X-squared = 16.297, df = 8, p-value = 0.03832
tibble(
  Statistic = round(hl_lab$statistic, 3),
  df        = hl_lab$parameter,
  `p-value` = format.pval(hl_lab$p.value, digits = 3, eps = 0.001)
) |>
  kable(caption = "Hosmer-Lemeshow Goodness-of-Fit Test") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Hosmer-Lemeshow Goodness-of-Fit Test
Statistic df p-value
16.297 8 0.0383

Interpretation: The Hosmer-Lemeshow (HL) test evaluates whether the observed event proportions within deciles of predicted probability agree with the model’s predicted probabilities. Although the Hosmer–Lemeshow test was statistically significant (p = 0.038), it suggests minor miscalibration and this is likely due to the large sample size, and the model still appears to fit reasonably well based on other diagnostics.

4c. (5 pts) Create a calibration plot showing observed vs. predicted probabilities by decile. Comment on whether the model appears well calibrated.

brfss_logistic |>
  mutate(
    pred_prob   = fitted(mod_mlr_full),
    obs_outcome = as.numeric(fmd) - 1,
    decile      = ntile(pred_prob, 10)
  ) |>
  group_by(decile) |>
  summarise(
    mean_pred = mean(pred_prob),
    mean_obs  = mean(obs_outcome),
    n         = n(),
    .groups   = "drop"
  ) |>
  ggplot(aes(x = mean_pred, y = mean_obs)) +
  geom_abline(slope = 1, intercept = 0, color = "red", linetype = "dashed",
              linewidth = 0.8) +
  geom_point(aes(size = n), color = "steelblue", alpha = 0.85) +
  geom_line(color = "steelblue", linewidth = 0.7) +
  scale_size_continuous(name = "Decile n", range = c(3, 8)) +
  scale_x_continuous(labels = scales::percent_format()) +
  scale_y_continuous(labels = scales::percent_format()) +
  labs(
    title    = "Calibration Plot: Observed vs. Predicted Probability of FMD",
    subtitle = "Each point = one predicted probability decile; dashed line = perfect calibration",
    x        = "Mean Predicted Probability (within decile)",
    y        = "Observed Proportion with FMD (within decile)"
  ) +
  theme_minimal()

Interpretation: The calibration plot compares the model’s predicted probabilities of frequent mental distress (FMD) with the observed proportions across deciles of predicted risk. The blue line (observed values) follows the red dashed line (perfect calibration) fairly closely, and it shows that the model’s predictions are well‑aligned with actual outcomes.

There is some mild deviation at a few deciles where the observed points fall slightly above or below the ideal line but overall it shows reasonable calibration, especially given the large sample size.

4d. (10 pts) Compute and plot the ROC curve using pROC::roc(). Report the AUC. Based on the AUC value, how would you describe the model’s discrimination ability (poor, acceptable, excellent, outstanding)?

roc_lab <- roc(
  response  = brfss_logistic$fmd,
  predictor = fitted(mod_mlr_full),
  levels    = c("No", "Yes"),
  direction = "<"
)

auc_lab <- auc(roc_lab)

# ROC curve plot
ggroc(roc_lab, color = "#2166ac", linewidth = 1.3) +
  geom_abline(slope = 1, intercept = 1, linetype = "dashed",
              color = "red", linewidth = 0.8) +
  annotate("text", x = 0.35, y = 0.15,
           label = paste0("AUC = ", round(auc_lab, 3)),
           size = 5, color = "#2166ac", fontface = "bold") +
  labs(
    title    = "ROC Curve for Frequent Mental Distress — Lab Model",
    subtitle = "Blue curve = model; red dashed = chance (AUC = 0.50)",
    x        = "Specificity (1 − False Positive Rate)",
    y        = "Sensitivity (True Positive Rate)"
  ) +
  theme_minimal()

# Print AUC with confidence interval
ci_auc <- ci.auc(roc_lab)
tibble(
  AUC        = round(as.numeric(auc_lab), 3),
  `95% CI Lower` = round(ci_auc[1], 3),
  `95% CI Upper` = round(ci_auc[3], 3)
) |>
  kable(caption = "AUC with 95% Bootstrap Confidence Interval") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
AUC with 95% Bootstrap Confidence Interval
AUC 95% CI Lower 95% CI Upper
0.736 0.716 0.757

Interpretation: An AUC of 0.736 falls in the range with acceptable discrimination.It means the model can reasonably distinguish between individuals with and without frequent mental distress, performing better than chance but not at the level of an excellent one.

Lab Summary

Overall interpretation:

Frequent mental distress among U.S. adults is influenced by a combination of behavioral, demographic, and socioeconomic factors. In the fully adjusted model, shorter sleep duration, current smoking, lower income, and a greater number of physically unhealthy days each independently increased the odds of FMD, while exercise showed no meaningful association after adjustment. Model performance was solid, with acceptable discrimination (AUC = 0.736) and reasonable calibration, despite a slightly significant Hosmer–Lemeshow test that likely reflects the large sample size rather than substantive misfit. Interaction testing showed that the protective effect of sleep did not differ significantly by sex, and predicted‑probability plots confirmed parallel declines in FMD risk for men and women as sleep increased. Overall, the findings highlight sleep, smoking cessation, and physical health improvement as key important points for population‑level mental health interventions.

End of Lab Activity