Introduction

Binary logistic regression handles two-category outcomes. But many epidemiologic outcomes have more than two categories:

General health: Excellent / Very Good / Good / Fair / Poor
Disease stage: I / II / III / IV
Treatment choice: Drug A / Drug B / Drug C / No treatment
BMI category: Underweight / Normal / Overweight / Obese

When the response variable has \(K > 2\) categories, we use polytomous logistic regression (also called multinomial logistic regression). When the categories have a natural order, we can take advantage of that ordering with ordinal logistic regression.

Today’s roadmap (Kleinbaum Ch. 23): 1. Polytomous (multinomial) logistic regression for nominal outcomes 2. Ordinal logistic regression for ordered outcomes 3. The proportional odds assumption 4. Choosing between models

Why Not Just Run Multiple Binary Models?

Suppose general health has 3 categories: Excellent, Good, Poor. A naive approach would be to dichotomize and run a binary logistic regression repeatedly:

Excellent vs. not
Good vs. not
Poor vs. not

Problems with this approach:

Information loss. Dichotomizing throws away the distinction between the other categories.
Inconsistent samples. Each model uses a different comparison group.
No coherent probability model. Predicted probabilities won’t sum to 1.
Inefficiency. Standard errors are larger than necessary.

A polytomous model fits all categories simultaneously in a single likelihood, sharing information across comparisons.

Setup and Data

library(tidyverse)
library(haven)
library(janitor)
library(broom)
library(knitr)
library(kableExtra)
library(gtsummary)
library(nnet)        # multinom() for polytomous regression
library(MASS)        # polr() for ordinal regression
library(brant)       # Brant test for proportional odds
library(ggeffects)

options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))

brfss_full <- read_xpt(
  "C:/Users/userp/OneDrive/Рабочий стол/HSTA553/R files/LLCP2020.XPT"
) |>
  clean_names()

Building the Outcome: General Health

The BRFSS variable genhlth asks: “Would you say that in general your health is…”

1 = Excellent, 2 = Very Good, 3 = Good, 4 = Fair, 5 = Poor

We collapse this into a 3-level ordinal outcome to keep models tractable for class:

brfss_poly <- brfss_full |>
  mutate(
    # 3-level ordinal outcome
    genhlth_3 = case_when(
      genhlth %in% c(1, 2) ~ "Excellent/VG",
      genhlth == 3         ~ "Good",
      genhlth %in% c(4, 5) ~ "Fair/Poor",
      TRUE                 ~ NA_character_
    ),
    # Ordered factor for ordinal regression
    genhlth_ord = factor(genhlth_3,
      levels = c("Excellent/VG", "Good", "Fair/Poor"),
      ordered = TRUE),
    # Unordered factor for polytomous regression
    genhlth_nom = factor(genhlth_3,
      levels = c("Excellent/VG", "Good", "Fair/Poor")),
    # Predictors
    age = age80,
    sex = factor(sexvar, levels = c(1, 2), labels = c("Male", "Female")),
    bmi = ifelse(bmi5 > 0, bmi5 / 100, NA_real_),
    exercise = factor(case_when(
      exerany2 == 1 ~ "Yes",
      exerany2 == 2 ~ "No",
      TRUE          ~ NA_character_
    ), levels = c("No", "Yes")),
    income_cat = case_when(
      income2 %in% 1:8 ~ as.numeric(income2),
      TRUE             ~ NA_real_
    ),
    smoker = factor(case_when(
      smokday2 %in% c(1, 2) ~ "Current",
      smokday2 == 3          ~ "Former/Never",
      TRUE                   ~ NA_character_
    ), levels = c("Former/Never", "Current"))
  ) |>
  filter(
    !is.na(genhlth_ord), !is.na(age), age >= 18, !is.na(sex),
    !is.na(bmi), !is.na(exercise), !is.na(income_cat), !is.na(smoker)
  )

set.seed(1220)
brfss_poly <- brfss_poly |>
  dplyr::select(genhlth_ord, genhlth_nom, age, sex, bmi,
                exercise, income_cat, smoker) |>
  slice_sample(n = 5000)

saveRDS(brfss_poly,
  "C:/Users/userp/OneDrive/Рабочий стол/HSTA553/R files/brfss_polytomous_2020.rds")

brfss_poly |>
  count(genhlth_ord) |>
  mutate(pct = round(100 * n / sum(n), 1)) |>
  kable(col.names = c("General Health", "N", "%"),
        caption = "Outcome Distribution") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Outcome Distribution
General Health	N	%
Excellent/VG	2380	47.6
Good	1624	32.5
Fair/Poor	996	19.9

Since genhlth_ord and genhlth_nom contain the same three levels (just coded as ordered vs. unordered factors), the distributions will be identical — but showing both makes the distinction between the two factor types explicit for the lecture, which is useful context for introducing polytomous vs. ordinal regression.

Part 1: Polytomous (Multinomial) Logistic Regression

The Model

For an outcome with \(K\) categories, we choose one as the reference (say, \(Y = 1\)) and fit \(K - 1\) logit equations:

\[ \log\left[\frac{\Pr(Y = k)}{\Pr(Y = 1)}\right] = \alpha_k + \beta_{k1} X_1 + \cdots + \beta_{kp} X_p, \quad k = 2, \ldots, K \]

Each comparison gets its own intercept and its own slopes. With 3 outcome categories and 5 predictors, that’s \(2 \times (1 + 5) = 12\) parameters.

The probabilities are recovered as:

\[ \Pr(Y = k \mid X) = \frac{\exp(\alpha_k + X^T \beta_k)}{1 + \sum_{j=2}^{K} \exp(\alpha_j + X^T \beta_j)} \]

By construction, the probabilities sum to 1.

Fitting in R: `nnet::multinom()`

mod_multi <- multinom(
  genhlth_nom ~ age + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  trace = FALSE
)

summary(mod_multi)

## Call:
## multinom(formula = genhlth_nom ~ age + sex + bmi + exercise + 
##     income_cat + smoker, data = brfss_poly, trace = FALSE)
## 
## Coefficients:
##           (Intercept)        age  sexFemale        bmi exerciseYes income_cat
## Good        -1.317306 0.01497067 -0.1417121 0.05087197   -0.500636 -0.1693992
## Fair/Poor   -1.283880 0.02579060 -0.1257209 0.06741312   -1.338926 -0.3932631
##           smokerCurrent
## Good          0.4117899
## Fair/Poor     0.3436415
## 
## Std. Errors:
##           (Intercept)         age  sexFemale         bmi exerciseYes income_cat
## Good        0.2689762 0.002189706 0.06782701 0.005768280  0.08178912 0.01748213
## Fair/Poor   0.3296881 0.002916356 0.08583048 0.006778269  0.09155273 0.02090747
##           smokerCurrent
## Good         0.07723392
## Fair/Poor    0.09720041
## 
## Residual Deviance: 9287.557 
## AIC: 9315.557

Interpreting the Coefficients

The output gives two sets of coefficients, one for each non-reference category:

“Good” vs. “Excellent/VG”
“Fair/Poor” vs. “Excellent/VG”

Each \(\beta\) is interpreted as a log relative risk ratio (RRR). Exponentiating gives the relative risk ratio (sometimes called a relative odds ratio) for being in category \(k\) vs. the reference, per one-unit change in \(X\).

tidy(mod_multi, conf.int = TRUE, exponentiate = TRUE) |>
  dplyr::select(y.level, term, estimate, conf.low, conf.high, p.value) |>
  mutate(across(c(estimate, conf.low, conf.high), \(x) round(x, 3)),
         p.value = format.pval(p.value, digits = 3)) |>
  kable(col.names = c("Outcome", "Predictor", "RRR", "Lower", "Upper", "p"),
        caption = "Multinomial Logistic Regression: RRRs vs. Excellent/VG") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Multinomial Logistic Regression: RRRs vs. Excellent/VG
Outcome	Predictor	RRR	Lower	Upper	p
Good	(Intercept)	0.268	0.158	0.454	9.71e-07
Good	age	1.015	1.011	1.019	8.10e-12
Good	sexFemale	0.868	0.760	0.991	0.036679
Good	bmi	1.052	1.040	1.064	< 2e-16
Good	exerciseYes	0.606	0.516	0.712	9.30e-10
Good	income_cat	0.844	0.816	0.874	< 2e-16
Good	smokerCurrent	1.510	1.297	1.756	9.73e-08
Fair/Poor	(Intercept)	0.277	0.145	0.529	9.85e-05
Fair/Poor	age	1.026	1.020	1.032	< 2e-16
Fair/Poor	sexFemale	0.882	0.745	1.043	0.142987
Fair/Poor	bmi	1.070	1.056	1.084	< 2e-16
Fair/Poor	exerciseYes	0.262	0.219	0.314	< 2e-16
Fair/Poor	income_cat	0.675	0.648	0.703	< 2e-16
Fair/Poor	smokerCurrent	1.410	1.165	1.706	0.000407

Example interpretation. Holding all else constant, current smokers have a relative risk ratio of 1.41 (95% CI: 1.17–1.71) for reporting “Fair/Poor” health vs. “Excellent/VG” compared to former/never smokers. In other words, the relative risk of being in the “Fair/Poor” category (rather than “Excellent/VG”) is about 41% higher among current smokers.

Predicted Probabilities

ggpredict(mod_multi, terms = "income_cat [1:8]") |>
  plot() +
  labs(title = "Predicted Probability of Each Health Category by Income",
       x = "Income Category (1 = lowest, 8 = highest)",
       y = "Predicted Probability") +
  theme_minimal()

Interpretation. Holding all other predictors at their means, the predicted probability of reporting “Fair/Poor” health drops sharply across income levels — from 63% at the lowest income category to 17% at the highest. The predicted probability of “Excellent/VG” health shows the opposite pattern, rising from 11% to 49% across the same range. The probability of “Good” health remains comparatively stable (roughly 26%–35%), reflecting that income primarily shifts respondents between the two extremes rather than into or out of the “Good” category.

Part 2: Ordinal Logistic Regression

When categories are ordered, the polytomous model wastes information. It treats “Good” as just as different from “Excellent” as it is from “Fair/Poor”. An ordinal model exploits the ordering and produces fewer parameters and easier interpretation.

The Proportional Odds (Cumulative Logit) Model

The most common ordinal model is the proportional odds model, also called the cumulative logit model. Rather than modeling the probability of each category separately (as in the multinomial model), it models the probability of being at or below a given category. This respects the ordering of the outcome — “Excellent/VG” < “Good” < “Fair/Poor” in terms of health decline — and yields a single, consistent estimate of each predictor’s effect across the entire outcome scale.

The tradeoff for this parsimony is an additional assumption: the proportional odds assumption, which requires that the effect of each predictor is the same regardless of which cut-point is being modeled. This should be checked after fitting.

Define cumulative probabilities:

\[ \gamma_k = \Pr(Y \leq k \mid X) \]

The model is:

\[ \log\left[\frac{\Pr(Y \leq k)}{\Pr(Y > k)}\right] = \alpha_k - \beta_1 X_1 - \cdots - \beta_p X_p \]

Key features:

One intercept per cut-point (\(k = 1, \ldots, K - 1\))
A single set of slopes \(\beta\), shared across all cut-points
The “proportional odds” assumption: the effect of \(X\) on the log-odds of being at or below category \(k\) is the same for every \(k\)

With \(K = 3\) and 5 predictors, that’s \(2 + 5 = 7\) parameters (vs. 12 for multinomial).

In plain terms: imagine income predicts health. The proportional odds assumption says that the boost in odds of being in “Excellent/VG or better” (vs. “Good or worse”) from moving up one income level is the same as the boost in odds of being in “Excellent/VG or Good” (vs. “Fair/Poor”). The effect of income doesn’t change depending on which boundary you’re crossing — it’s a single, consistent shift up or down the health scale.

Fitting in R: `MASS::polr()`

mod_ord <- polr(
  genhlth_ord ~ age + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  Hess = TRUE
)

summary(mod_ord)

## Call:
## polr(formula = genhlth_ord ~ age + sex + bmi + exercise + income_cat + 
##     smoker, data = brfss_poly, Hess = TRUE)
## 
## Coefficients:
##                  Value Std. Error t value
## age            0.01797   0.001867   9.627
## sexFemale     -0.12885   0.056835  -2.267
## bmi            0.04973   0.004534  10.967
## exerciseYes   -0.92167   0.064426 -14.306
## income_cat    -0.26967   0.014113 -19.107
## smokerCurrent  0.29313   0.064304   4.558
## 
## Intercepts:
##                   Value    Std. Error t value 
## Excellent/VG|Good   0.0985   0.2200     0.4478
## Good|Fair/Poor      1.8728   0.2212     8.4650
## 
## Residual Deviance: 9318.274 
## AIC: 9334.274

Odds Ratios and Interpretation

tidy(mod_ord, conf.int = TRUE, exponentiate = TRUE) |>
  filter(coef.type == "coefficient") |>
  dplyr::select(term, estimate, conf.low, conf.high) |>
  mutate(across(c(estimate, conf.low, conf.high), \(x) round(x, 3))) |>
  kable(col.names = c("Predictor", "OR", "Lower", "Upper"),
        caption = "Ordinal Logistic Regression: Cumulative ORs") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Ordinal Logistic Regression: Cumulative ORs
Predictor	OR	Lower	Upper
age	1.018	1.014	1.022
sexFemale	0.879	0.786	0.983
bmi	1.051	1.042	1.060
exerciseYes	0.398	0.351	0.451
income_cat	0.764	0.743	0.785
smokerCurrent	1.341	1.182	1.521

Interpretation. An OR of \(e^{\beta}\) for a predictor means: a one-unit increase in \(X\) multiplies the odds of being in a worse health category (i.e., \(Y > k\) vs. \(Y \leq k\)) by \(e^{\beta}\), and this is true at every cut-point. Here, \(\beta\) is the estimated coefficient from the model (the log-odds), and \(e^{\beta}\) is simply that coefficient exponentiated — converting it from the log-odds scale to an odds ratio that is easier to interpret.

For example, if smoking has OR = 1.6, then current smokers have 1.6 times the odds of reporting Good or worse health (vs. Excellent/VG), AND 1.6 times the odds of reporting Fair/Poor (vs. Excellent/VG or Good).

Predicted Probabilities

ggpredict(mod_ord, terms = "income_cat [1:8]") |>
  plot() +
  labs(title = "Ordinal Model: Predicted Probability of Each Health Category",
       x = "Income Category", y = "Predicted Probability") +
  theme_minimal()

Part 3: Checking the Proportional Odds Assumption

The proportional odds assumption is strong. It states that the effect of each predictor on the log-odds of being at or below any cut-point is constant across all cut-points — in other words, one \(\beta\) summarizes the predictor’s effect across the entire outcome scale. If this holds, the ordinal model is both valid and parsimonious. If it fails, that single \(\beta\) is a misleading average of effects that actually differ across thresholds, and a more flexible model is needed.

There are two complementary ways to evaluate this assumption: a visual check and a formal test.

Visual Check: Empirical Logits

The idea is to fit separate binary logistic regressions at each cumulative cut-point and compare the coefficients. If the proportional odds assumption holds, the log-odds coefficients should be roughly the same at both cut-points — the two sets of points should overlap or sit close together. Large separation between cut-points for a given predictor is a warning sign.

cut1 <- glm(I(as.numeric(genhlth_ord) <= 1) ~ age + sex + bmi + exercise + income_cat + smoker,
            data = brfss_poly, family = binomial)
cut2 <- glm(I(as.numeric(genhlth_ord) <= 2) ~ age + sex + bmi + exercise + income_cat + smoker,
            data = brfss_poly, family = binomial)

bind_rows(
  broom::tidy(cut1, conf.int = TRUE) |> mutate(cutpoint = "\u2264 Excellent/VG"),
  broom::tidy(cut2, conf.int = TRUE) |> mutate(cutpoint = "\u2264 Good")
) |>
  filter(term != "(Intercept)") |>
  ggplot(aes(x = estimate, y = term, color = cutpoint, shape = cutpoint)) +
  geom_vline(xintercept = 0, linetype = "dashed", color = "gray50") +
  geom_pointrange(aes(xmin = conf.low, xmax = conf.high),
                  position = position_dodge(width = 0.5)) +
  labs(
    title = "Visual Check: Proportional Odds Assumption",
    subtitle = "Coefficients should overlap across cut-points if assumption holds",
    x = "Log-Odds Coefficient",
    y = "Predictor",
    color = "Cut-point",
    shape = "Cut-point"
  ) +
  theme_minimal()

Predictors whose point estimates are far apart between cut-points (e.g., smokerCurrent, sexFemale) are candidates for violating the assumption — confirm with the Brant test below.

Formal Test: Brant Test

Omnibus p > 0.05: assumption holds for the model overall
Predictor-level p < 0.05: assumption fails for that variable

brant_out <- brant(mod_ord)

## -------------------------------------------- 
## Test for X2  df  probability 
## -------------------------------------------- 
## Omnibus      35.87   6   0
## age      0.06    1   0.81
## sexFemale    0.89    1   0.34
## bmi      7.1 1   0.01
## exerciseYes  10.58   1   0
## income_cat   10.54   1   0
## smokerCurrent    7.76    1   0.01
## -------------------------------------------- 
## 
## H0: Parallel Regression Assumption holds

brant_tbl <- as.data.frame(brant_out) |>
  tibble::rownames_to_column("Predictor") |>
  mutate(
    X2_num = X2,
    p_num = probability,
    X2 = round(X2, 2),
    probability = ifelse(p_num < 0.001, "< 0.001", sprintf("%.3f", p_num)),
    Fails = dplyr::case_when(
      Predictor == "Omnibus" & p_num < 0.05 ~ "Yes (overall)",
      Predictor == "Omnibus" & p_num >= 0.05 ~ "No (overall)",
      p_num < 0.05 ~ "Yes",
      TRUE ~ ""
    )
  ) |>
  dplyr::select(Predictor, X2, df, probability, Fails)

kable(
  brant_tbl,
  col.names = c("Predictor", "X^2", "df", "p-value", "Assumption Violated?"),
  caption   = "Brant Test for Proportional Odds Assumption",
  align     = c("l", "r", "r", "r", "c")
) |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) |>
  row_spec(0, bold = TRUE) |>
  row_spec(1, bold = TRUE, background = "#f0f0f0")

Brant Test for Proportional Odds Assumption
Predictor	X^2	df	p-value	Assumption Violated?
Omnibus	35.87	6	< 0.001	Yes (overall)
age	0.06	1	0.814
sexFemale	0.89	1	0.345
bmi	7.10	1	0.008	Yes
exerciseYes	10.58	1	0.001	Yes
income_cat	10.54	1	0.001	Yes
smokerCurrent	7.76	1	0.005	Yes

What If It Fails?

Use the multinomial model instead (loses parsimony, gains flexibility)
Use a partial proportional odds model (some \(\beta\)s vary across cut-points)
Collapse outcome categories
Use a different link (continuation-ratio, adjacent-category)

Part 4: Comparing the Two Models

tibble(
  Feature = c("Outcome type", "Number of equations", "Parameters (3 cats, 5 preds)",
              "Key assumption", "Best when"),
  `Multinomial (multinom)` = c("Nominal or ordinal",
                                "K - 1 = 2",
                                "12",
                                "None beyond independence",
                                "Categories unordered or PO fails"),
  `Ordinal (polr)`         = c("Ordinal only",
                                "1 (with K - 1 cut-points)",
                                "7",
                                "Proportional odds",
                                "Categories ordered AND PO holds")
) |>
  kable(caption = "Multinomial vs. Ordinal Logistic Regression") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Multinomial vs. Ordinal Logistic Regression
Feature	Multinomial (multinom)	Ordinal (polr)
Outcome type	Nominal or ordinal	Ordinal only
Number of equations	K - 1 = 2	1 (with K - 1 cut-points)
Parameters (3 cats, 5 preds)	12	7
Key assumption	None beyond independence	Proportional odds
Best when	Categories unordered or PO fails	Categories ordered AND PO holds

Likelihood Comparison

Both models are fit by maximum likelihood. We can compare their fits with AIC (lower = better):

tibble(
  Model = c("Multinomial", "Ordinal (proportional odds)"),
  AIC   = c(AIC(mod_multi), AIC(mod_ord)),
  df    = c(mod_multi$edf, length(coef(mod_ord)) + length(mod_ord$zeta))
) |>
  mutate(AIC = round(AIC, 1)) |>
  kable(caption = "Model Comparison") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Model Comparison
Model	AIC	df
Multinomial	9315.6	14
Ordinal (proportional odds)	9334.3	8

Tradeoff: The ordinal model is more parsimonious. If proportional odds holds, it has lower AIC and tighter confidence intervals. If PO fails, the multinomial model is more honest.

Summary

Concept	Key Point
Polytomous outcome	Use `nnet::multinom()` for \(K > 2\) categories
Coefficients	\(K - 1\) sets, each interpreted as log relative risk ratios vs. reference
Ordinal outcome	Use `MASS::polr()` to exploit ordering
Cumulative logit	Single set of \(\beta\)s describes all cut-points
Proportional odds	Strong assumption; check with `brant()`
Model choice	Ordinal if ordered AND PO holds; otherwise multinomial

Next Lecture (April 21)

Logistic regression applications and case studies
Putting it all together: model building strategy
Reporting results for publication

References

Kleinbaum, D. G., Kupper, L. L., Nizam, A., & Rosenberg, E. S. (2013). Applied Regression Analysis and Other Multivariable Methods (5th ed.), Chapter 23.
Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.), Chapter 8.
Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.). Wiley.

Part 5: In-Class Lab Activity

EPI 553 — Polytomous and Ordinal Regression Lab Due: End of class, April 16, 2026, NUrsultan

Instructions

Complete the four tasks below using brfss_polytomous_2020.rds. Submit a knitted HTML file via Brightspace.

Data

Variable	Description	Type
`genhlth_ord`	General health (Excellent/VG < Good < Fair/Poor)	Ordered factor (outcome)
`genhlth_nom`	General health, unordered	Factor (outcome)
`age`	Age in years	Numeric
`sex`	Male / Female	Factor
`bmi`	Body mass index	Numeric
`exercise`	Exercised in past 30 days (No/Yes)	Factor
`income_cat`	Household income (1-8)	Numeric
`smoker`	Former/Never vs. Current	Factor

library(tidyverse)
library(broom)
library(knitr)
library(kableExtra)
library(gtsummary)
library(nnet)
library(MASS)
library(brant)
library(ggeffects)

options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))

brfss_poly <- readRDS("C:/Users/userp/OneDrive/Рабочий стол/HSTA553/R files/brfss_polytomous_2020.rds")

Task 1: Explore the Outcome (15 points)

1a. (5 pts) Create a frequency table of genhlth_ord showing N and percentage for each category.

brfss_poly |>
  count(genhlth_ord) |>
  mutate(percent = round(100 * n / sum(n), 1)) |>
  kable(
    col.names = c("General Health", "N", "Percent"),
    caption = "Frequency Distribution of General Health"
  ) |>
  kable_styling(full_width = FALSE)

Frequency Distribution of General Health
General Health	N	Percent
Excellent/VG	2380	47.6
Good	1624	32.5
Fair/Poor	996	19.9

1b. (5 pts) Create a stacked bar chart showing the distribution of general health by smoker status.

brfss_poly |>
  ggplot(aes(x = smoker, fill = genhlth_ord)) +
  geom_bar(position = "fill") +
  scale_y_continuous(labels = scales::percent_format()) +
  labs(
    title = "Distribution of General Health by Smoking Status",
    x = "Smoking Status",
    y = "Proportion",
    fill = "General Health"
  ) +
  theme_minimal()

1c. (5 pts) Use tbl_summary() to create a descriptive table stratified by genhlth_ord with at least 4 predictors.

brfss_poly |>
  dplyr::select(genhlth_ord, age, sex, bmi, exercise, income_cat, smoker) |>
  tbl_summary(
    by = genhlth_ord,
    statistic = list(
      all_continuous() ~ "{mean} ({sd})",
      all_categorical() ~ "{n} ({p}%)"
    ),
    digits = all_continuous() ~ 1
  ) |>
  bold_labels()

Characteristic	Excellent/VG N = 2,380¹	Good N = 1,624¹	Fair/Poor N = 996¹
IMPUTED AGE VALUE COLLAPSED ABOVE 80	54.7 (16.5)	57.9 (16.3)	60.8 (14.6)
sex
Male	1,315 (55%)	906 (56%)	496 (50%)
Female	1,065 (45%)	718 (44%)	500 (50%)
bmi	27.4 (5.2)	29.1 (6.4)	30.0 (8.0)
exercise	1,999 (84%)	1,161 (71%)	470 (47%)
income_cat
1	39 (1.6%)	72 (4.4%)	114 (11%)
2	69 (2.9%)	92 (5.7%)	110 (11%)
3	124 (5.2%)	128 (7.9%)	151 (15%)
4	174 (7.3%)	188 (12%)	150 (15%)
5	198 (8.3%)	197 (12%)	124 (12%)
6	354 (15%)	249 (15%)	111 (11%)
7	443 (19%)	289 (18%)	112 (11%)
8	979 (41%)	409 (25%)	124 (12%)
smoker
Former/Never	1,692 (71%)	1,015 (63%)	597 (60%)
Current	688 (29%)	609 (38%)	399 (40%)
¹ Mean (SD); n (%)

Task 2: Multinomial Logistic Regression (20 points)

2a. (5 pts) Fit a multinomial logistic regression model predicting genhlth_nom from at least 4 predictors using multinom().

mod_multi <- multinom(
  genhlth_nom ~ age + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  trace = FALSE
)

summary(mod_multi)

## Call:
## multinom(formula = genhlth_nom ~ age + sex + bmi + exercise + 
##     income_cat + smoker, data = brfss_poly, trace = FALSE)
## 
## Coefficients:
##           (Intercept)        age  sexFemale        bmi exerciseYes income_cat
## Good        -1.317306 0.01497067 -0.1417121 0.05087197   -0.500636 -0.1693992
## Fair/Poor   -1.283880 0.02579060 -0.1257209 0.06741312   -1.338926 -0.3932631
##           smokerCurrent
## Good          0.4117899
## Fair/Poor     0.3436415
## 
## Std. Errors:
##           (Intercept)         age  sexFemale         bmi exerciseYes income_cat
## Good        0.2689762 0.002189706 0.06782701 0.005768280  0.08178912 0.01748213
## Fair/Poor   0.3296881 0.002916356 0.08583048 0.006778269  0.09155273 0.02090747
##           smokerCurrent
## Good         0.07723392
## Fair/Poor    0.09720041
## 
## Residual Deviance: 9287.557 
## AIC: 9315.557

2b. (10 pts) Report the relative risk ratios (exponentiated coefficients) with 95% CIs in a clean table.

tidy(mod_multi, conf.int = TRUE, exponentiate = TRUE) |>
  dplyr::select(y.level, term, estimate, conf.low, conf.high, p.value) |>
  mutate(
    across(c(estimate, conf.low, conf.high), ~ round(.x, 3)),
    p.value = ifelse(p.value < 0.001, "<0.001", round(p.value, 3))
  ) |>
  kable(
    col.names = c("Outcome", "Predictor", "RRR", "Lower 95% CI", "Upper 95% CI", "p-value"),
    caption = "Multinomial Logistic Regression: Relative Risk Ratios"
  ) |>
  kable_styling(full_width = FALSE)

Multinomial Logistic Regression: Relative Risk Ratios
Outcome	Predictor	RRR	Lower 95% CI	Upper 95% CI	p-value
Good	(Intercept)	0.268	0.158	0.454	<0.001
Good	age	1.015	1.011	1.019	<0.001
Good	sexFemale	0.868	0.760	0.991	0.037
Good	bmi	1.052	1.040	1.064	<0.001
Good	exerciseYes	0.606	0.516	0.712	<0.001
Good	income_cat	0.844	0.816	0.874	<0.001
Good	smokerCurrent	1.510	1.297	1.756	<0.001
Fair/Poor	(Intercept)	0.277	0.145	0.529	<0.001
Fair/Poor	age	1.026	1.020	1.032	<0.001
Fair/Poor	sexFemale	0.882	0.745	1.043	0.143
Fair/Poor	bmi	1.070	1.056	1.084	<0.001
Fair/Poor	exerciseYes	0.262	0.219	0.314	<0.001
Fair/Poor	income_cat	0.675	0.648	0.703	<0.001
Fair/Poor	smokerCurrent	1.410	1.165	1.706	<0.001

2c. (5 pts) Interpret the RRR for one predictor in the “Fair/Poor vs. Excellent/VG” comparison.

#Holding all other variables constant, current smokers have 1.41 times higher relative risk of being in the Fair/Poor health category rather than the Excellent/Very Good category compared to former/never smokers (RRR = 1.41, 95% CI: 1.17–1.71, p < 0.001). This indicates that smoking is significantly associated with worse self-rated health. In other words, current smokers are more likely to report poorer general health relative to better health.

Task 3: Ordinal Logistic Regression (25 points)

3a. (5 pts) Fit an ordinal logistic regression model with the same predictors using polr().

mod_ord <- polr(
  genhlth_ord ~ age + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  Hess = TRUE
)

summary(mod_ord)

## Call:
## polr(formula = genhlth_ord ~ age + sex + bmi + exercise + income_cat + 
##     smoker, data = brfss_poly, Hess = TRUE)
## 
## Coefficients:
##                  Value Std. Error t value
## age            0.01797   0.001867   9.627
## sexFemale     -0.12885   0.056835  -2.267
## bmi            0.04973   0.004534  10.967
## exerciseYes   -0.92167   0.064426 -14.306
## income_cat    -0.26967   0.014113 -19.107
## smokerCurrent  0.29313   0.064304   4.558
## 
## Intercepts:
##                   Value    Std. Error t value 
## Excellent/VG|Good   0.0985   0.2200     0.4478
## Good|Fair/Poor      1.8728   0.2212     8.4650
## 
## Residual Deviance: 9318.274 
## AIC: 9334.274

3b. (5 pts) Report the cumulative ORs with 95% CIs.

tidy(mod_ord, conf.int = TRUE, exponentiate = TRUE) |>
  dplyr::filter(coef.type == "coefficient") |>
  dplyr::select(term, estimate, conf.low, conf.high) |>
  mutate(
    across(c(estimate, conf.low, conf.high), ~ round(.x, 3))
  ) |>
  kable(
    col.names = c("Predictor", "OR", "Lower 95% CI", "Upper 95% CI"),
    caption = "Ordinal Logistic Regression: Cumulative Odds Ratios"
  ) |>
  kable_styling(full_width = FALSE)

Ordinal Logistic Regression: Cumulative Odds Ratios
Predictor	OR	Lower 95% CI	Upper 95% CI
age	1.018	1.014	1.022
sexFemale	0.879	0.786	0.983
bmi	1.051	1.042	1.060
exerciseYes	0.398	0.351	0.451
income_cat	0.764	0.743	0.785
smokerCurrent	1.341	1.182	1.521

3c. (5 pts) Interpret one OR in plain language, making sure to mention the “at every cut-point” property. #Holding all other variables constant, current smokers have approximately 1.34 times higher odds of being in a worse general health category compared to former/never smokers (OR = 1.34, 95% CI: 1.18–1.52). Because this is an ordinal logistic regression model, this effect applies at every cut-point of the outcome, meaning it applies both to Excellent/Very Good versus Good/Fair-Poor and to Excellent/Very Good/Good versus Fair/Poor. This indicates that smoking is consistently associated with poorer health across all levels of the outcome.

3d. (10 pts) Use ggpredict() to plot predicted probabilities of each health category across a continuous predictor of your choice.

ggpredict(mod_ord, terms = "income_cat [1:8]") |>
  plot() +
  labs(
    title = "Predicted Probability of General Health by Income Category",
    x = "Income Category",
    y = "Predicted Probability",
    color = "General Health"
  ) +
  theme_minimal()

#The predicted probability plot shows a clear relationship between income and general health. As income increases, the probability of reporting Fair/Poor health decreases substantially, while the probability of reporting Excellent/Very Good health increases. The probability of being in the “Good” category rises slightly at middle income levels and then decreases at higher income levels. Overall, this pattern indicates a strong socioeconomic gradient, where higher income is associated with better self-reported health.

Task 4: Check Assumptions and Compare (15 points)

4a. (5 pts) Run the Brant test for proportional odds. Does the assumption hold?

brant(mod_ord)

## -------------------------------------------- 
## Test for X2  df  probability 
## -------------------------------------------- 
## Omnibus      35.87   6   0
## age      0.06    1   0.81
## sexFemale    0.89    1   0.34
## bmi      7.1 1   0.01
## exerciseYes  10.58   1   0
## income_cat   10.54   1   0
## smokerCurrent    7.76    1   0.01
## -------------------------------------------- 
## 
## H0: Parallel Regression Assumption holds

#The Brant test shows that the proportional odds assumption is violated overall, as the omnibus test is statistically significant (χ² = 35.87, df = 6, p < 0.001). This indicates that the relationship between predictors and the outcome is not consistent across all cut-points. In particular, BMI, exercise, income, and smoking show evidence of violating the assumption, while age and sex do not. Therefore, the proportional odds assumption does not fully hold for this model.

4b. (5 pts) Compare the AIC of the multinomial and ordinal models. Which fits better?

#The multinomial logistic regression model has a lower AIC compared to the ordinal logistic regression model, indicating that it provides a better fit to the data. Therefore, the multinomial model fits the data better than the ordinal model.

4c. (5 pts) Based on your results, which model would you recommend reporting? Justify in 2-3 sentences.

#I would recommend reporting the multinomial logistic regression model. Although the outcome variable is ordered, the Brant test indicates that the proportional odds assumption is violated, meaning the ordinal model may not adequately capture the relationships between predictors and general health. In addition, the multinomial model provides a better fit to the data and allows for more flexible estimation of predictor effects across outcome categories.

Completion credit (25 points): Awarded for a complete, good-faith attempt. Total: 75 + 25 = 100 points.

End of Lab Activity

Polytomous and Ordinal Logistic Regression

EPI 553 — Principles of Statistical Inference II

Muntasir Masum

April 16, 2026

Introduction

Why Not Just Run Multiple Binary Models?

Setup and Data

Building the Outcome: General Health

Part 1: Polytomous (Multinomial) Logistic Regression

The Model

Fitting in R: `nnet::multinom()`

Interpreting the Coefficients

Predicted Probabilities

Part 2: Ordinal Logistic Regression

The Proportional Odds (Cumulative Logit) Model

Fitting in R: `MASS::polr()`

Odds Ratios and Interpretation

Predicted Probabilities

Part 3: Checking the Proportional Odds Assumption

Visual Check: Empirical Logits

Formal Test: Brant Test

What If It Fails?

Part 4: Comparing the Two Models

Likelihood Comparison

Summary

Next Lecture (April 21)

References

Part 5: In-Class Lab Activity

Instructions

Data

Task 1: Explore the Outcome (15 points)

Task 2: Multinomial Logistic Regression (20 points)

Task 3: Ordinal Logistic Regression (25 points)

Task 4: Check Assumptions and Compare (15 points)

Polytomous and Ordinal Logistic Regression

EPI 553 — Principles of Statistical Inference II

Muntasir Masum

April 16, 2026

Introduction

Why Not Just Run Multiple Binary Models?

Setup and Data

Building the Outcome: General Health

Part 1: Polytomous (Multinomial) Logistic Regression

The Model

Fitting in R: nnet::multinom()

Interpreting the Coefficients

Predicted Probabilities

Part 2: Ordinal Logistic Regression

The Proportional Odds (Cumulative Logit) Model

Fitting in R: MASS::polr()

Odds Ratios and Interpretation

Predicted Probabilities

Part 3: Checking the Proportional Odds Assumption

Visual Check: Empirical Logits

Formal Test: Brant Test

What If It Fails?

Part 4: Comparing the Two Models

Likelihood Comparison

Summary

Next Lecture (April 21)

References

Part 5: In-Class Lab Activity

Instructions

Data

Task 1: Explore the Outcome (15 points)

Task 2: Multinomial Logistic Regression (20 points)

Task 3: Ordinal Logistic Regression (25 points)

Task 4: Check Assumptions and Compare (15 points)

Fitting in R: `nnet::multinom()`

Fitting in R: `MASS::polr()`