Generalized Linear Models
Exploratory Data Analysis For Epidemiology
Learning objectives for this lesson:
- Recognise when an outcome has more than two categories or is a count or a rate, and place multinomial, ordinal, Poisson, and negative binomial regression within the generalised linear model family alongside linear and logistic regression
- Select an appropriate model for a multi-category outcome (multinomial, proportional-odds, adjacent-category, or continuation-ratio) based on study objectives and data, fit each model, and evaluate the assumptions on which each is based
- Interpret odds ratio and relative risk ratio estimates from each multi-category model and compute predicted probabilities from them
- Distinguish among simple counts, rates with person-time denominators, population rates, and area-based counts, and describe the Poisson distribution and its mean = variance property
- Specify and interpret a Poisson regression model including the offset term, and interpret incidence rate ratios (IRR) from exponentiated Poisson coefficients
- Evaluate Poisson models using Pearson, deviance, and Anscombe residuals, and distinguish apparent from real overdispersion and apply appropriate corrections
- Compare negative binomial regression models (NB-1, NB-2) to Poisson regression, and apply zero-inflated, hurdle, and zero-truncated models to handle excess zeros
- Choose the right member of the GLM family for a given outcome by sketching the link function and the comparison it implies, and report effects with the interpretation that model warrants
This course was developed by Dr. Kiffer G. Card, Faculty of Health Sciences, Simon Fraser University based on Dohoo, I. R., Martin, S. W., & Stryhn, H. (2012). Methods in Epidemiologic Research. VER Inc.
Glossary: Key Terms, People & Concepts
📚 Reference page, available throughout the lesson
This glossary collects the key concepts, people, and ideas you will meet in this lesson. Use it as a reference while you work through the material, or as a review before assessments. Type in the search box to filter entries.
MASS::polr fits proportional-odds models; VGAM::vglm handles a broader family including partial-PO and continuation-ratio.
Introduction & Overview of Models
Introduction and Overview
An earlier lesson covered logistic regression for binary outcomes. This lesson extends the framework to outcomes with more than two categories, whether nominal (no natural order) or ordinal (ordered). The four content sections walk through the major models in order: an overview of the four-model toolkit (this section), the multinomial logistic model for nominal outcomes (a later section), the proportional-odds model for ordinal outcomes including how to test the proportional-odds assumption (a later section), and finally adjacent-category and continuation-ratio models as alternatives when proportional-odds doesn't hold (a later section). For an epidemiology-focused overview that maps all four models onto the same dataset, see Ananth & Kleinbaum (1997); the open-access encyclopedia entries on ordinal regression and multinomial logistic regression summarise the same toolkit at an undergraduate level.
Learning Objectives
- Distinguish nominal from ordinal outcomes and explain why each calls for a different modelling strategy.
- Map the four logits used by multinomial, proportional-odds, adjacent-category, and continuation-ratio models.
- Use the Apgar-score example to anticipate how each model will partition the same outcome.
- Choose an initial model based on the structure and ordering of the outcome categories.
When Outcomes Have More Than Two Categories
In many epidemiological studies the outcome variable has more than two categories. These outcomes fall into two broad types: nominal data, where the categories have no natural ordering (e.g., type of disease, preferred clinic), and ordinal data, where the categories are ordered (e.g., pain severity: none, mild, moderate, severe).
The choice of model depends on whether the outcome is nominal or ordinal. Nominal data require multinomial logistic regression or log-linear models. Ordinal data can be analysed with the same multinomial model (ignoring the ordering), but more efficient approaches exploit the ordering: proportional-odds, adjacent-category, and continuation-ratio models (McCullagh, 1980; Ananth & Kleinbaum, 1997).
The Apgar Score Example
Throughout Chapter 17, the authors use Apgar scores as a running example. Apgar scores (measured at birth) are recoded into four ordinal categories. The research question is whether the number of prenatal visits is associated with Apgar score category.
| Apgar Category | Code | Prenatal Visits < 6 | Prenatal Visits ≥ 6 | Total |
|---|---|---|---|---|
| 1–6 (Low) | 0 | 47 | 25 | 72 |
| 7 | 1 | 48 | 42 | 90 |
| 8 | 2 | 59 | 72 | 131 |
| 9–10 (High) | 3 | 134 | 227 | 361 |
| Total | 288 | 366 | 654 |
Overview of the Four Models
Each of the four models for multi-category outcomes uses a different formulation of the logit (log-odds). Understanding the logit structure is the key to understanding each model.
Compares each outcome category to a baseline category. For J categories, the model estimates J−1 sets of coefficients. Each set describes how predictors relate to the log-odds of being in category j versus the baseline.
No assumptions about ordering are made, so this model is appropriate for both nominal and ordinal outcomes (though it is less efficient for ordinal data).
Based on cumulative probabilities. The logit compares the probability of being at or above category j versus below it. A single coefficient per predictor applies at every cutpoint, the proportional-odds assumption.
This is the most common ordinal logistic model and is more parsimonious than the multinomial model (McCullagh, 1980).
Compares each category to the adjacent (next lower) category. This model is a constrained version of the multinomial model where the coefficient for categories n levels apart equals n times the coefficient for adjacent categories.
Like the proportional-odds model, it estimates a single β1 per predictor.
Compares each category to all lower categories combined. This model is especially appropriate when the outcome represents sequential stages that must be “passed through” to reach higher levels (e.g., number of attempts to achieve certification).
Can be fit as a series of separate binary logistic regressions with appropriately recoded outcome variables.
1. Nominal outcome data differ from ordinal outcome data in that:
2. How many sets of coefficients does a multinomial logistic model estimate for J outcome categories?
3. Which model assumes the effect of a predictor is the same across all cutpoints?
✎ Reflection
Think of an ordinal outcome variable from your own field of study. What are the categories, and which of the four models introduced here do you think would be most appropriate? Why?
Multinomial Logistic Regression
Introduction and Overview
An earlier section mapped the four-model landscape. This section walks into the first one in detail: the multinomial logistic model. This is the most general option: it works on any categorical outcome, ordered or not, but you pay for that generality with a coefficient for every comparison and many more parameters to interpret.
Learning Objectives
- Set up a multinomial logistic model as J−1 simultaneous binary logits against a chosen baseline.
- Interpret exponentiated coefficients as relative risk ratios for non-baseline versus baseline categories.
- Compute predicted probabilities for every outcome category from the joint set of linear predictors.
- Recognise the independence-of-irrelevant-alternatives (IIA) assumption and its implications.
The Multinomial Logistic Model
The multinomial logistic model simultaneously fits J−1 separate logistic models, each comparing one category to a chosen baseline. All parameters are estimated jointly, so the model accounts for the correlation among the comparisons.
Predicted Probabilities
The predicted probability for each outcome category is computed from the set of linear predictors. Let Xβ(j) denote the linear predictor for category j.
These probabilities always sum to 1 across all categories. Each predicted probability depends on all sets of coefficients, rather than only the coefficients for that category.
These formulas write category 0 as the baseline for illustration. In the Apgar results below, the high category (9–10, coded 3) is used as the reference instead. Which category you call the baseline is a labelling choice: it re-expresses the coefficients but leaves the fitted probabilities and the overall fit unchanged.
Interpreting Odds Ratios
Exponentiated coefficients from a multinomial model are technically ratios of relative risks (RRR), not true odds ratios. Each exp(β(j)) gives the ratio of the probability of being in category j relative to the baseline, for a one-unit change in the predictor.
Read intuitively, a relative-risk ratio of 0.24 means that for mothers with six or more prenatal visits, the risk of the low category measured against the high baseline category is about a quarter of what it is for mothers with fewer visits. The name relative-risk ratio, rather than odds ratio, is a reminder that the comparison is built from probabilities taken against the baseline, not from odds.
Because the multinomial model estimates separate coefficients for each comparison, the effects can differ across categories. For ordinal outcomes, you would typically expect a gradient, with more pronounced effects for the categories furthest from the baseline.
Predicted Probabilities
Predicted probabilities from a multinomial model vary by the values of all predictors. To communicate results, it is often useful to compute predicted probabilities at specific covariate patterns (e.g., prenatal visits < 6 vs ≥ 6) and present them in a table or graph.
Testing significance can be done with Wald tests (for individual coefficients) or likelihood-ratio tests (for overall effects). Because the multinomial model has J−1 coefficients per predictor, an overall LRT that tests all J−1 simultaneously is generally preferred over examining individual coefficients.
Independence of Irrelevant Alternatives (IIA)
The multinomial logistic model assumes IIA: the odds of choosing one category over another are independent of what other categories are available. If this assumption is violated, adding or removing a category would change the odds between the remaining categories.
Two tests are available: the Hausman-McFadden test and the Small-Hsiao test. However, these tests often give conflicting results, and IIA violations are primarily a concern for nominal data where alternatives are genuinely substitutable (e.g., choosing a mode of transport). For ordinal data, the independence of irrelevant alternatives is rarely a practical concern.
In the Apgar score example, the multinomial model (with category 3 [9–10] as baseline) produced the following key results for prenatal visits (≥6 vs <6):
- Category 0 vs 3: OR = 0.24, so those with ≥6 visits have 76% lower relative risk of a low Apgar score
- Category 1 vs 3: OR = 0.65, a 35% lower relative risk of Apgar = 7
- Category 2 vs 3: OR = 0.72, a 28% lower relative risk of Apgar = 8
The gradient (0.24 → 0.65 → 0.72) shows the strongest effect for the lowest Apgar category, as expected for an ordinal outcome.
Note on IIA Tests
The Hausman-McFadden and Small-Hsiao tests for IIA often give conflicting results and are not always reliable. In practice, IIA is mainly a concern for nominal (unordered) outcomes with genuinely substitutable alternatives. For ordinal outcomes, it is rarely problematic. Regression diagnostics can be performed by fitting ordinary logistic models for pairs of categories and using standard diagnostic techniques.
Alternative-Specific Data
In some situations, predictors may vary across alternatives rather than (or in addition to) varying across observations. For example, in a study of clinic choice, the distance to each clinic varies by alternative. Special formulations of the multinomial model (conditional logit or mixed logit) accommodate such alternative-specific data.
1. In multinomial logistic regression, the exponentiated coefficients represent:
2. The IIA assumption states that:
3. How are predicted probabilities computed from a multinomial model?
✎ Reflection
Consider the Apgar score example. Why do you think the OR for the lowest Apgar category (0.24) is more extreme than for the middle categories? What does this gradient tell us about prenatal care and birth outcomes?
Proportional-Odds Model
Introduction and Overview
An earlier section used multinomial logistic regression to fit a model that ignores any ordering in the outcome categories. This section takes the more parsimonious route: when the categories are ordered, the proportional-odds model uses far fewer parameters by assuming the effect of each predictor is the same across all category cut-points. The trade-off is that you have to verify the proportional-odds assumption holds.
Learning Objectives
- Express the proportional-odds (cumulative logit) model in terms of an underlying latent continuous variable and cutpoints.
- Interpret a single odds ratio as the effect of a predictor across every dichotomisation of the ordinal outcome.
- Test the proportional-odds assumption using the score (Brant) test or by comparing nested models.
- Diagnose what to do when proportional-odds clearly fails for a given predictor.
The Most Common Ordinal Model
The proportional-odds model (also called the cumulative logit model or ordinal logistic regression) is the most widely used model for ordinal outcomes. It was formalised by McCullagh (1980) and is based on the idea of an underlying continuous latent variable that is divided into the observed ordinal categories by a series of cutpoints, the structure of the ordered logit model.
The latent variable Si is divided by cutpoints (τ1, τ2, …, τJ−1) into J observed categories. If Si falls between τj−1 and τj, the observation is classified into category j.
The Proportional-Odds Logit
The model takes the form of a cumulative logit: logit(p(Y ≥ j)) = β0j + βX. The key feature is that the intercept varies across cutpoints (giving parallel lines on a logit scale) but the slope coefficients are the same for every cutpoint. This means a single OR summarises the effect of each predictor across all levels of the outcome (McCullagh, 1980).
Proportional odds in plain words. The predictor multiplies the odds by the same amount wherever you split the ordered outcome. If six or more prenatal visits multiply the odds of scoring above the lowest Apgar category by 1.59, they multiply the odds of reaching the top category by that same 1.59. You get a single odds ratio, and it applies at every cutpoint. That economy is the reason to reach for the model, and the proportional-odds assumption is the one thing you have to check before trusting it.
In the Apgar score example, the proportional-odds model yields an OR of 1.59 for prenatal visits (≥6 vs <6). This means that individuals with 6 or more prenatal visits have 1.59 times the odds of being at or above any given Apgar category, compared to those with fewer visits. This single OR applies at every cutpoint (0 vs 1+, 0–1 vs 2+, and 0–2 vs 3).
The course dataset phaa_survey_clean.csv has both an unordered multi-category variable (region, 5 Lower-Mainland regions) and an ordered one (education, 5 levels). The full annotated script is in r-activities/HSCI_410_Lesson_6_Ordinal_and_Multinomial_Models.R.
library(nnet); library(MASS); library(brant)
phaa <- read.csv("phaa_survey_clean.csv", stringsAsFactors = FALSE)
phaa$region <- relevel(factor(phaa$region), ref = "Vancouver")
phaa$education <- factor(phaa$education,
levels = c("Less than high school", "High school", "Some college",
"Bachelor's", "Graduate degree"),
ordered = TRUE)
# 1. MULTINOMIAL: nominal outcome (region)
mn <- multinom(region ~ age + gender + smoker, data = phaa, trace = FALSE)
exp(coef(mn)) # relative-risk ratios
z <- summary(mn)$coefficients / summary(mn)$standard.errors
round((1 - pnorm(abs(z))) * 2, 3) # p-values
# 2. PROPORTIONAL-ODDS: ordered outcome (education)
po <- polr(education ~ age + gender + smoker, data = phaa, Hess = TRUE)
exp(cbind(OR = coef(po), confint(po))) # OR + 95% CI
# 3. Brant test: is the proportional-odds assumption defensible?
brant(po)
# 4. Multinomial fit of the SAME ordered outcome, the fall-back to compare
mn_edu <- multinom(education ~ age + gender + smoker, data = phaa, trace = FALSE)
AIC(po, mn_edu) # lower AIC = better-fitting modelReading the multinomial output. The relative-risk ratios from multinom() compare each region to the reference (Vancouver). For polr(), a single OR per predictor applies between every adjacent pair of education levels; that is the proportional-odds assumption that brant() tests. If the Brant overall p-value is < .05, compare AIC(po, mn_edu) and prefer the multinomial fit when its AIC is clearly lower (or fit a partial-PO model in the stretch).
R Reflect on what you just ran
Use the questions below to interpret the output you produced. Look at your console / plot before answering.
1. From exp(coef(mn)) and the matching p-value matrix, pick one non-reference region (anything but Vancouver) and one predictor (e.g., smokerYes). Report the relative-risk ratio, and state in one sentence what it says about the relative likelihood of living in that region versus Vancouver.
2. From exp(cbind(OR = coef(po), confint(po))), report the OR for age and its 95% CI. Under the proportional-odds assumption, translate this OR into a sentence about being at-or-above any given education level.
3. Look at the brant(po) output. What is the overall p-value, and do any individual predictors flag a violation (p < .05)? Based on this, would you keep po or fall back to mn_edu (compare AICs)?
brant() typically returns an overall p-value around 0.01–0.05; if the global is significant, some individual predictors will also show p < .05, indicating violation of the proportional-odds assumption. If violations are flagged: (a) compare AIC of po vs. mn_edu (the multinomial alternative); (b) if mn_edu has substantially lower AIC, prefer the multinomial; (c) alternatively, fit a partial proportional odds model that relaxes the assumption only for the violating predictors. The cleanest practical move when the assumption is violated and the categories are ordered is the partial proportional-odds model; its category-specific slopes let the flagged predictors act differently at each cutpoint while the outcome's ordering is preserved.Testing the Proportional-Odds Assumption
The proportional-odds assumption is that the effect of each predictor is the same at every cutpoint. If violated, the model may give misleading results. Several tests are available:
Three main approaches exist:
- Approximate LRT: Compare the log-likelihoods of the proportional-odds model and the multinomial model. A significant difference suggests the proportional-odds assumption is violated.
- Wolfe-Gould approximate LRT: Based on J−1 separate binary logistic models at each cutpoint. Sum the log-likelihoods and compare to the proportional-odds model.
- Brant (Wald) test: Provides both an overall test and individual tests for each predictor, showing which specific variables violate the assumption (Brant, 1990).
If the proportional-odds assumption is violated, a generalised ordinal logistic regression model allows separate coefficients at each cutpoint. This model is equivalent to fitting J−1 separate binary logistic regressions simultaneously. It is more flexible but less parsimonious than the proportional-odds model (Williams, 2006).
A compromise approach is the partial proportional-odds model, which relaxes the proportional-odds assumption for selected predictors only (those that fail the Brant test) while maintaining it for the rest (Peterson & Harrell, 1990; Williams, 2006). This provides a good balance between flexibility and parsimony. Other alternatives include the stereotype logistic model and the heterogeneous choice logistic model.
⚠ The Proportional-Odds Assumption in Practice
The proportional-odds assumption is often violated in practice, especially with many predictors or when the outcome categories represent very different phenomena. Always test this assumption before reporting results from a proportional-odds model (Brant, 1990). If violated, consider a partial proportional-odds model or generalised ordinal logistic regression (Peterson & Harrell, 1990; Williams, 2006).
Brant Test Results Example
The Brant test provides both an overall test and predictor-specific tests. Here is an example of how results might be presented:
| Predictor | χ² | df | P-value | Assumption Holds? |
|---|---|---|---|---|
| Prenatal visits | 2.14 | 2 | 0.343 | Yes |
| Maternal age | 8.92 | 2 | 0.012 | No |
| Parity | 1.03 | 2 | 0.598 | Yes |
| Overall | 12.45 | 6 | 0.053 | Borderline |
In this example, only maternal age violates the assumption. A partial proportional-odds model that allows maternal age to have different effects at each cutpoint (while constraining prenatal visits and parity) would be appropriate.
Regression Diagnostics
Regression diagnostics for the proportional-odds model can be conducted by fitting binary logistic models at each cutpoint and applying the diagnostic techniques from Chapter 16 (residual analysis, influence measures, goodness-of-fit tests).
1. The proportional-odds model assumes:
2. If the proportional-odds assumption is violated for some but not all predictors, which model can be used?
3. The latent variable in a proportional-odds model represents:
✎ Reflection
Why do you think the proportional-odds assumption is so often violated in practice? Can you think of a scenario in your own research where you would expect the effect of a predictor to differ across cutpoints?
Adjacent-Category & Continuation-Ratio Models
Introduction and Overview
The proportional-odds model is the workhorse for ordered outcomes, but its proportional-odds assumption is genuinely restrictive and frequently fails in real data. This section closes the lesson with two alternatives that relax that assumption in different ways: the adjacent-category model and the continuation-ratio model. Each is appropriate for a different kind of ordering and a different research question.
Learning Objectives
- Specify the adjacent-category model and test its constraint against an unconstrained multinomial fit.
- Specify the continuation-ratio model and recognise the sequential-stage outcomes it suits.
- Fit a continuation-ratio model as a series of binary logistic regressions on recoded data.
- Choose between proportional-odds, adjacent-category, and continuation-ratio models based on the question and the data.
Adjacent-Category Model
The adjacent-category model compares the probability of being in category j versus category j−1 (the next lower category). It is a constrained version of the multinomial logistic model: the constraint is that the coefficient for categories n levels apart equals n times the coefficient for adjacent categories.
Like the proportional-odds model, the adjacent-category model estimates a single β1 per predictor, making it more parsimonious than the unconstrained multinomial model. The validity of this constraint can be tested by comparing the adjacent-category model to the unconstrained multinomial model using a likelihood-ratio test (LRT) (Ananth & Kleinbaum, 1997).
For the Apgar score data, the LRT comparing the adjacent-category model to the unconstrained multinomial model yielded χ² = 6.76, df = 5, P = 0.239. Since this is not significant, the adjacent-category model is a valid simplification of the multinomial model for these data.
Continuation-Ratio Model
The continuation-ratio model compares the probability of being in category j versus all lower categories combined. It is particularly useful when the outcome represents sequential stages that must be “passed through” to reach higher levels (Ananth & Kleinbaum, 1997).
When Is the Continuation-Ratio Model Appropriate?
The continuation-ratio model is ideal for outcomes where each level must be reached before the next can be attained. Examples include: number of attempts to pass an exam, stages of disease progression where remission must occur before relapse, or sequential rounds of a selection process. It is NOT appropriate when movements between categories are not sequential (e.g., Apgar scores, where a baby does not “pass through” each score level).
Fitting the Continuation-Ratio Model
The continuation-ratio model can be fit as a series of separate binary logistic regressions with a recoded outcome variable. For each comparison:
- Y = 1 for the level of interest
- Y = 0 for all lower levels
- Observations at higher levels are excluded (treated as missing)
Consider an example with 4 categories representing the number of attempts to gain admission to medical school (1, 2, 3, 4+):
| Original Category | Y1 (1 vs 0) | Y2 (2 vs 0–1) | Y3 (3 vs 0–2) |
|---|---|---|---|
| 0 (1 attempt) | 0 | 0 | 0 |
| 1 (2 attempts) | 1 | 0 | 0 |
| 2 (3 attempts) | – | 1 | 0 |
| 3 (4+ attempts) | – | – | 1 |
You can fit either a constrained version (equal ORs across levels, tested by LRT) or an unconstrained version (separate ORs at each level). The constrained version is more parsimonious and can be compared to the unconstrained version using a likelihood-ratio test.
The adjacent-category model is appropriate when the comparison of interest is between neighbouring categories of an ordinal outcome. It is a natural choice when you believe the effect of a predictor operates by shifting individuals one category at a time. The model can be validated by comparing it to the unconstrained multinomial model via LRT.
The continuation-ratio model is most appropriate when the outcome represents sequential stages that must be passed through in order. Each category must be reached before the next can be attained. Examples include: successive attempts at an exam, sequential rounds of treatment, or stages of career advancement. If categories can be reached without passing through lower levels, this model is not appropriate.
When one model is a constrained (nested) version of another, the likelihood-ratio test can be used to compare them. The test statistic is −2(lnLconstrained − lnLunconstrained), which follows a χ² distribution with degrees of freedom equal to the difference in the number of parameters. A significant result suggests the constraint is not valid and the more complex model is needed.
Decision Guide: Choosing Among the Four Models
Step 1: Is the outcome nominal or ordinal? If nominal, use multinomial logistic regression.
Step 2: If ordinal, does the outcome represent sequential stages? If yes, consider the continuation-ratio model.
Step 3: If not sequential, fit the proportional-odds model and test the assumption. If it holds, use proportional-odds.
Step 4: If the proportional-odds assumption fails, consider the adjacent-category model, partial proportional-odds, or generalised ordinal logistic regression.
Step 5: Compare nested models using LRT to select the most parsimonious adequate model.
1. In the adjacent-category model, the coefficient for categories n levels apart is:
2. The continuation-ratio model is most appropriate when:
3. If an LRT comparing the adjacent-category model to the multinomial model is NOT significant, this suggests:
✎ Reflection
Can you think of an example from public health or epidemiology where a continuation-ratio model would be more appropriate than a proportional-odds model? What makes the outcome sequential in your example?
Introduction & The Poisson Distribution
Introduction and Overview
Earlier lessons covered regression for continuous, binary, and ordered/multi-category outcomes. This lesson takes on the next major outcome type: counts. Number of disease cases, hospital admissions, parasites, or events per unit time are all counts, and they need their own GLM family (Nelder & Wedderburn, 1972). The four content sections walk through this in order: the Poisson distribution and why it's the natural starting point (this section), Poisson regression with the offset term that lets us model rates as well as counts (a later section), residuals and the most consequential complication, overdispersion (a later section), and finally negative binomial and zero-adjusted models when standard Poisson breaks down (a later section). The offset term is the key conceptual bridge to the rate-based cohort designs you met in earlier courses and lessons.
Learning Objectives
- Distinguish simple counts, person-time rates, population rates, and area-based counts in epidemiological data.
- State the assumptions of the Poisson distribution, including the variance-equals-mean property.
- Compute Poisson probabilities and recognise the shape of the distribution at small versus large means.
- Identify when count data are likely to violate Poisson assumptions and why it matters.
Why Model Count and Rate Data?
Many outcomes in epidemiology are measured as counts, such as the number of disease cases in a region, the number of doctor visits per year, or the number of parasites on a host animal. These outcomes differ fundamentally from continuous outcomes (modelled with linear regression) and binary outcomes (modelled with logistic regression). Count data require their own family of statistical models because they are discrete, non-negative, and often right-skewed (Coxe, West, & Aiken, 2009).
Chapter 18 introduces the statistical tools for modelling count and rate data, beginning with the Poisson distribution and Poisson regression, and extending to negative binomial and zero-adjusted models for situations where the basic Poisson assumptions are violated.
Types of Count and Rate Data
Before selecting an analytical approach, it is essential to understand which type of count or rate data you are working with. There are four main types encountered in epidemiological research:
The Poisson Distribution
The Poisson distribution is the foundational probability distribution for modelling count data. It describes the probability of observing a given number of events in a fixed interval of time or space, assuming events occur independently at a constant average rate. It is the foundational distribution for the Poisson model.
From raindrops to per-minute counts to the Poisson PMF. Next ▶ advances scenes.
A 6-scene visualization of count and rate data: random events at constant rate λ, per-minute counting, the histogram filling, the Poisson PMF emerging, and the bridge to rates and rate ratios.
In this formula, Y is the count of events, μ (mu) is both the mean and the expected number of events, and e is the base of the natural logarithm (≈ 2.718). The factorial in the denominator (y!) ensures the probabilities are correctly normalised. The Poisson distribution is defined for non-negative integers: y = 0, 1, 2, 3, …
Key Property: Mean = Variance
The defining property of the Poisson distribution is that the mean equals the variance: E(Y) = Var(Y) = μ (Coxe et al., 2009). This single-parameter property means that as the expected count increases, so does the variability. This assumption is central to Poisson regression, and when it is violated (variance > mean), we have overdispersion, which requires alternative approaches covered in later sections.
The Poisson distribution is particularly useful for modelling rare events in large populations. When the probability of an event is small and the number of trials (or opportunities) is large, the Poisson distribution provides an excellent approximation. Examples include the number of rare disease cases in a large population, or the number of equipment failures over an extended operating period.
The Poisson distribution is appropriate when: (1) events are independent of one another; (2) the rate at which events occur is constant over the observation period; (3) two events cannot occur at exactly the same instant; and (4) the probability of an event in a short interval is proportional to the length of the interval. In practice, these assumptions are often approximately met in epidemiological settings.
When μ is small (e.g., μ < 3), the distribution is noticeably right-skewed, with most observations clustering near zero and a long right tail. As μ increases, the distribution becomes more symmetric and begins to resemble a normal distribution. By the time μ ≥ 20, a normal approximation with mean μ and variance μ is often adequate.
1. Which type of count data uses person-time in the denominator?
2. What is the key property of the Poisson distribution?
3. The Poisson distribution is most appropriate for modelling:
Reflection
How might the type of count data (simple counts vs. rates) influence your choice of analytical approach in an epidemiological study you're familiar with?
offset(log(person_years))) is required for a chronic-disease registry where each patient contributes different person-time. Confusing the two leads to substantial bias: treating rate data as counts ignores the denominator, while treating count data as rates over-corrects for non-existent variation in exposure time.Poisson Regression Model & Interpretation
Introduction and Overview
An earlier section introduced the Poisson distribution as the probability model behind count data. This section turns the distribution into a regression. The log-linear formulation (Nelder & Wedderburn, 1972) lets us link counts to predictors on a multiplicative scale, and the offset term is the trick that converts a count model into a rate model, precisely what you need to handle person-time denominators from the rate-based cohort designs you saw in earlier courses and lessons.
Learning Objectives
- Write down a Poisson regression model with a log link and identify its linear predictor.
- Use an offset term to convert a count model into a rate model with person-time denominators.
- Interpret exponentiated Poisson coefficients as multiplicative rate ratios.
- Translate a fitted Poisson regression into incidence rates and predicted counts for new covariate values.
The Expected Count
The starting point for Poisson regression is the relationship between the expected number of events and the underlying rate. If an individual (or group) is observed for n units of person-time and the event rate is λ (lambda), the expected count is:
Here, n represents the person-time at risk (e.g., person-years of follow-up) and λ is the incidence rate. The expected count is simply the product of the time at risk and the rate at which events occur. Different subjects may contribute different amounts of person-time, which must be accounted for in the model.
The Log-Linear Model
Poisson regression uses a log link function to relate the expected count to a linear combination of predictors. Taking the natural logarithm of both sides of the expected count equation and incorporating predictors gives us the Poisson regression model:
The term ln(n) is the offset, a fixed term in the model that is included, rather than estimated, to account for the fact that different observations may have different amounts of exposure (person-time). The β coefficients describe how the log of the expected count (or rate) changes with the predictors. Fixing that coefficient at exactly 1 encodes a simple idea: a subject followed for twice as long is expected to accumulate about twice as many events, so the offset simply carries the denominator of the rate rather than being a quantity the data must estimate.
The Offset Term
The offset is one of the most important concepts in Poisson regression. It transforms the model from one that predicts counts to one that effectively predicts rates (Coxe et al., 2009).
Modelling Counts (Without Offset)
When no offset is included, the model predicts the expected count directly:
ln(E(Y)) = β0 + β1X1 + …
This is appropriate when all observations have the same amount of exposure or follow-up time. For instance, if all herds are observed for exactly one year, the count of disease cases directly reflects the rate. In practice, this situation is relatively uncommon, since most epidemiological studies have subjects with varying follow-up times.
Modelling Rates (With Offset)
When the offset ln(n) is included, the model effectively predicts the rate rather than the raw count:
ln(E(Y)) = ln(n) + β0 + β1X1 + …
This is equivalent to modelling ln(E(Y)/n) = β0 + β1X1 + …, where E(Y)/n is the expected rate. The offset accounts for the fact that subjects with longer follow-up times are expected to accumulate more events simply by virtue of being observed longer. This is the standard approach when follow-up times vary across subjects.
Interpreting Poisson Regression Coefficients
In Poisson regression, the exponentiated coefficient eβ is interpreted as an incidence rate ratio (IRR), consistent with the wider generalised linear model framework. This is analogous to the odds ratio in logistic regression but applies to rates rather than odds.
Incidence Rate Ratio (IRR)
For a one-unit increase in the predictor X1, the incidence rate is multiplied by eβ1. If β1 = 0.30, then IRR = e0.30 = 1.35, meaning the rate increases by 35% for each one-unit increase in X1. An IRR > 1 indicates an increased rate; an IRR < 1 indicates a decreased rate; and an IRR = 1 indicates no association.
Suppose we model the number of mastitis cases per herd over one year, with herd size as a predictor and cow-years at risk as the offset. The Poisson regression yields βherd size = 0.012.
Interpretation: e0.012 = 1.012, so for each additional cow in the herd, the incidence rate of mastitis increases by 1.2%. A herd with 100 more cows would have an expected rate ratio of e0.012×100 = e1.2 = 3.32 compared to the baseline, a 3.32-fold higher mastitis rate. Notice that the scaling is done on the coefficient scale first: multiply the coefficient by 100, then exponentiate. Multiplying the rate ratio itself by 100 (1.012 times 100) would be wrong, because rate ratios compound by multiplication rather than adding up.
The companion dataset phaa_followup.csv records how many GP visits each participant had during their follow-up. Because follow-up time varies, we need an offset of log(fu_years) to turn the count into a rate. The full annotated script is in r-activities/HSCI_410_Lesson_7_Count_and_Rate_Data.R.
library(MASS); library(AER)
phaa <- read.csv("phaa_followup.csv", stringsAsFactors = FALSE)
phaa$smoker <- factor(phaa$smoker, levels = c("No","Yes"))
# 1. A peek at the count outcome
summary(phaa$gp_visits)
hist(phaa$gp_visits, breaks = 30,
main = "GP visits during follow-up", xlab = "Visits")
# 2. Poisson regression with offset to model the RATE per person-year
fit_rate <- glm(gp_visits ~ age + smoker + hypertension
+ offset(log(fu_years)),
family = poisson, data = phaa)
summary(fit_rate)
exp(coef(fit_rate)) # incidence-rate ratios
exp(confint(fit_rate))
# 3. Goodness of fit: Pearson chi^2 / df ~ 1 = good
sum(residuals(fit_rate, type = "pearson")^2) / fit_rate$df.residual
# 4. Formal overdispersion test
dispersiontest(fit_rate)
# 5. Negative binomial fall-back when Poisson is overdispersed
fit_nb <- glm.nb(gp_visits ~ age + smoker + hypertension
+ offset(log(fu_years)), data = phaa)
AIC(fit_rate, fit_nb)
cbind(Poisson = exp(coef(fit_rate)),
NegBin = exp(coef(fit_nb)))Why the offset isn't a predictor. Because its coefficient is fixed at 1, offset(log(fu_years)) shifts the intercept onto the rate scale without using a degree of freedom. The IRR for smokerYes tells you the multiplicative effect of smoking on the visit rate, holding age and hypertension constant. If dispersiontest() is significant, prefer glm.nb(); CIs widen but the point estimates are usually similar.
R Reflect on what you just ran
Use the questions below to interpret the output you produced. Look at your console / plot before answering.
1. From exp(coef(fit_rate)) and exp(confint(fit_rate)), report the IRR (and 95% CI) for smokerYes. Translate it into one sentence: how does the rate of GP visits per person-year differ between smokers and non-smokers?
exp(coef(fit_rate)) typically returns an IRR for smokerYes of about 1.40, 95% CI roughly (1.25, 1.55). Interpretation: smokers have an incidence rate of GP visits per person-year about 40% higher than non-smokers, holding other covariates constant. The CI excludes 1, so the effect is statistically significant.2. Compute sum(residuals(fit_rate, type = "pearson")^2) / fit_rate$df.residual. Is it close to 1, or much larger? Also report the p-value from dispersiontest(fit_rate). What do those two pieces of evidence say about overdispersion?
sum(residuals(fit_rate, type="pearson")^2) / fit_rate$df.residual typically returns a value around 1.4–2.0, meaningfully larger than 1, suggesting overdispersion. dispersiontest(fit_rate) gives a small p-value (typically < 0.01), confirming statistically significant overdispersion. Both pieces of evidence point to the same conclusion: the Poisson assumption (variance = mean) is violated, and the standard errors from Poisson are likely too small (CIs too narrow, p-values inflated).3. From AIC(fit_rate, fit_nb) and the side-by-side IRR comparison, which model do you prefer? Are the point estimates similar between Poisson and NegBin? What typically changes when you switch (point estimates, CIs, or both)?
AIC(fit_rate, fit_nb) typically shows AIC much lower for the negative binomial model (often hundreds of points lower), strongly favouring NegBin. The point estimates for IRRs are very similar between Poisson and NegBin, which is expected because both are estimating the conditional mean structure. What changes are the standard errors: NegBin SEs are larger (because they account for the extra variance from the over-dispersion parameter), so the CIs widen and p-values become less extreme. The lesson: overdispersion doesn't bias the point estimates much, but mis-specifying it gives over-confident inference.Poisson Regression for Relative Risk Estimation
An important application of Poisson regression is estimating relative risks (RR) directly from binary outcome data. When the outcome is rare, the Poisson model can provide estimates of the RR that are more interpretable than the odds ratios from logistic regression. This approach typically uses robust (sandwich) standard errors to account for the fact that binary data do not truly follow a Poisson distribution.
1. What is the purpose of the offset term in Poisson regression?
2. The exponentiated Poisson regression coefficient (eβ) is interpreted as:
3. Using Poisson regression to estimate relative risks from binary data is appropriate when:
Reflection
Consider a study where participants have very different follow-up times. How would using an offset term change your interpretation compared to simply modelling raw counts?
offset(log(person_years)) in a Poisson model fixes the coefficient on log(person-years) at exactly 1, effectively modelling the rate (events per person-year) rather than the raw count. Without an offset, the model would estimate a coefficient on log(person-years) that absorbs the relationship between exposure time and outcome, leaving it biased and not interpretable. With the offset, the IRRs read as multipliers on the rate (smokers have 1.4-fold higher rate of events per unit time), which is the unit of inference most epidemiologic questions want. Comparing to a count model without offset: a participant with 5 events in 1 year is the same as one with 5 events in 5 years, which is obviously wrong; the offset corrects this.Evaluating Poisson Models & Overdispersion
Introduction and Overview
An earlier section fit the Poisson model. This section turns to evaluating it. Poisson regression makes a strong assumption, that the variance equals the mean, which real count data frequently violate (Ver Hoef & Boveng, 2007). Overdispersion is the most common diagnosis you'll make on a count model, and addressing it is what a later section will be about. First, though, you need the residuals and goodness-of-fit tools to detect it.
Learning Objectives
- Compute and interpret Pearson, deviance, and Anscombe residuals from a Poisson model.
- Apply the deviance and Pearson chi-squared statistics as overall goodness-of-fit tests.
- Define overdispersion in terms of the variance-to-mean ratio and explain why it inflates Type I error.
- Choose between quasi-Poisson, scale-corrected, and negative binomial responses to overdispersion.
Residuals for Poisson Models
Just as in linear regression, residuals are the primary tool for evaluating how well a Poisson model fits the observed data. However, because the variance of a Poisson variable depends on its mean, raw residuals (observed − expected) are not directly comparable across observations. Several types of standardised residuals have been developed:
Pearson residuals standardise the raw residual by dividing by the square root of the expected value:
This accounts for the Poisson assumption that Var(Y) = μ. If the model fits well, Pearson residuals should have approximately mean 0 and variance 1. The sum of squared Pearson residuals follows an approximate χ² distribution and can be used as an overall goodness-of-fit test.
Deviance residuals are based on the contribution of each observation to the overall model deviance (the log-likelihood ratio comparing the fitted model to a saturated model). They are defined as:
di = sign(yi − μ̂i) × √[2(yi ln(yi/μ̂i) − (yi − μ̂i))]
Deviance residuals tend to be more normally distributed than Pearson residuals, especially when some expected counts are small. This makes them preferable for normal probability plots and other diagnostic displays.
Anscombe residuals use a transformation of the observed counts designed to make the residuals as close to normally distributed as possible. They apply a cube-root transformation to both the observed and expected values. Anscombe residuals are particularly useful when checking the normality assumption of residuals in Poisson models, and they complement Pearson and deviance residuals in a thorough model evaluation.
Goodness of Fit
The overall fit of a Poisson model can be assessed using the sum of squared Pearson residuals, which approximately follows a χ² distribution with (n − p) degrees of freedom, where n is the number of observations and p is the number of estimated parameters. A significant test statistic suggests the model does not fit the data adequately.
An important diagnostic is the dispersion parameter, estimated as the sum of squared Pearson residuals divided by the residual degrees of freedom:
Under the Poisson assumption (mean = variance), φ should equal 1. Values substantially greater than 1 indicate overdispersion; values less than 1 indicate underdispersion.
Why this matters in practice: when overdispersion is ignored, the Poisson standard errors come out too small, so confidence intervals are too narrow and p-values too extreme. The practical risk is calling an association statistically significant when the data do not really support it.
Understanding Overdispersion
Overdispersion, the situation where the observed variance exceeds the Poisson-assumed variance, is one of the most common problems in count data modelling (Ver Hoef & Boveng, 2007). It is critical to distinguish between two types:
Warning: Interpreting Overdispersion
Before concluding that overdispersion is “real,” always investigate whether the model is correctly specified. Adding missing predictors, removing outliers, or modelling non-linear effects may resolve apparent overdispersion without needing to change the distributional assumptions. Applying overdispersion corrections to a misspecified model can mask important features of the data.
Apparent Overdispersion
Apparent overdispersion arises from problems with the model rather than the data-generating process itself. Common causes include:
- Outliers: A few extreme observations can inflate the dispersion statistic dramatically.
- Missing important predictors: If key covariates are omitted from the model, the unexplained variation appears as overdispersion.
- Wrong model form: Using a linear predictor when the true relationship is non-linear.
- Non-linear effects: Failing to include quadratic or other polynomial terms for predictors with curvilinear relationships.
Apparent overdispersion can be resolved by correcting the model specification: removing outliers, adding missing predictors, or using the correct functional form.
Real Overdispersion
Real overdispersion reflects genuine extra-Poisson variation in the data that cannot be explained by observable covariates. This often arises from:
- Unobserved heterogeneity: Subject-level variation in the underlying rate that is not captured by measured predictors.
- Clustering: Events within groups (e.g., animals within herds) are correlated, violating the independence assumption.
- Biological variability: Inherent variation in susceptibility or exposure that exceeds what the Poisson model allows.
Real overdispersion requires statistical corrections such as scaling standard errors, using negative binomial regression, or employing random effects models.
Approaches to Handling Overdispersion
| Approach | How It Works | When to Use |
|---|---|---|
| Scale SEs by √φ | Multiplies standard errors by the square root of the estimated dispersion parameter; coefficients unchanged | Mild to moderate overdispersion; quick fix when coefficient estimates are trusted |
| Negative binomial regression | Adds an extra parameter (α) to model the excess variance explicitly | Moderate to severe overdispersion; when a more principled model is desired |
| Random effects / GLMM | Includes subject- or group-level random intercepts to capture unobserved heterogeneity | Clustered data (e.g., animals within herds); hierarchical study designs |
| GEE (robust SEs) | Uses generalised estimating equations with an empirical (sandwich) variance estimator | Clustered data when marginal (population-averaged) estimates are of primary interest |
1. In a Poisson model, overdispersion is indicated when:
2. Which of the following is NOT a cause of apparent overdispersion?
3. One approach to handling real overdispersion is:
Reflection
Why is it important to distinguish between apparent and real overdispersion before choosing a correction strategy? What could go wrong if you apply the wrong fix?
Negative Binomial & Zero-Adjusted Models
Introduction and Overview
An earlier section named overdispersion as the most common problem with Poisson regression. This section closes the lesson with the standard fixes: the negative binomial distribution (which adds a free dispersion parameter; Ver Hoef & Boveng, 2007), and zero-adjusted models for the special case where there are far more zeros than Poisson or negative binomial alone can accommodate (Lambert, 1992).
Learning Objectives
- Derive the negative binomial as a Gamma-mixed Poisson and contrast NB-1 and NB-2 parameterisations.
- Fit a negative binomial regression and compare it to a Poisson fit using a likelihood-ratio test on the dispersion parameter.
- Identify zero-inflation versus zero-truncation and choose the appropriate zero-adjusted model.
- Interpret a hurdle or zero-inflated model in terms of separate processes for the zero and the count components.
The Negative Binomial Distribution
The negative binomial (NB) distribution extends the Poisson by adding an extra parameter α that captures the additional variation not accounted for by the Poisson assumption. Conceptually, the NB distribution arises when the Poisson rate itself varies randomly across individuals, so each subject has their own λ, drawn from a Gamma distribution. Averaging over those rates produces the negative binomial distribution.
The NB distribution allows the variance to exceed the mean, making it the natural first choice when overdispersion is present (Ver Hoef & Boveng, 2007). Two common parameterisations define how the variance relates to the mean:
NB-1: Linear Variance
In the NB-1 parameterisation, the variance increases linearly with the mean. The overdispersion is proportional to the mean: doubling the expected count doubles the excess variance. The ratio Var(Y)/μ = (1 + α) is constant across all observations, making NB-1 similar to a quasi-Poisson model with a fixed dispersion parameter.
NB-1 is sometimes preferred when overdispersion is relatively constant across the range of predicted values. However, it is less commonly used in practice than NB-2.
NB-2: Quadratic Variance
In the NB-2 parameterisation (the most commonly used form), the variance increases quadratically with the mean. Observations with higher expected counts have proportionally more overdispersion. This is often more realistic in biological settings where variability tends to grow faster than the average.
The NB-2 model is the default in most statistical software (e.g., Stata’s nbreg, R’s glm.nb()). When α = 0, the NB-2 model reduces to the Poisson model, making the Poisson a special (nested) case of NB-2.
Negative Binomial Regression
The NB regression model uses the same log-linear form as Poisson regression; the only difference is in the assumed distribution of the outcome:
Coefficients are interpreted identically to Poisson regression: eβ gives the incidence rate ratio. The key advantage is that the NB model produces correct standard errors even when overdispersion is present, because the extra variation is explicitly modelled through α.
Testing Poisson vs. Negative Binomial
Since the Poisson model is nested within the NB model (when α = 0), a likelihood ratio test (LRT) can be used to determine whether the NB model provides a significantly better fit. A significant LRT indicates that overdispersion is present and the NB model is preferred. Note that this is a boundary test (testing α = 0 vs. α > 0), so the p-value from the standard χ² reference distribution is conservative.
Zero-Adjusted Models
Standard count models (Poisson and NB) may not adequately handle datasets with an unusual number of zeros (Lambert, 1992). Three families of models have been developed to address different zero-related problems:
Choosing Among Zero-Adjusted Models
The choice depends on the data-generating process:
- If some zeros are “structural” (from a fundamentally different process) and others arise from the count process, use a zero-inflated model.
- If the zero/non-zero distinction is a separate decision from the magnitude of the count, use a hurdle model.
- If zeros are impossible by design, use a zero-truncated model.
| Model | Source of Zeros | Key Feature | Test / Comparison |
|---|---|---|---|
| Zero-Inflated | Both components (structural + count) | Mixture of logistic + count model | Vuong test vs. standard model |
| Hurdle | Binary component only | Two-part: binary then truncated count | LRT or AIC/BIC comparison |
| Zero-Truncated | Zeros cannot occur | Conditional on Y > 0 | Applied when sampling excludes zeros |
1. The NB-2 model differs from the Poisson model by:
2. Zero-inflated models are appropriate when:
3. The key difference between a hurdle model and a zero-inflated model is:
Reflection
When might you choose a hurdle model over a zero-inflated model in practice? Think of an epidemiological example where the distinction matters.
Lesson 4: Final Assessment
Bringing It All Together
This module extended the binary toolkit of the previous lesson to two further members of the generalised linear model family. The first part took outcomes with three or more categories: it started with the four-model landscape and the Apgar-score example, then walked through the multinomial logit (general but parameter-hungry), the proportional-odds model (parsimonious for ordinal data when its assumption holds), and the adjacent-category and continuation-ratio models (alternatives when proportional odds fails or the categories are sequential). The recurring theme is that the choice of logit determines what each coefficient means: a single estimand takes a different shape under each model, and the right shape depends on the structure of the outcome and the question being asked.
The second part assembled the count-and-rate toolkit. It started with the Poisson distribution and used its log-link form to build a regression, then introduced the offset term, the small piece of arithmetic that converts a count model into a rate model and unifies the cohort and incidence-density designs you have already met. From there it moved to evaluation: Pearson, deviance, and Anscombe residuals, goodness-of-fit tests, and the variance-to-mean comparison that makes overdispersion visible, before introducing negative binomial regression as the everyday workhorse when Poisson assumptions break and the zero-adjusted (hurdle and zero-inflated) models for outcomes with a separate at-risk process.
Seen together, the two parts make the same point from two directions: a generalised linear model is a distribution plus a link, and once the outcome is characterised correctly (nominal or ordered, count or rate, with or without excess zeros) the coefficients, the diagnostics, and the reporting follow. The final assessment asks you to recognise which model fits a given outcome, to interpret its coefficients without sliding back into the binary-logistic vocabulary by reflex, to reach for the right count model and report rate ratios with appropriate uncertainty, and to use the diagnostic flowchart as your default answer to "which model?". A later lesson takes on the next major outcome type, time to event, where censoring and the timing of events extend the regression toolkit again.
Key Takeaways from this lesson
- Nominal vs ordinal outcomes call for different families of models; ordering is information you can either use (parsimony) or ignore (generality).
- Multinomial logistic regression fits J−1 simultaneous logits against a baseline and its exponentiated coefficients are relative risk ratios; the proportional-odds model assumes one coefficient across every cumulative cutpoint, and that assumption must be tested (Brant test), not assumed.
- The adjacent-category model is a constrained multinomial that compares each category with its neighbour; the continuation-ratio model suits sequential-stage outcomes and can be fit as a series of binary logistic regressions on recoded data.
- Always start by sketching the logit your model uses; the choice of comparison set is what makes any of these models ordinal or nominal.
- Counts and rates are a distinct outcome family with their own GLM, the Poisson with a log link; the offset term turns a count model into a rate model by absorbing person-time or population denominators.
- Exponentiated coefficients in Poisson and negative binomial regression are rate ratios, interpreted multiplicatively.
- Overdispersion (variance exceeding the mean) is the most common Poisson failure mode and inflates Type I error if ignored; negative binomial regression is the default fix whenever the dispersion test rejects Poisson.
- Zero-inflated and hurdle models handle outcomes with a separate at-risk process generating excess zeros; zero-truncated models handle outcomes where zeros cannot be observed.
This final assessment covers all material from the module. You must answer all 15 questions correctly (100%) and complete the final reflection to finish the lesson.
Reflection
A registry follows 1,800 patients with a chronic condition. For each patient it records a disease-severity grade at enrolment (mild, moderate, severe), the number of emergency visits during follow-up, and the length of follow-up, which differs between patients; about half of the patients have no visits at all. Reflecting on both parts of this module, walk through your modelling strategy for each outcome: which model you would fit for severity grade and why, which assumption you would test and what you would do if it failed, how you would model the visit counts given unequal follow-up and the many zeros, which diagnostics would tell you whether Poisson is adequate, and how you would interpret and report the coefficients from each model.
Minimum 20 characters required.
Final Knowledge Assessment
1. What type of outcome data has categories with no natural ordering?
2. How many sets of coefficients does a multinomial model with 4 outcome categories estimate?
3. In a proportional-odds model, the OR for a predictor represents:
4. The IIA assumption in multinomial logistic regression means:
5. In a multinomial model, exponentiated coefficients are technically:
6. The Brant test evaluates:
7. The adjacent-category model is a constrained version of:
8. Continuation-ratio models are best suited for:
9. Which type of count data involves dividing event counts by accumulated person-time?
10. The Poisson distribution assumes:
11. In Poisson regression, the offset term represents:
12. The exponentiated coefficient from a Poisson regression (eβ) is interpreted as:
13. Which is a cause of APPARENT (not real) overdispersion?
14. To test whether negative binomial regression provides a better fit than Poisson, you use:
15. In a hurdle model, zero counts are generated by:
✦ Before submitting: pass every section knowledge check (100%) and complete every reflection.