1.1 What Are Zero-Inflated Models?

In regression, zero-inflated models combine two components: a point mass at zero and a second distribution for the outcome. Predictors can enter both components through separate linear predictors and link functions. In this sense, zero-inflated regression models are technically defined as generalized linear mixture models.

Although they are often introduced for count data, the general framework is not limited to counts. For example, Liu et al. (2019) review zero-inflated models for non-negative continuous outcomes, while Ospina and Ferrari (2012) develop zero-or-one-inflated beta regression for proportions.

This tutorial focuses on count data, where an observed count can arise from two distinct latent processes:

1. A latent extra-zero state, whose zeros are often called structural zeros.
2. A count-generating state, which produces sampling zeros as well as positive counts.

What does this actually mean? Well, suppose you are counting clinic visits. A zero can mean at least two different things:

• the person was never really in a position to generate a visit during the study window
• the person could have generated a visit, but happened not to

A zero from the first route is called a structural zero; one from the second is a sampling zero. A standard one-process count model, such as Poisson regression, treats both people as coming from the same count-generating process. A zero-inflated model represents both routes as latent possibilities, but the data do not reveal which route produced any individual zero.

1.2 When Do We Need Zero-Inflated Models?

The key challenge zero-inflated models try to address is excess zeros in the data. However, the useful question is not “does my dataset contain a lot of zeros?”, but do the zeros plausibly come from more than one process?

This distinction will become clearer in the following sections, but for now these are some helpful examples to keep in mind:

• Healthcare use: Some patients are effectively unable to generate events because they never access services (structural zeros)—for example, people living in remote areas with no nearby clinics, or those excluded by policy or eligibility criteria. Other patients could use services but happen not to during the study window (sampling zeros)—for example, healthy patients who simply do not visit in that period or patients whose condition does not require care.
• Disease surveillance: Some locations truly have zero prevalence for a pathogen because the pathogen cannot survive there or has been eradicated (structural zeros). Other locations harbor the pathogen but a given survey fails to detect it during that sampling round (sampling zeros) — e.g., low prevalence or imperfect tests.
• Specialist treatments or prescriptions: Some patients will never be eligible for a specific specialist-only procedure or advanced drug (structural zeros) — think of treatments restricted by age, comorbidity, or insurance coverage. Other patients are eligible but simply don’t receive the treatment during the study window (sampling zeros) because they don’t yet need it, they defer care, or the treatment is rarely used in short follow-up.

In each case, the zero may carry information about the process that generated the observation. That still leaves an important modelling caveat: excess zeros are not the same thing as proven structural zeros. A flexible one-process count model can sometimes reproduce many zeros without needing a separate latent class.

2.1 Generalized Linear Models

Before fitting zero-inflated models, it helps to revisit generalized linear models (GLMs). Counts are non-negative integers, usually right-skewed, and often have variance that changes with the mean. Ordinary linear regression does not encode those features: it can predict negative values and assumes a constant-variance normal error structure. GLMs allow us to retain a regression-style mean model while using a response distribution suited to the outcome.

A GLM connects a linear predictor to the mean of an outcome distribution through a link function. Three ingredients define that structure:

1. Linear predictor (): a linear combination of predictor values and coefficients
2. Link function (): the transformation connecting the linear predictor to the conditional mean
3. Response distribution: a distribution suited to the outcome, such as Bernoulli for a binary response or Poisson for a count

Zero-inflated count models use this structure twice: a binary component models membership in the latent extra-zero group, and a count component models the expected outcome among observations in the count-generating group. The original zero-inflated Poisson regression paper framed this as a practical way to model excess zeros in manufacturing defects (Lambert 1992); the same idea now appears across health, ecology, insurance, and many other settings.

Logistic and Poisson regression provide the two building blocks. A short exponential-family refresher first recovers the logit link for a Bernoulli response; the following count-model section then applies the same logic to Poisson regression and its extensions.

2.2 The Exponential Family

The exponential family is a collection of probability distributions with a shared mathematical form. This might sound abstract, but it includes most distributions you encounter in practice: normal, Poisson, binomial, gamma, and many others.

A density or probability mass function in this family can be written as:

The notation separates the parts that govern the distribution:

• is the natural parameter;
• is a dispersion parameter, with controlling its scale;
• determines the mean and variance through its first two derivatives;
• supplies the remaining terms needed to define a valid distribution.

Even though the formula looks daunting, the important part to keep in mind is that this format essentially standardises how we work with different distributions. In particular, it can be proved that for all distributions in this family, the mean and variance are:

This common structure is what allows GLMs to pair different response distributions with a linear predictor. For a fuller treatment, McCullagh and Nelder give the classic mathematical account, while Dobson and Barnett provide a more introductory route through the same material (McCullagh and Nelder 1989; Dobson and Barnett 2018).

2.3 Count-Model Building Blocks

Poisson regression is the count-data counterpart to the Bernoulli example above: its exponential-family form produces a log link, while its response distribution supplies the mean-variance relationship. That separation matters because the link and the variance assumption can succeed or fail for different reasons.

The models therefore follow a deliberate sequence. We start with Poisson regression, check its conditional variance and zero predictions, move to negative binomial regression when the count process is overdispersed, and only then ask whether a separate zero-inflation process remains useful. Each step preserves the regression-style mean model while relaxing a specific distributional restriction.

3.1 Medicare Data and Outcome

The medcare dataset from the catdata package is a reduced teaching extract of the 4,406-person sample used by Deb and Trivedi, drawn from the 1987 National Medical Expenditure Survey (Schauberger and Tutz 2025; Deb and Trivedi 1997). It records healthcare use among people aged 66 and over who were covered by Medicare. The outcome, ofp, is each person’s number of physician office visits during the 1987 NMES observation year.

I like this example because the case for a zero-inflated model is not obvious from one very dramatic spike at zero. The data instead make us work through the same questions that often arise in the real world: how variable are the counts, which patients have no visits, and can a standard count model reproduce the observed zeros?

The extract omits survey weights, strata, and cluster identifiers, so the analysis treats the supplied records as independent and reports model-based rather than design-based uncertainty (for fuller provenance, assumptions, and software versions, jump ahead to Section 6.3).

The analysis uses the following packages and variables:

# Load required packages
library(glmmTMB)
library(lme4)
library(DHARMa)
library(performance)
library(parameters)
library(ggplot2)
library(dplyr)
library(catdata)

# Load the medical care dataset
data(medcare)

# Examine the structure of the data
head(medcare)

  ofp hosp healthpoor healthexcellent numchron male age married school
1   5    1          0               0        2    1 6.9       1      6
2   1    0          0               0        2    0 7.4       1     10
3  13    3          1               0        4    0 6.6       0     10
4  16    1          1               0        2    1 7.6       1      3
5   3    0          0               0        2    0 7.9       1      6
6  17    0          1               0        5    0 6.6       0      7

glimpse(medcare)

Rows: 4,406
Columns: 9
$ ofp             <int> 5, 1, 13, 16, 3, 17, 9, 3, 1, 0, 0, 44, 2, 1, 19, 19, …
$ hosp            <int> 1, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, …
$ healthpoor      <int> 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, …
$ healthexcellent <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ numchron        <int> 2, 2, 4, 2, 2, 5, 0, 0, 0, 0, 1, 5, 1, 1, 1, 0, 1, 2, …
$ male            <int> 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, …
$ age             <dbl> 6.9, 7.4, 6.6, 7.6, 7.9, 6.6, 7.5, 8.7, 7.3, 7.8, 6.6,…
$ married         <int> 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, …
$ school          <int> 6, 10, 10, 3, 6, 7, 8, 8, 8, 8, 8, 15, 8, 8, 12, 8, 8,…

The variables used in this section are:

• ofp: Number of physician office visits (our count outcome)
• healthpoor: Individual has poor health (reference: average health)
• healthexcellent: Individual has excellent health
• numchron: Number of chronic conditions
• male: Gender (female = 0, male = 1)
• age: Age in decades (6.6 represents 66 years)
• married: Marital status (married = 1, else = 0)

3.2 Distribution of Visits and Observed Zeros

The outcome summary keeps the sample size, centre, spread, and frequency of zeros in one place.

stats_tbl <- tibble::tibble(
  Metric = c(
    "Participants",
    "Zero-visit records",
    "Proportion with zero visits",
    "Median visits",
    "Mean visits",
    "Variance",
    "Variance-to-mean ratio",
    "Maximum visits"
  ),
  Value = c(
    format(nrow(medcare), big.mark = ","),
    format(sum(medcare$ofp == 0), big.mark = ","),
    sprintf("%.1f%%", 100 * mean(medcare$ofp == 0)),
    sprintf("%.0f", median(medcare$ofp)),
    sprintf("%.2f", mean(medcare$ofp)),
    sprintf("%.2f", var(medcare$ofp)),
    sprintf("%.2f", var(medcare$ofp) / mean(medcare$ofp)),
    sprintf("%.0f", max(medcare$ofp))
  )
)

knitr::kable(stats_tbl)

There are three key insights we can draw from Table 1:

• 683 people (15.5%) have no recorded visits, and zero is the most frequent single count. That is a noticeable concentration, but its size alone does not establish zero-inflation. A Poisson or negative binomial distribution can also generate zeros.

• The outcome is considerably right-skewed: the median is four visits, the mean is 5.77, and the maximum is 89 visits.

• The variance is 7.91 times the mean. This marginal variance-to-mean ratio warns us that a Poisson model may be too restrictive, although the relevant assumption is conditional equidispersion after accounting for the predictors. We will test that on the fitted model rather than declaring overdispersion from the raw outcome alone.

3.2.1 Zero Patterns Across Patient Characteristics

By exploring the distribution of zero-visit records, we notice that they are not distributed uniformly across the observed patient characteristics.

Show visualization code

p1 <- ggplot(medcare, aes(x = ofp)) +
  geom_histogram(
    binwidth = 1,
    boundary = -0.5,
    fill = "#4C78A8",
    color = "white"
  ) +
  coord_cartesian(xlim = c(-0.5, 20.5)) +
  labs(
    title = "Most participants recorded relatively few visits",
    x = "Physician office visits",
    y = "Participants"
  ) +
  theme_minimal() +
  theme(plot.title = element_text(face = "bold"))

health_zeros <- medcare |>
  mutate(health_status = case_when(
    healthpoor == 1 ~ "Poor",
    healthexcellent == 1 ~ "Excellent",
    TRUE ~ "Average"
  )) |>
  mutate(
    health_status = factor(
      health_status,
      levels = c("Poor", "Average", "Excellent")
    )
  ) |>
  group_by(health_status) |>
  summarise(prop_zeros = mean(ofp == 0), .groups = "drop")

p2 <- health_zeros |>
  ggplot(aes(x = health_status, y = prop_zeros)) +
  geom_col(fill = "#F58518", width = 0.7) +
  scale_y_continuous(
    labels = function(x) paste0(round(100 * x), "%"),
    limits = c(0, 0.27)
  ) +
  labs(
    title = "Zeros vary by reported health",
    x = "Self-reported health",
    y = "Zero-visit records"
  ) +
  theme_minimal() +
  theme(plot.title = element_text(face = "bold"))

chronic_zeros <- medcare |>
  group_by(numchron) |>
  summarise(
    prop_zeros = mean(ofp == 0),
    participants = n(),
    .groups = "drop"
  ) |>
  filter(participants >= 20)

p3 <- chronic_zeros |>
  ggplot(aes(x = numchron, y = prop_zeros)) +
  geom_col(fill = "#54A24B", width = 0.75) +
  scale_x_continuous(breaks = chronic_zeros$numchron) +
  scale_y_continuous(
    labels = function(x) paste0(round(100 * x), "%"),
    limits = c(0, 0.32)
  ) +
  labs(
    title = "Zeros fall as chronic conditions increase",
    x = "Chronic conditions",
    y = "Zero-visit records"
  ) +
  theme_minimal() +
  theme(plot.title = element_text(face = "bold"))

gridExtra::grid.arrange(
  p1,
  p2,
  p3,
  layout_matrix = rbind(c(1, 1), c(2, 3))
)

The descriptive pattern is clearest for chronic conditions. About 29% of people with no chronic conditions recorded zero visits, compared with 9% of those with two conditions and 6% of those with three. The final bars are based on smaller groups, so the overall decline matters more than every local change.

Self-reported health shows a less linear pattern. The zero proportion is highest for people reporting excellent health (24%), followed by average health (15%) and poor health (11%). These are unadjusted comparisons: age, sex, marital status, and chronic conditions differ between the groups. We therefore cannot read the bars as isolated health-status effects, still less as evidence that any particular zero is structural.

What the plots do establish is heterogeneity. The probability of a zero changes with observed patient characteristics, which makes a separate zero process plausible enough to investigate. It does not yet tell us whether a zero-inflated model will outperform an overdispersed one-process model.

# Create binary indicator: 1 = zero visits, 0 = any visits
medcare$no_visits <- ifelse(medcare$ofp == 0, 1, 0)

# Fit logistic regression
logistic_model <- glm(
  no_visits ~ healthpoor + healthexcellent + numchron + age + male + married,
  data = medcare,
  family = binomial()
)

logistic_table <- model_parameters(
  logistic_model,
  exponentiate = TRUE
) |>
  filter(Parameter != "(Intercept)") |>
  transmute(
    Predictor = recode(
      Parameter,
      healthpoor = "Poor vs average health",
      healthexcellent = "Excellent vs average health",
      numchron = "Chronic conditions (per condition)",
      age = "Age (per decade)",
      male = "Male vs female",
      married = "Married vs not married"
    ),
    `Odds ratio` = sprintf("%.2f", Coefficient),
    `95% CI` = sprintf("%.2f–%.2f", CI_low, CI_high),
    `p-value` = format.pval(p, digits = 2, eps = 0.001)
  )

knitr::kable(logistic_table, align = c("l", "r", "c", "r"))

3.3 Poisson Baseline

The standard starting point for count data is a Poisson regression. Even when we expect it to fail, this model provides a useful baseline:

• What relationships would a one-process count model estimate?
• How badly does the variance assumption break?
• How many zeros would this simpler model expect?

For this application, the important Poisson assumptions are:

1. Conditional distribution: Given the predictors, each person’s count follows a Poisson distribution.
2. Conditional equidispersion: The conditional variance equals the conditional mean,
3. Independent records: Participants are independent after conditioning on the modelled predictors.
4. Comparable exposure: Each count covers the same observation window; otherwise the model needs an exposure offset.

We will first interpret the conditional-mean estimates, then check whether the Poisson variance and zero probabilities fit the data. Unlike the logistic model, which asked whether a person recorded no visits, the Poisson model uses the entire count and asks how the average number of visits changes with the predictors. It therefore provides a baseline model for the count process.

# Fit Poisson model
count_formula <- ofp ~ healthpoor + healthexcellent + numchron +
  age + male + married

poisson_model <- glm(
  count_formula,
  data = medcare,
  family = poisson()
)

poisson_table <- model_parameters(
  poisson_model,
  exponentiate = TRUE
) |>
  filter(Parameter != "(Intercept)") |>
  transmute(
    Predictor = recode(
      Parameter,
      healthpoor = "Poor vs average health",
      healthexcellent = "Excellent vs average health",
      numchron = "Chronic conditions (per condition)",
      age = "Age (per decade)",
      male = "Male vs female",
      married = "Married vs not married"
    ),
    `Rate ratio` = sprintf("%.2f", Coefficient),
    `95% CI` = sprintf("%.2f–%.2f", CI_low, CI_high),
    `p-value` = format.pval(p, digits = 2, eps = 0.001)
  )

knitr::kable(poisson_table, align = c("l", "r", "c", "r"))

Table 3 presents the results of the Poisson regression. As a quick reminder, a Poisson rate ratio describes a proportional difference in the expected count: the average number of visits predicted for people with the same included characteristics.

Here, under the Poisson mean model, poor health is associated with a 29% higher expected visit count than average health, while excellent health is associated with a 30% lower count. Each additional chronic condition is associated with an 18% increase. Men have an expected count about 10% lower than women, and a decade of age is associated with a 5% reduction. The estimate for marital status is essentially null.

3.4 Where the Poisson Model Fails

The fitted mean structure does not establish that the Poisson distribution describes the conditional variation. The Pearson residuals provide a direct assessment of equidispersion after adjustment for the predictors.

overdispersion_check <- check_overdispersion(poisson_model)

overdispersion_table <- tibble::tibble(
  Metric = c(
    "Pearson chi-squared",
    "Residual degrees of freedom",
    "Dispersion ratio",
    "p-value"
  ),
  Value = c(
    format(
      round(overdispersion_check$chisq_statistic),
      big.mark = ","
    ),
    format(
      overdispersion_check$residual_df,
      big.mark = ","
    ),
    sprintf("%.2f", overdispersion_check$dispersion_ratio),
    format.pval(overdispersion_check$p_value, digits = 2, eps = 0.001)
  )
)

knitr::kable(overdispersion_table)

The dispersion ratio is 7.20, well above the value of 1 expected under a Poisson model. The observed visit counts therefore vary much more than the model permits under the Poisson assumption. This overdispersion makes the model-based standard errors too small and the corresponding confidence intervals and p-values too optimistic. The rate ratios still summarize the fitted mean structure, but this means that the Poisson model does not provide reliable inference for these data.

Overdispersion identifies the Poisson model as inadequate but does not identify the source of the extra variation. The next one-process candidate is therefore negative binomial regression, which relaxes equidispersion without introducing a latent extra-zero group.

3.5 Negative-Binomial Fit and Zero-Frequency Comparison

The negative binomial baseline retains count_formula, so it makes the same conditional comparisons as the Poisson model. Its nbinom2 variance, , allows variation to increase faster than the mean.

nb_model <- glmmTMB(
  count_formula,
  data = medcare,
  family = nbinom2()
)

The fitted dispersion parameter is 1.13. This finite value allows substantially more conditional variation than Poisson; the Poisson limit would require . The model still treats every participant as arising from one count-generating process.

Zero prediction isolates one important consequence of the different variance assumptions. For participant , the fitted Poisson probability of zero is

whereas the fitted NB2 probability is

Summing the person-specific probabilities gives the expected number of zero-visit records under each model.

# Observed zeros
observed_zeros <- sum(medcare$ofp == 0)

# Predicted zeros from Poisson model
poisson_pred <- predict(poisson_model, type = "response")
poisson_zero_prob <- exp(-poisson_pred)  # P(Y=0) for Poisson
expected_zeros_poisson <- sum(poisson_zero_prob)

# Predicted zeros from negative binomial model
nb_mean <- predict(nb_model, type = "conditional")
nb_dispersion <- predict(nb_model, type = "disp")
nb_zero_prob <- dnbinom(0, mu = nb_mean, size = nb_dispersion)
expected_zeros_nb <- sum(nb_zero_prob)

zero_comparison <- tibble::tibble(
  Model = c("Poisson", "Negative binomial (NB2)"),
  `Expected zeros` = round(c(
    expected_zeros_poisson,
    expected_zeros_nb
  )),
  `Observed minus expected` = observed_zeros - `Expected zeros`,
  `Expected / observed` = sprintf(
    "%.1f%%",
    100 * `Expected zeros` / observed_zeros
  )
)

knitr::kable(
  zero_comparison,
  align = c("l", "r", "r", "r")
)

The fitted Poisson model expects approximately 37 zero-visit records, only 5.4% of the 683 observed. NB2 expects approximately 623, or 91.2% of the observed number. Allowing extra variation within one count process therefore explains most of what initially looked like a zero surplus.

The remaining shortfall is about 60 zero records. That discrepancy keeps a separate extra-zero process under consideration, but it does not prove that such a process exists. Misspecification of the conditional mean, other forms of heterogeneity, or a different count distribution could also affect zero prediction.

The baseline analysis has therefore narrowed rather than settled the question. Poisson is too restrictive, while NB2 captures most of the dispersion and zero frequency without a latent class. Section 4 introduces the extra-zero mechanism so that its additional structure can be compared with this stronger one-process benchmark.

4.1 A Latent Two-Process Model

Unlike the negative binomial baseline, which keeps every participant in one count process, a zero-inflated model allows either of two states to generate each observed count. For participant , we introduce as a binary indicator of that state.

• means the extra-zero state is active. In this state, the recorded number of visits must be zero.
• means the count-generating state is active. This state can produce either zero visits or a positive number of visits.

The state is latent because it is not recorded in the data. When a participant has zero visits, we do not know which state generated that zero. The idea is to model the probability that an individual patient will fall under each state.

This probability may vary across patient profiles. Let contain the predictors used for this part of the model, and let be the resulting probability that participant is in the extra-zero state. We write

In other words, takes the value 1 with probability , based on the participant’s zero-inflation predictors .

Now let be the observed number of visits and let contain the predictors used for the count component. The outcome model formalizes the two states described above:

When the extra-zero state is active, by definition the number of visits must be zero. In the count-generating state, visits follow , a count distribution with mean . The placeholder represents a family-specific dispersion parameter when one is required: it is absent for Poisson and equals for the NB2 model used later.

In essence, and forgetting about notation for a moment, the central idea is that the extra-zero state always produces zero, whereas the count-generating state can produce either zero or a positive count. Figure 4 illustrates this two-stage process.

For the Medicare data, the two components therefore ask different questions. The zero-inflation component asks which patient profiles receive a higher probability of the extra-zero state. The count component asks how expected visits change within the count-generating state.

4.2 Mixture Probabilities and Likelihood

Zeros are ambiguous because either latent state can produce them. Positive counts are not: they can come only from the count-generating state. This asymmetry gives the model information about both components. It estimates them jointly from all observations rather than classifying the zeros first and then fitting separate regressions.

Let denote the probability that the count component assigns to the value . The possible routes are:

• If , the observation may come directly from the extra-zero state, with probability , or from the count-generating state producing zero, with probability .
• If , the observation has one possible route: the count-generating state, with probability .

The likelihood contribution is the probability that the model assigns to the outcome observed for participant . The two cases above give

In each case, is the probability that the model assigns to participant ’s observed outcome. For a zero, the contribution adds the probabilities of both routes instead of choosing one. Across the sample, these contributions form a single objective for estimating the count, zero-inflation, and dispersion parameters together.

Joint estimation also explains why the preliminary logistic regression in Table 2 is not the fitted zero-inflation component. The preliminary regression models the observed event , regardless of which state produced the zero. The fitted zero-inflation component models the latent event . These are different outcomes: the probability of an observed zero includes both routes, whereas is the model probability of the extra-zero state.

Because the state remains unobserved, a zero raises the practical question: given that we observed a zero, what is the probability that it came from the extra-zero state rather than from the count-generating state? The posterior probability below answers that question. Bayes’ rule first expresses it in terms of model probabilities:

The first numerator term equals one because the extra-zero state always produces zero. The prior model probability of that state is by definition of the zero-inflation component. The denominator contains both routes to an observed zero: the extra-zero route and a zero from the count-generating state. For a positive count, the posterior probability of the extra-zero state is zero because that state cannot produce a positive value.

For a concrete example with a Poisson count component, suppose the model assigns observations with a particular covariate profile an extra-zero probability and a count-component mean .

• Within the count-generating state, the probability of zero visits is , or 4.1%.
• Combining both routes gives an overall zero probability of , or 18.5%.
• If an observation with this profile equals zero, the posterior probability of the extra-zero state is , or 81%.

This posterior probability therefore updates uncertainty about the latent state. Because the count component considers zero unlikely for this profile, observing zero shifts much of the model’s probability toward the extra-zero state. This soft assignment helps estimate the two components jointly.

4.3 Mixture Mean and Variance

Once the model has estimated the extra-zero probability and the behaviour of the count-generating state, we can ask what it predicts for observed visits. Here, is the mean of the chosen count distribution for participant within the count-generating state. Let denote the corresponding variance:

In a single-process count model, would also be the expected observed count. In a zero-inflated model, however, the model assigns some probability to an always-zero state. We must average over both states to obtain expected observed visits.

The extra-zero state contributes zero with probability , while the count-generating state contributes an average of with probability . Therefore,

This conditional mixture mean is conditional on the observed predictors and , but averages over the unobserved state . It is the quantity we need when comparing expected visits across patient profiles. Reporting instead would describe only the count-generating state and would overstate expected observed visits whenever .

The mean alone does not tell us how observations are distributed around that average. Two fitted distributions can predict the same expected number of visits while assigning very different probabilities to zero, moderate, and high counts. The mixture variance summarizes this spread, so it helps us judge whether the fitted distribution represents the observed pattern of visits.

The count distribution determines . A Poisson count state has , giving a zero-inflated Poisson (ZIP) model. An NB2 count state has , giving a zero-inflated negative-binomial (ZINB) model. The resulting mixture variances are shown below.

Model	Count-state zero probability	Count-state variance	Conditional mixture variance
ZIP
ZINB (NB2)

Both models have conditional mixture mean . Their difference lies in the spread around that mean. ZIP uses a Poisson count state, whereas ZINB adds the term and therefore allows overdispersion within the count-generating state. As , that additional variation disappears and the ZINB expressions approach their ZIP counterparts.

For the Medicare case study, the mean identity tells us how to calculate expected observed visits, while the variance distinguishes what each count distribution can represent. The next section connects and to predictors and asks how changes in either component affect the overall mean.

4.4 Regression and Overall Effects

We have seen that the overall mean depends on both the count-state mean and the extra-zero probability . The next question is then: how does a predictor change that mean?

In a zero-inflated model, a predictor may change expected visits within the count-generating state, the probability of the extra-zero state, or both. Its component coefficient therefore need not describe its association with observed visits.

Let contain the count-component predictors and let contain their coefficients. Let contain the zero-inflation predictors and let contain their coefficients. In glmmTMB, the zero-inflation component always uses a logit link. The count-component link is more flexible: the log link is the default for the Poisson and NB2 distributions used here, as it is for most count distributions available in glmmTMB, but it can be changed. The models in this tutorial therefore use

The log link keeps positive, while the logit link keeps between zero and one. A predictor may appear in either equation or in both.

Count component: For a predictor in , is the rate ratio for a one-unit increase in , holding the other count-component predictors fixed. It describes how changes within the count-generating state.

Zero-inflation component: For a predictor in , is the odds ratio for the extra-zero state. A value above one means that a one-unit increase in raises the modelled odds of that state, holding the other zero-inflation predictors fixed.

Neither coefficient alone answers the practical question: how do expected observed visits change? That quantity still requires both components, which as we have seen in the previous section, is given by:

In essence, the two pathways can reinforce or offset one another. Suppose a one-unit increase in a predictor raises from 4 to 5 but also raises from 0.10 to 0.30. The count-state mean increases by 25%, yet the overall mean falls slightly, from to . The higher extra-zero probability more than offsets the higher count-state mean. If rises while falls, the two pathways instead reinforce one another.

Because the pathways can reinforce or offset one another, component coefficient tables do not show how expected observed visits change. Section 5 therefore uses response-scale predictions of rather than stopping at rate ratios and odds ratios. Before fitting those models, however, we must decide which predictors belong in each component.

4.5 Choosing Predictors

As we have seen in the previous section, placing a variable in one formula but not the other restricts which part of the outcome distribution it may change, and as such is an important choice to make.

The two components pose different questions:

1. Which variables describe visit intensity within the count-generating state?
2. Which variables plausibly describe the probability of the extra-zero state?

The count component retains the baseline adjustment set used in the Poisson analysis:

ofp ~ healthpoor + healthexcellent + numchron + age + male + married

These predictors represent health need and demographic differences that may be associated with visit rates. Their inclusion defines the conditional comparison; it does not turn the resulting coefficients into causal effects.

Two available variables are deliberately left out. hosp counts hospital stays during the same observation period as the office visits, so treating it as a prior predictor would blur the temporal ordering between two concurrent measures of healthcare use. school records years of education and could reasonably enter as a socioeconomic covariate. We leave it out to keep the teaching specification focused. The count formula is a starting point, and a fuller analysis would compare other substantively plausible alternatives.

Ideally, the zero-inflation formula would include direct measures of why someone cannot or does not enter the visit-generating process, such as eligibility restrictions, insurance coverage, or geographic access. The medcare data unfortunately contain no such measures. Its health and demographic variables can describe systematic differences in the estimated extra-zero probability, but they cannot identify the mechanism behind those differences.

For the initial ZIP and ZINB comparison, the zero-inflation component uses the more compact formula:

~ numchron + male + married

Chronic conditions, sex, and marital status showed the clearest adjusted associations with an observed zero in Table 2. That result provides an empirical starting point, but it’s not a direct measurement of extra-zero membership: as established in the mixture-probability subsection, observed zeros combine both possible routes to zero.

These three predictors therefore enter both components initially, allowing each to be associated with visit intensity and with the estimated extra-zero probability. Including or excluding a predictor imposes a substantive restriction, and the available variables provide only indirect support for either choice. In the next section we will therefore also compare some alternative zero-inflation specifications.

5.1 Fit Matched Candidate Specifications

For the sake of simplicity and exposition, every specification considered here uses the same count formula:

ofp ~ healthpoor + healthexcellent + numchron + age + male + married

The Poisson baseline uses the Poisson family with no zero component, while the negative-binomial baseline uses NB2 with no zero component. This subsection adds only the candidates needed for the first comparisons:

• Compact ZIP: Poisson count family with ~ numchron + male + married as its zero formula.
• Intercept-only ZINB: NB2 count family with ~ 1 as its zero formula.
• Compact ZINB: NB2 count family with ~ numchron + male + married as its zero formula.

Additional zero-component formulas are deferred until their rationale is introduced.

1compact_zi_formula <- ~ numchron + male + married

# Compact zero-inflated Poisson model
zip_model <- glmmTMB(
  count_formula,
2  ziformula = compact_zi_formula,
  data = medcare,
  family = poisson()
)

# Intercept-only zero-inflated NB2 model
zinb_intercept_model <- glmmTMB(
  count_formula,
3  ziformula = ~ 1,
  data = medcare,
  family = nbinom2()
)

zinb_compact_model <- glmmTMB(
  count_formula,
  ziformula = compact_zi_formula, 
  data = medcare,
4  family = nbinom2()
)

We compare these specifications in matched pairs because each pair isolates one modelling decision: the count family, the addition of a constant extra-zero probability, or whether that probability varies with patient characteristics.

It’s important to keep in mind that the resulting fit statistics provide relative evidence among the specifications, but they neither identify a distinct extra-zero population nor replace distributional diagnostics, which are still needed to assess whether the chosen count distribution adequately represents the data.

5.2 Separate Count-Family and Zero-Component Evidence

The fitted specifications now let us separate the three decisions posed at the start of this section. We first compare Poisson with NB2, using both the ordinary and zero-inflated versions, to choose the count family. We then return to the ordinary NB2 baseline and add the simplest possible extra-zero component before asking whether that component should vary across patient characteristics.

Both AIC and BIC ask whether an improvement in likelihood is worth the additional parameters. Lower values favour a specification, but only differences among models fitted to the same observations with compatible likelihoods are meaningful. AIC emphasises expected predictive accuracy, whereas BIC is more reluctant to add parameters at this sample size, so comparing them shows how sensitive the result is to the penalty for complexity. These scores can rank the candidates considered here; the residual checks later will show whether the leading models are actually able to reproduce the data.

How large do these differences need to be? There is no universal pass-or-fail cutoff. Values that are close together indicate little separation between two specifications, whereas gaps of hundreds or thousands show a clear preference for the model with the lower value.

models <- list(
  Poisson = poisson_model,
  NB2 = nb_model,
  `Compact ZIP` = zip_model,
  `Intercept-only ZINB` = zinb_intercept_model,
  `Compact ZINB` = zinb_compact_model
)

comparison_table <- tibble::tibble(
  Model = names(models),
  AIC = vapply(models, AIC, numeric(1)),
  BIC = vapply(models, BIC, numeric(1)),
  RMSE = vapply(
    models,
    function(model) {
      predictions <- predict(model, type = "response")
      sqrt(mean((medcare$ofp - predictions)^2))
    },
    numeric(1)
  )
) |>
  mutate(
    across(c(AIC, BIC), ~ round(.x, 1)),
    RMSE = round(RMSE, 3)
  )

knitr::kable(comparison_table, align = c("l", "r", "r", "r"))

We start with the count family. Moving from Poisson to NB2 lowers AIC from 37093.4 to 24576.1 when neither model has an extra-zero component. Holding the compact zero formula fixed leads to the same conclusion: moving from ZIP to compact ZINB lowers AIC from 33115.3 to 24458.0. BIC tells the same story in both comparisons. These gaps leave little doubt that Poisson variation is too restrictive here, whether or not the model includes an extra-zero component.

The next question is whether NB2 needs an extra-zero component at all. The cleanest starting point is intercept-only ZINB: it keeps the NB2 count process unchanged and assigns every participant the same extra-zero probability. Compared with ordinary NB2, its AIC is only 1.7 points lower, while its BIC is 4.7 points higher. The small AIC gap and the reversal under BIC suggest that a constant extra-zero probability adds little to the ordinary NB2 model.

What changes when the extra-zero probability can vary between participants? Relative to intercept-only ZINB, the compact ZINB lowers AIC by 116.4 points and BIC by 97.2 points. Both gaps are large: allowing the zero component to vary with chronic conditions, sex, and marital status improves the fitted distribution substantially. It does not, however, show that the zeros came from a separate, identifiable group of patients.

The RMSE column appears to disagree with AIC and BIC because it assesses only the predicted average. For each participant, RMSE compares the observed count with the model’s expected count; it does not consider how much probability the model assigned to that count or to other possible counts. Compact ZIP therefore achieves the lowest RMSE, 6.488, even though its AIC and BIC are much higher than those of the NB2 models. Two models can predict similar averages while assigning very different probabilities to zeros, moderate counts, and unusually high counts. If prediction became a primary goal, repeated cross-validation should assess a score for the entire predictive distribution, such as held-out log predictive density or ranked probability score, alongside RMSE.

The comparison therefore narrows the next decision: retain NB2 variation in the count component, then ask which covariate-dependent zero formula, if any, provides a defensible working specification.

5.3 Choose the Zero-Component Specification

The preceding comparison showed that adding a constant extra-zero probability to NB2 changes fit very little, whereas allowing that probability to vary across participants produces a much larger improvement. That result leaves an important specification question: which participant characteristics should be allowed to change the extra-zero probability?

Starting from the intercept-only ZINB, we compare three ways of allowing that probability to vary:

• The excellent-health specification represents a narrow health-need explanation;
• The compact specification carries forward the strongest adjusted associations with an observed zero in these data;
• The shared-adjustment specification allows every count predictor to have a separate zero-component association.

All four models retain the same NB2 count family and count formula, so the comparison concerns only the zero-component specification. This small candidate set tests whether the compact model’s improvement depends on its particular formula and whether the broader specification earns its additional complexity.

zinb_spec_models <- list(
  `Intercept-only` = zinb_intercept_model,
  `Excellent-health only` = glmmTMB(
    count_formula,
    ziformula = ~ healthexcellent,
    data = medcare,
    family = nbinom2()
  ),
  Compact = zinb_compact_model,
  `Shared adjustment set` = glmmTMB(
    count_formula,
    ziformula = ~ healthpoor + healthexcellent + numchron +
      age + male + married,
    data = medcare,
    family = nbinom2()
  )
)

zinb_model <- zinb_spec_models[["Compact"]]

zinb_spec_table <- tibble::tibble(
  Specification = names(zinb_spec_models),
  `Zero-inflation formula` = c(
    "~ 1",
    "~ healthexcellent",
    sub("^~", "~ ", format(compact_zi_formula)),
    "~ healthpoor + healthexcellent + numchron + age + male + married"
  ),
  `Substantive restriction` = c(
    "Keeps the extra-zero probability constant across participants.",
    "Allows an excellent-health difference, but none by the other predictors; average and poor health share the same fitted probability.",
    "Allows differences by chronic-condition count, sex, and marital status; excludes age and additional self-rated-health differences.",
    "Allows each count predictor its own zero-component association; imposes the fewest restrictions at the cost of three additional coefficients."
  ),
  AIC = vapply(zinb_spec_models, AIC, numeric(1)),
  BIC = vapply(zinb_spec_models, BIC, numeric(1))
) |>
  mutate(
    across(c(AIC, BIC), ~ round(.x, 1))
  ) |>
  arrange(AIC)

knitr::kable(zinb_spec_table, align = c("l", "l", "l", "r", "r"))

The four zero formulas are not all nested, so no single likelihood-ratio test can compare all of them. We therefore use the information-criterion framework introduced above: AIC for relative in-sample comparison and BIC as a stronger parsimony check.

The results give us little reason to carry forward the intercept-only or excellent-health-only models. As the preceding comparison showed, the intercept-only model receives little support beyond ordinary NB2. Allowing the extra-zero probability to vary only with excellent health does no better: its AIC is 24576.0 and its BIC is 24639.9, both higher than the intercept-only model’s values of 24574.4 and 24631.9. A single excellent-health indicator is therefore too narrow to reproduce the improvement seen with the compact formula.

That leaves the compact and shared-adjustment specifications. The shared-adjustment model lowers AIC by only 0.2 points, from 24458.0 to 24457.8, but raises BIC by 18.9 points. I therefore keep the compact formula: the three extra coefficients in the shared-adjustment model make almost no difference to AIC and incur a much larger BIC penalty.

The compact formula was chosen partly after examining adjusted associations with an observed zero in this same dataset, and remains a provisional teaching specification.

Given the results, for the rest of the article, we carry forward ~ numchron + male + married as the provisional compact ZINB working specification. The confidence intervals and p-values reported later treat this formula as if it had been fixed before the analysis. They do not account for the predictor-selection step.

Finally, before relying on these comparisons, it’s good practice to check whether each candidate produced a numerically trustworthy solution. A successful optimizer termination, a positive-definite Hessian, and finite standard errors indicate that the software found a well-defined optimum. Large zero-component coefficients can instead signal a nearly flat likelihood in which the extra-zero probability approaches zero or one.

candidate_glmmtmb_models <- c(
  list(
    NB2 = nb_model,
    `Compact ZIP` = zip_model
  ),
  setNames(
    zinb_spec_models,
    paste(names(zinb_spec_models), "ZINB")
  )
)

check_estimation_stability <- function(model) {
  coefficient_blocks <- summary(model)$coefficients
  standard_errors <- unlist(lapply(
    coefficient_blocks,
    function(block) {
      if (is.null(block) || nrow(block) == 0) {
        return(numeric(0))
      }
      block[, "Std. Error"]
    }
  ))
  zero_coefficients <- unname(fixef(model)$zi)

  # diagnose() prints a short message, so capture it while retaining its result.
  diagnose_message <- capture.output(
    diagnose_ok <- glmmTMB::diagnose(
      model,
      check_zstats = FALSE,
      explain = FALSE
    )
  )

  tibble::tibble(
    optimizer_ok = model$fit$convergence == 0,
    hessian_ok = isTRUE(model$sdr$pdHess),
    finite_se = length(standard_errors) > 0 &&
      all(is.finite(standard_errors)),
    extreme_zero_coefficient = length(zero_coefficients) > 0 &&
      any(abs(zero_coefficients) > 10),
    diagnose_ok = isTRUE(diagnose_ok)
  )
}

stability_checks <- dplyr::bind_rows(
  lapply(candidate_glmmtmb_models, check_estimation_stability),
  .id = "Model"
) |>
  mutate(
    all_checks_pass = optimizer_ok & hessian_ok & finite_se &
      !extreme_zero_coefficient & diagnose_ok
  )

compact_checks <- stability_checks |>
  filter(Model == "Compact ZINB")

stability_status <- bind_rows(
  tibble::tibble(
    Scope = "All comparison candidates",
    Check = "Candidates with one or more flagged checks",
    Result = sprintf(
      "%d of %d",
      sum(!stability_checks$all_checks_pass),
      nrow(stability_checks)
    )
  ),
  tibble::tibble(
    Scope = "Compact ZINB",
    Check = c(
      "Optimizer termination",
      "Positive-definite Hessian",
      "Finite standard errors",
      "Zero-component coefficient magnitude",
      "Supplementary diagnose() checks"
    ),
    Result = c(
      ifelse(compact_checks$optimizer_ok, "Passed", "Flagged"),
      ifelse(compact_checks$hessian_ok, "Passed", "Flagged"),
      ifelse(compact_checks$finite_se, "Passed", "Flagged"),
      ifelse(
        compact_checks$extreme_zero_coefficient,
        "Flagged",
        "No coefficient with |estimate| > 10"
      ),
      ifelse(compact_checks$diagnose_ok, "Passed", "Flagged")
    )
  )
)

knitr::kable(stability_status, align = c("l", "l", "l"))

All six glmmTMB candidates passed these checks, including the compact ZINB shown above. Their likelihood comparisons and coefficient estimates therefore show no evident numerical warning.

5.4 Check Distributional Fit

We now examine how closely the compact ZINB reproduces the observed visit counts. DHARMa simulates outcomes from the fitted model and compares each observation with its simulated distribution. It expresses the result as a scaled residual between 0 and 1; a well-calibrated model should produce residuals that are approximately uniform across that interval.

Three standard checks provide the first overview:

1. Zero frequency: Does the model reproduce the number of observed zeros?
2. Dispersion: Is the observed residual spread consistent with the simulations?
3. Overall uniformity: Does the fitted conditional distribution reproduce the response more broadly?

We begin with zero frequency because it was the specific weakness of the negative-binomial model. NB remains as a benchmark for that comparison; the other diagnostics show what changes when the compact ZINB repairs the zero count. I use 1,000 simulations, rather than DHARMa’s default of 250, to reduce Monte Carlo variation and support the bootstrap outlier check below. A fixed seed makes the results reproducible, and the same simulations supply every diagnostic and figure in this subsection.

nb_residuals <- simulateResiduals(
  fittedModel = nb_model,
  n = 1000,
  seed = 123,
  plot = FALSE
)

zinb_residuals <- simulateResiduals(
  fittedModel = zinb_model,
  n = 1000,
  seed = 123,
  plot = FALSE
)

nb_zero_test <- testZeroInflation(nb_residuals, plot = FALSE)
nb_dispersion_test <- testDispersion(nb_residuals, plot = FALSE)
nb_uniformity_test <- testUniformity(nb_residuals, plot = FALSE)

zinb_zero_test <- testZeroInflation(zinb_residuals, plot = FALSE)
zinb_dispersion_test <- testDispersion(zinb_residuals, plot = FALSE)
zinb_uniformity_test <- testUniformity(zinb_residuals, plot = FALSE)

format_diagnostic <- function(statistic, p_value, digits = 2) {
  p_text <- format.pval(p_value, digits = 2, eps = 0.001)
  p_text <- if (startsWith(p_text, "<")) {
    sub("<", "< ", p_text, fixed = TRUE)
  } else {
    paste("=", p_text)
  }

  sprintf(paste0("%.", digits, "f (p %s)"), statistic, p_text)
}

diagnostic_table <- tibble::tibble(
  Check = c("Zero frequency", "Dispersion", "Overall uniformity"),
  `Statistic shown` = c(
    "Observed/simulated zeros",
    "Observed/simulated spread",
    "KS D (uniformity)"
  ),
  `Negative binomial` = c(
    format_diagnostic(nb_zero_test$statistic, nb_zero_test$p.value),
    format_diagnostic(
      nb_dispersion_test$statistic,
      nb_dispersion_test$p.value
    ),
    format_diagnostic(
      nb_uniformity_test$statistic,
      nb_uniformity_test$p.value,
      digits = 3
    )
  ),
  `Compact ZINB` = c(
    format_diagnostic(zinb_zero_test$statistic, zinb_zero_test$p.value),
    format_diagnostic(
      zinb_dispersion_test$statistic,
      zinb_dispersion_test$p.value
    ),
    format_diagnostic(
      zinb_uniformity_test$statistic,
      zinb_uniformity_test$p.value,
      digits = 3
    )
  )
)

knitr::kable(diagnostic_table, align = c("l", "l", "r", "r"))

The zero-frequency row returns to the discrepancy that motivated the ZINB. The negative-binomial model produces too few zeros: the observed number is 1.10 times its simulated expectation (). For the compact ZINB, the ratio falls to 0.97 (). The extra-zero component therefore brings the overall number of zeros into line with the simulations. The remaining rows broaden the assessment to residual spread and the full residual distribution.

The dispersion row reverses the comparison. NB’s ratio of 1.06 is consistent with its simulations, whereas the ZINB ratio is 1.18 (). The ZINB residual spread differs from its simulated expectation by about 18%, a noticeable as well as statistically significant departure.

Both models also depart from uniformity. The Kolmogorov–Smirnov (KS) statistic is the largest vertical gap between the empirical cumulative distribution of the residuals and the uniform reference distribution. For the compact ZINB, corresponds to a maximum gap of 2.9 percentage points. Its -value of 0.0013 is the probability, under uniform residuals, of obtaining a maximum gap at least this large. The zero component repairs the zero frequency, while the dispersion and uniformity results identify broader discrepancies to investigate.

The standard DHARMa display in Figure 5 brings these results together. The left panel is a uniform QQ plot: residuals from a calibrated model should follow the diagonal. The accompanying labels report the KS uniformity, dispersion, and outlier checks; of these, the KS test corresponds directly to the QQ comparison.

In the right panel, I replace DHARMa’s default quantile regressions with a single smooth line showing how the average residual changes across rank-transformed fitted values. The line should remain close to 0.5 throughout. A red star marks an observation whose count falls below every simulated value or above every simulated value for that participant. The later covariate plots follow the 25th, 50th, and 75th residual quantiles separately.

Show overall residual-plot code

plot(
  zinb_residuals,
  quantreg = FALSE
)

The QQ curve lies close to the diagonal, and the right-panel smoother stays near 0.5 through most of the fitted range. Across the 4,406 observations, the diagnostic statistics pick up small but consistent departures. The next checks trace those departures through the observed count frequencies and covariates.

zinb_outlier_test <- testOutliers(
  zinb_residuals,
  type = "bootstrap",
  nBoot = 1000,
  plot = FALSE
)

outlier_table <- tibble::tibble(
  Quantity = c(
    "Observed counts beyond all simulated values",
    "Expected 95% frequency range under the fitted model",
    "Bootstrap comparison"
  ),
  Result = c(
    sprintf(
      "%d of %s (%.2f%%)",
      unname(zinb_outlier_test$statistic),
      format(
        unname(zinb_outlier_test$parameter),
        big.mark = ",",
        scientific = FALSE
      ),
      100 * unname(zinb_outlier_test$estimate)
    ),
    sprintf(
      "%.2f%%–%.2f%%",
      100 * unname(zinb_outlier_test$conf.int[1]),
      100 * unname(zinb_outlier_test$conf.int[2])
    ),
    paste(
      "p",
      if (zinb_outlier_test$p.value < 0.001) {
        "< 0.001"
      } else {
        paste("=", format.pval(zinb_outlier_test$p.value, digits = 2))
      }
    )
  )
)

knitr::kable(outlier_table, align = c("l", "l"))

We next return from scaled residuals to the visit counts themselves. Figure 6 compares the observed frequency of each count with the range generated by the same simulations, using the earlier NB model as a benchmark.

Show count-frequency check code

summarise_count_frequencies <- function(residuals, model_name) {
  simulated <- residuals$simulatedResponse
  observed <- medcare$ofp
  displayed_counts <- 0:15
  count_labels <- c(as.character(displayed_counts), "16+")

  simulated_counts <- c(
    lapply(displayed_counts, function(value) {
      colSums(simulated == value)
    }),
    list(colSums(simulated >= 16))
  )

  tibble::tibble(
    Model = model_name,
    Count = factor(count_labels, levels = count_labels),
    Observed = c(
      vapply(
        displayed_counts,
        function(value) sum(observed == value),
        numeric(1)
      ),
      sum(observed >= 16)
    ),
    `Simulated median` = vapply(
      simulated_counts,
      median,
      numeric(1)
    ),
    Lower = vapply(
      simulated_counts,
      quantile,
      numeric(1),
      probs = 0.025,
      names = FALSE
    ),
    Upper = vapply(
      simulated_counts,
      quantile,
      numeric(1),
      probs = 0.975,
      names = FALSE
    )
  )
}

count_frequency_check <- bind_rows(
  summarise_count_frequencies(
    nb_residuals,
    "Negative-binomial benchmark"
  ),
  summarise_count_frequencies(
    zinb_residuals,
    "Compact ZINB"
  )
) |>
  mutate(
    Model = factor(
      Model,
      levels = c(
        "Negative-binomial benchmark",
        "Compact ZINB"
      )
    )
  )

ggplot(count_frequency_check, aes(x = Count)) +
  geom_linerange(
    aes(
      ymin = Lower,
      ymax = Upper,
      colour = "95% simulation interval"
    ),
    linewidth = 0.8
  ) +
  geom_point(
    aes(y = `Simulated median`, colour = "Simulated median"),
    shape = 95,
    size = 5
  ) +
  geom_point(
    aes(y = Observed, colour = "Observed frequency"),
    size = 2
  ) +
  facet_wrap(vars(Model), ncol = 1) +
  scale_colour_manual(
    values = c(
      "95% simulation interval" = "#5B8DB8",
      "Simulated median" = "#1F4E79",
      "Observed frequency" = "#B23A33"
    )
  ) +
  labs(
    x = "Physician office visits",
    y = "Number of participants",
    colour = NULL
  ) +
  theme_minimal(base_size = 11) +
  theme(
    legend.position = "bottom",
    panel.grid.minor = element_blank()
  )

The NB panel shows the discrepancy already identified earlier: the observed zero count lies above its simulation interval, while counts of one and two are overproduced. The compact ZINB brings the zero and low-count frequencies much closer to the data. However, this marginal agreement can still conceal conditional problems because it pools participants with different fitted distributions.

To locate those problems, Figure 7 groups the ZINB residuals by age and chronic-condition count. A uniform residual distribution would place the three displayed quantiles on the dashed 0.25, 0.50, and 0.75 reference lines within every group.

Show covariate-residual check code

age_levels <- c(
  "66–69", "70–74", "75–79", "80–84", "85–89", "90+"
)
chronic_levels <- c("0", "1", "2", "3", "4", "5", "6+")

residual_covariates <- tibble::tibble(
  Residual = zinb_residuals$scaledResiduals,
  `Age band` = cut(
    round(medcare$age * 10),
    breaks = c(65, 69, 74, 79, 84, 89, Inf),
    labels = age_levels
  ),
  `Chronic conditions` = factor(
    ifelse(medcare$numchron >= 6, "6+", medcare$numchron),
    levels = chronic_levels
  )
)

age_quantiles <- residual_covariates |>
  group_by(`Age band`) |>
  summarise(
    Q25 = quantile(Residual, 0.25),
    Median = median(Residual),
    Q75 = quantile(Residual, 0.75),
    .groups = "drop"
  ) |>
  rename(Level = `Age band`)

chronic_quantiles <- residual_covariates |>
  group_by(`Chronic conditions`) |>
  summarise(
    Q25 = quantile(Residual, 0.25),
    Median = median(Residual),
    Q75 = quantile(Residual, 0.75),
    .groups = "drop"
  ) |>
  rename(Level = `Chronic conditions`)

to_long_quantiles <- function(data, panel_name) {
  bind_rows(
    transmute(
      data,
      Panel = panel_name,
      Level = as.character(Level),
      Quantile = "25th",
      Value = Q25
    ),
    transmute(
      data,
      Panel = panel_name,
      Level = as.character(Level),
      Quantile = "Median",
      Value = Median
    ),
    transmute(
      data,
      Panel = panel_name,
      Level = as.character(Level),
      Quantile = "75th",
      Value = Q75
    )
  )
}

covariate_quantiles <- bind_rows(
  to_long_quantiles(age_quantiles, "Age band (years)"),
  to_long_quantiles(chronic_quantiles, "Chronic conditions")
) |>
  mutate(
    Level = factor(Level, levels = c(age_levels, chronic_levels)),
    Quantile = factor(
      Quantile,
      levels = c("25th", "Median", "75th")
    )
  )

ggplot(
  covariate_quantiles,
  aes(x = Level, y = Value, colour = Quantile, group = Quantile)
) +
  geom_hline(
    yintercept = c(0.25, 0.5, 0.75),
    linetype = "dashed",
    colour = "#8A8A8A",
    linewidth = 0.5
  ) +
  geom_line(linewidth = 0.8) +
  geom_point(size = 2) +
  facet_wrap(vars(Panel), scales = "free_x", ncol = 2) +
  scale_colour_manual(
    values = c(
      "25th" = "#0072B2",
      "Median" = "#222222",
      "75th" = "#D55E00"
    )
  ) +
  scale_y_continuous(
    limits = c(0, 1),
    breaks = c(0, 0.25, 0.5, 0.75, 1)
  ) +
  labs(
    x = NULL,
    y = "DHARMa scaled-residual quantile",
    colour = "Within-group quantile"
  ) +
  theme_minimal(base_size = 11) +
  theme(
    legend.position = "bottom",
    panel.grid.minor = element_blank()
  )

The age quantiles drift below their references among the oldest participants, rather than continuing the near-flat pattern implied by a single linear age term. Chronic-condition groups also vary, particularly at the low and high ends of the observed range. The plots locate the clearest departures in these parts of the data. Either the count mean, the extra-zero probability, or a combination of the two could produce the patterns.

The remaining binary predictors could hide group-specific departures or suggest an omitted interaction. Figure 8 therefore compares the residual distributions across health status, sex, and marital status. For uniform residuals, each box should have a median near 0.5 and its lower and upper edges near 0.25 and 0.75.

Show grouped-residual check code

health_status <- factor(
  ifelse(
    medcare$healthpoor == 1,
    "Poor",
    ifelse(medcare$healthexcellent == 1, "Excellent", "Average")
  ),
  levels = c("Average", "Poor", "Excellent")
)

group_residuals <- bind_rows(
  tibble::tibble(
    Panel = "Health status",
    Level = as.character(health_status),
    Residual = zinb_residuals$scaledResiduals
  ),
  tibble::tibble(
    Panel = "Sex",
    Level = ifelse(medcare$male == 1, "Male", "Female"),
    Residual = zinb_residuals$scaledResiduals
  ),
  tibble::tibble(
    Panel = "Marital status",
    Level = ifelse(
      medcare$married == 1,
      "Married",
      "Not married"
    ),
    Residual = zinb_residuals$scaledResiduals
  )
) |>
  mutate(
    Panel = factor(
      Panel,
      levels = c("Health status", "Sex", "Marital status")
    ),
    Level = factor(
      Level,
      levels = c(
        "Average", "Poor", "Excellent",
        "Female", "Male",
        "Not married", "Married"
      )
    )
  )

ggplot(group_residuals, aes(x = Level, y = Residual)) +
  geom_hline(
    yintercept = c(0.25, 0.5, 0.75),
    linetype = "dashed",
    colour = "#8A8A8A",
    linewidth = 0.5
  ) +
  geom_boxplot(
    fill = "#D9E8F5",
    colour = "#1F4E79",
    width = 0.62,
    outlier.alpha = 0.15,
    outlier.size = 0.7
  ) +
  facet_wrap(vars(Panel), scales = "free_x", ncol = 3) +
  scale_y_continuous(
    limits = c(0, 1),
    breaks = c(0, 0.25, 0.5, 0.75, 1)
  ) +
  labs(
    x = NULL,
    y = "DHARMa scaled residual"
  ) +
  theme_minimal(base_size = 11) +
  theme(panel.grid.minor = element_blank())

The group distributions are broadly similar, with no particular health, sex, or marital-status interaction standing out as the next addition. The age and chronic-condition patterns offer clearer directions for further work. A fuller analysis could compare substantively motivated nonlinear terms in the count mean, the extra-zero equation, or both, and evaluate them with out-of-sample data or resampling. Adding a quadratic term chosen from these same residual plots would instead combine model development and assessment in one step.

Together, these standard checks show that the compact ZINB reproduces the overall zero and low-count frequencies much better than the negative-binomial benchmark. The remaining departures are meaningful but limited in magnitude: residual spread is about 18% above its simulated expectation, the largest uniformity gap is 2.9 percentage points, and 20 of 4,406 observations (0.45%) fall beyond all their simulated values. The age and chronic-condition plots locate further discrepancies mainly at the ends of their observed ranges. The small p-values make these patterns unlikely to be simulation noise; their absolute size and location determine how much they matter in practice.

These results support a calculated working-model choice, and for the sake of this tutorial, I retain the compact ZINB as an illustrative working model because it is effectively tied for the lowest AIC, has the lowest BIC, repairs the main zero-frequency discrepancy, and directly represents the two-process question of interest. A fuller analysis would compare substantively motivated nonlinear terms, interactions, and alternative specifications, preferably with out-of-sample assessment. The next subsection interprets the fitted components while carrying these diagnostic limitations into the conclusions.

5.5 Interpreting the Zero-Inflated Model

A zero-inflated regression offers two related views of the fitted associations. We first examine and separately to see whether each predictor acts through the count-generating state, the latent extra-zero state, or both. We then recombine the two components on the observed response scale, where their associations may reinforce or offset one another. The first view explains how the model represents the data; the second gives the practical interpretation in expected visits.

count_df <- model_parameters(
  zinb_model,
  component = "conditional",
  effects = "fixed",
  exponentiate = TRUE
) |>
  filter(Parameter != "(Intercept)") |>
  transmute(
    Predictor = recode(
      Parameter,
      healthpoor = "Poor vs average health",
      healthexcellent = "Excellent vs average health",
      numchron = "Chronic conditions (per condition)",
      age = "Age (per decade)",
      male = "Male vs female",
      married = "Married vs not married"
    ),
    `Conditional-mean ratio` = sprintf("%.2f", Coefficient),
    `95% CI` = sprintf("%.2f–%.2f", CI_low, CI_high),
    `p-value` = format.pval(p, digits = 2, eps = 0.001)
  )

knitr::kable(count_df, align = c("l", "r", "c", "r"), row.names = FALSE)

zi_df <- model_parameters(
  zinb_model,
  component = "zero_inflated",
  effects = "fixed",
  exponentiate = TRUE
) |>
  filter(Parameter != "(Intercept)") |>
  transmute(
    Predictor = recode(
      Parameter,
      healthpoor = "Poor vs average health",
      healthexcellent = "Excellent vs average health",
      numchron = "Chronic conditions (per condition)",
      age = "Age (per decade)",
      male = "Male vs female",
      married = "Married vs not married"
    ),
    `Odds ratio` = sprintf("%.2f", Coefficient),
    `95% CI` = sprintf("%.2f–%.2f", CI_low, CI_high),
    `p-value` = format.pval(p, digits = 2, eps = 0.001)
  )

knitr::kable(zi_df, align = c("l", "r", "c", "r"), row.names = FALSE)

average_response_prediction <- function(model, newdata) {
  mean(predict(model, newdata = newdata, type = "response"))
}

# For chronic conditions, compare each eligible participant's observed count
# with that count plus one. Excluding the observed maximum avoids extrapolation.
chronic_reference <- medcare |>
  filter(numchron < max(numchron))
chronic_comparison <- chronic_reference |>
  mutate(numchron = numchron + 1)

# Binary contrasts compare two versions of every record while keeping the
# remaining observed predictors unchanged.
female_scenario <- medcare |>
  mutate(male = 0)
male_scenario <- medcare |>
  mutate(male = 1)
not_married_scenario <- medcare |>
  mutate(married = 0)
married_scenario <- medcare |>
  mutate(married = 1)

estimate_response_contrasts <- function(model) {
  tibble::tibble(
    Contrast_id = c("chronic", "sex", "marriage"),
    Contrast = c(
      "Adding one chronic condition among participants with 0–7 conditions",
      "Male vs female",
      "Married vs not married"
    ),
    Reference = c(
      average_response_prediction(model, chronic_reference),
      average_response_prediction(model, female_scenario),
      average_response_prediction(model, not_married_scenario)
    ),
    Comparison = c(
      average_response_prediction(model, chronic_comparison),
      average_response_prediction(model, male_scenario),
      average_response_prediction(model, married_scenario)
    )
  ) |>
    mutate(
      Difference = Comparison - Reference,
      Ratio = Comparison / Reference
    )
}

overall_point_contrasts <- estimate_response_contrasts(zinb_model)

# bootMer requires a numeric statistic. Stopping here records an unstable refit
# as a bootstrap failure rather than allowing it to enter the intervals.
bootstrap_response_statistic <- function(model) {
  fit_is_stable <-
    identical(model$fit$convergence, 0L) &&
    isTRUE(model$sdr$pdHess)

  if (!fit_is_stable) {
    stop("Bootstrap refit failed the convergence or Hessian check.")
  }

  contrasts <- estimate_response_contrasts(model)
  statistics <- c(contrasts$Difference, contrasts$Ratio)
  names(statistics) <- c(
    paste0(contrasts$Contrast_id, "_difference"),
    paste0(contrasts$Contrast_id, "_ratio")
  )
  statistics
}

n_boot <- 500L
bootstrap_seed <- 20260730L

# Leave one physical core free for Quarto and other desktop work. The fallback
# keeps the tutorial runnable when R cannot determine the physical core count.
physical_cores <- parallel::detectCores(logical = FALSE)
n_workers <- if (is.na(physical_cores)) {
  1L
} else {
  max(1L, physical_cores - 1L)
}

run_bootstrap_refits <- function() {
  bootstrap_cluster <- parallel::makeCluster(n_workers)
  on.exit(parallel::stopCluster(bootstrap_cluster), add = TRUE)

  # Windows uses separate R sessions, so each worker needs the modelling
  # packages and every object used when glmmTMB reconstructs the original fit.
  parallel::clusterEvalQ(
    bootstrap_cluster,
    suppressPackageStartupMessages({
      library(lme4)
      library(glmmTMB)
      library(dplyr)
    })
  )

  parallel::clusterExport(
    bootstrap_cluster,
    c(
      "zinb_model", "medcare",
      "chronic_reference", "chronic_comparison",
      "female_scenario", "male_scenario",
      "not_married_scenario", "married_scenario",
      "average_response_prediction", "estimate_response_contrasts",
      "bootstrap_response_statistic"
    ),
    envir = environment()
  )

  # bootMer simulates outcomes, refits the complete ZINB, and evaluates the six
  # response-scale statistics for every replicate. Each refit uses one thread;
  # independent replicates are distributed across the worker processes.
  lme4::bootMer(
    zinb_model,
    FUN = bootstrap_response_statistic,
    nsim = n_boot,
    seed = bootstrap_seed,
    use.u = FALSE,
    type = "parametric",
    parallel = "snow",
    ncpus = n_workers,
    cl = bootstrap_cluster
  )
}

bootstrap_result <- run_bootstrap_refits()
bootstrap_succeeded <- complete.cases(bootstrap_result$t)
n_boot_success <- sum(bootstrap_succeeded)
n_boot_failed <- n_boot - n_boot_success
bootstrap_failure_messages <- attr(bootstrap_result, "boot.fail.msgs")

# A high failure rate makes percentile intervals unreliable and signals a model
# stability problem that should be investigated rather than silently ignored.
if (n_boot_success < 0.95 * n_boot) {
  stop(
    "Fewer than 95% of parametric-bootstrap models converged with ",
    "a positive-definite Hessian."
  )
}

# Cache only the raw successful statistics and diagnostics. Formatting remains
# in the lightweight table cell so prose or layout edits do not rerun refits.
bootstrap_statistics <- bootstrap_result$t[bootstrap_succeeded, , drop = FALSE]
rm(bootstrap_result)

# Convert the cached bootstrap matrix into percentile intervals. This cell is
# intentionally lightweight: labels, rounding, and layout can change freely.
interval_endpoint <- function(column, probability) {
  quantile(
    bootstrap_statistics[, column],
    probs = probability,
    names = FALSE
  )
}

bootstrap_intervals <- tibble::tibble(
  Contrast_id = overall_point_contrasts$Contrast_id,
  Difference_low = vapply(
    paste0(Contrast_id, "_difference"),
    interval_endpoint,
    numeric(1),
    probability = 0.025
  ),
  Difference_high = vapply(
    paste0(Contrast_id, "_difference"),
    interval_endpoint,
    numeric(1),
    probability = 0.975
  ),
  Ratio_low = vapply(
    paste0(Contrast_id, "_ratio"),
    interval_endpoint,
    numeric(1),
    probability = 0.025
  ),
  Ratio_high = vapply(
    paste0(Contrast_id, "_ratio"),
    interval_endpoint,
    numeric(1),
    probability = 0.975
  )
)

overall_contrasts <- overall_point_contrasts |>
  left_join(bootstrap_intervals, by = "Contrast_id") |>
  transmute(
    Contrast,
    `Reference mean` = sprintf("%.2f", Reference),
    `Comparison mean` = sprintf("%.2f", Comparison),
    `Change (95% CI)` = sprintf(
      "%+.2f (%+.2f to %+.2f)",
      Difference,
      Difference_low,
      Difference_high
    ),
    `Ratio (95% CI)` = sprintf(
      "%.2f (%.2f–%.2f)",
      Ratio,
      Ratio_low,
      Ratio_high
    )
  )

knitr::kable(
  overall_contrasts,
  align = c("c", "c", "c", "r", "r"),
  row.names = FALSE
)

6.1 A Repeatable Workflow

The approach we used for the Medicare data, and that I have used in practice, can be broken down and adapted to other count outcomes:

1. Define the outcome and the processes that may generate it. Establish the observation window or exposure, decide what one count represents, and identify plausible reasons for both variation in the counts and the occurrence of zeros.
2. Establish a count-family baseline. Begin with a simple count model, then use dispersion and residual checks to decide whether the count distribution needs more flexibility. A negative-binomial model is often a useful next candidate when Poisson variation is too restrictive.
3. Test a basic extra-zero component. Within the selected count family, compare the ordinary model with a matched specification containing a constant extra-zero probability. This isolates whether a zero component adds enough to justify further development.
4. Choose zero-component predictors deliberately. Use subject-matter knowledge and direct measures of the proposed zero-generating process whenever possible. When several formulas are plausible, compare a small candidate set with clear substantive interpretations.
5. Check estimation stability. Confirm optimizer convergence, a positive-definite Hessian, finite standard errors, and the absence of extreme zero-component estimates before using fit statistics or residual diagnostics.
6. Check the complete fitted distribution. Use simulation diagnostics to examine zero frequency, dispersion, overall distributional fit, residual patterns across important predictors, and observed versus predicted count frequencies. For predictive work, add repeated out-of-sample assessment with a distribution-sensitive score, such as held-out log predictive density or ranked probability score, alongside RMSE.
7. Interpret the model in terms of the observed outcome. Explain the count and zero components separately, then recombine them on the response scale. Keep the interpretation conditional on the chosen specification, its remaining diagnostic departures, and the scientific meaning of the latent state.

6.2 Choosing an Appropriate Candidate

Given the workflow, you can use the table below to form a candidate set, then use matched comparisons, estimation checks, and simulation diagnostics to decide which specifications remain defensible. Note that the table covers only the four model families discussed in this article; other count distributions and hurdle-type specifications may also be worth considering!

6.3 Scope and Reproducibility

These qualifications define the boundaries of the Medicare example. Age is recorded in decades: 6.6 represents 66 years, as confirmed by the fuller AER::NMES1988 documentation. The available data provide no participant-specific observation times, and the models include no exposure offset. We therefore treat each participant’s visit count as observed over the same one-year period, the 1987 NMES observation year, and compare counts under this common observation window.

Without the original survey-design variables, the estimates describe associations within the supplied sample under model-based uncertainty. They should not be interpreted as design-based population estimates for US Medicare beneficiaries.

6.4 Further Reading

• For the Medicare example and its original economic application, Deb and Trivedi develop a finite-mixture analysis of healthcare use among older adults (Deb and Trivedi 1997).
• For model fitting, Brooks and colleagues explain glmmTMB’s parameterizations and its implementation of zero-inflated generalized linear models (Brooks et al. 2017).
• For model checking, Hartig documents the simulation-based residuals and diagnostic tests provided by DHARMa (Hartig 2026).
• For a broader practical guide in R, Zeileis, Kleiber, and Jackman compare Poisson, negative binomial, hurdle, and zero-inflated regression (Zeileis et al. 2008).
• For the methodological foundations, Lambert introduces zero-inflated Poisson regression, while Mullahy develops modified count models and hurdle-style formulations (Lambert 1992; Mullahy 1986).