25  Multinomial Logistic Regression, Ordinal Regression, and Poisson Regression

“It’s easy to lie with statistics; it is easier to lie without them.” - Frederick Mosteller

The previous chapters focused on binary logistic regression. In this chapter, we briefly introduce three related models for categorical and count responses:

These models extend the regression idea beyond quantitative responses while still using predictors to explain or predict an outcome.

25.1 Nominal Response

Logistic regression is most frequently used when the response variable has two categories. Sometimes, however, the response variable has more than two categories.

If those categories have no natural ordering, the response is nominal. For example, a consumer’s product choice among product A, product B, and product C is nominal because the categories are names rather than ordered levels.

Multinomial logistic regression models the probability of each response category by comparing each non-baseline category to a chosen baseline category.

NoteNominal versus ordinal responses
Response type Example Appropriate model
Nominal Product A, product B, product C Multinomial logistic regression
Ordinal Mild, moderate, severe Ordinal logistic regression
Count Number of store visits Poisson regression

The distinction matters because nominal models ignore ordering, ordinal models use ordering, and Poisson models are for nonnegative integer counts.

25.2 Multinomial Logistic Regression

For a response with \(J\) categories, multinomial logistic regression chooses one category as the baseline. The model then compares each other category to that baseline.

For example, if category 1 is the baseline, then the model fits log-odds equations such as

\[ \log\left(\frac{P(Y=2)}{P(Y=1)}\right) = \beta_{20}+\beta_{21}x_1+\cdots+\beta_{2,p-1}x_{p-1} \]

and

\[ \log\left(\frac{P(Y=3)}{P(Y=1)}\right) = \beta_{30}+\beta_{31}x_1+\cdots+\beta_{3,p-1}x_{p-1}. \]

Each non-baseline category gets its own set of coefficients.

Example 25.1  

ExamplePregnancy duration data

A study was undertaken to examine the association between several risk factors and pregnancy duration.

The response variable has three categories:

Code Pregnancy duration category
1 Preterm: less than 36 weeks
2 Intermediate: 36 to 37 weeks
3 Full term: 38 weeks or greater

The predictors are:

Variable Meaning
\(X_1\) Nutritional status; higher scores indicate better nutritional status
\(X_2\) Indicator for age less than or equal to 20
\(X_3\) Indicator for age greater than 30
\(X_4\) Indicator for alcohol history
\(X_5\) Indicator for smoking history

The age categories are represented by two indicator variables:

Age category \(X_2\) \(X_3\)
Less than or equal to 20 1 0
21 to 30 0 0
Greater than 30 0 1

Thus, age 21 to 30 is the reference group.

Example 25.2  

ExampleLoading and preparing the pregnancy data

We first read the data, assign useful variable names, and convert the response to a factor.

library(tidyverse)
library(tidymodels)
library(poissonreg)

tidymodels_prefer()

pregnancy_data <- read.table("CH14TA13.txt")

names(pregnancy_data) <- c(
  "id", "Y", "Y1", "Y2", "Y3",
  "X1", "X2", "X3", "X4", "X5"
)

pregnancy_data <- pregnancy_data |>
  mutate(
    Y = factor(
      Y,
      levels = c(1, 2, 3),
      labels = c("Preterm", "Intermediate", "FullTerm")
    )
  ) |>
  select(-id, -Y1, -Y2, -Y3)

pregnancy_data |>
  count(Y) |>
  knitr::kable()
Y n
Preterm 26
Intermediate 35
FullTerm 41

A few targeted plots are often easier to interpret than a large scatterplot matrix. We begin with the distribution of the response categories.

pregnancy_data |>
  count(Y) |>
  ggplot(aes(x = Y, y = n, fill = Y)) +
  geom_col(show.legend = FALSE) +
  labs(
    x = "Pregnancy duration category",
    y = "Number of observations",
    title = "Pregnancy duration category counts"
  ) +
  theme_minimal()

Bar chart showing the number of observations in each pregnancy duration category.

Next, we examine nutritional status by pregnancy duration category.

pregnancy_data |>
  ggplot(aes(x = Y, y = X1, color = Y)) +
  geom_boxplot(outlier.shape = NA, alpha = 0.15) +
  geom_jitter(width = 0.12, height = 0, alpha = 0.7) +
  labs(
    x = "Pregnancy duration category",
    y = "Nutritional status",
    color = "Category",
    title = "Nutritional status by pregnancy duration"
  ) +
  theme_minimal()

Boxplots and jittered points showing nutritional status by pregnancy duration category.

Finally, we look at the observed category proportions for the indicator predictors.

pregnancy_indicator_proportions <- pregnancy_data |>
  select(Y, X2, X3, X4, X5) |>
  pivot_longer(
    cols = X2:X5,
    names_to = "predictor",
    values_to = "indicator_value"
  ) |>
  mutate(
    predictor = recode(
      predictor,
      X2 = "Age <= 20",
      X3 = "Age > 30",
      X4 = "Alcohol history",
      X5 = "Smoking history"
    ),
    indicator_value = factor(indicator_value, levels = c(0, 1))
  ) |>
  count(predictor, indicator_value, Y) |>
  group_by(predictor, indicator_value) |>
  mutate(proportion = n / sum(n)) |>
  ungroup()

pregnancy_indicator_proportions |>
  ggplot(aes(x = indicator_value, y = proportion, fill = Y)) +
  geom_col() +
  facet_wrap(~ predictor) +
  scale_y_continuous(labels = scales::label_percent()) +
  labs(
    x = "Indicator value",
    y = "Observed category proportion",
    fill = "Category",
    title = "Pregnancy duration proportions by indicator value"
  ) +
  theme_minimal()

Faceted bar charts showing observed pregnancy duration proportions by binary predictor values.

Example 25.3  

ExampleFitting a multinomial logistic regression model

We can fit a multinomial logistic regression model using a tidymodels workflow.

pregnancy_recipe <- recipe(Y ~ X1 + X2 + X3 + X4 + X5, data = pregnancy_data)

pregnancy_model <- multinom_reg() |>
  set_engine("nnet")

pregnancy_workflow <- workflow() |>
  add_recipe(pregnancy_recipe) |>
  add_model(pregnancy_model)

pregnancy_fit <- pregnancy_workflow |>
  fit(data = pregnancy_data)

Because Preterm is the baseline category, the model fits one comparison for Intermediate versus Preterm and another for FullTerm versus Preterm.

The coefficients are on the log-odds scale.

pregnancy_fit |>
  extract_fit_engine() |>
  coef() |>
  knitr::kable(digits = 4)
(Intercept) X1 X2 X3 X4 X5
Intermediate -1.5170 0.0190 -0.0436 -0.1722 -0.9759 -0.2219
FullTerm -5.4754 0.0654 -2.9569 -2.0596 -2.0429 -2.4523

Exponentiating the coefficients gives odds ratios for each comparison.

pregnancy_fit |>
  extract_fit_engine() |>
  coef() |>
  exp() |>
  knitr::kable(digits = 4)
(Intercept) X1 X2 X3 X4 X5
Intermediate 0.2194 1.0192 0.9574 0.8418 0.3769 0.8010
FullTerm 0.0042 1.0676 0.0520 0.1275 0.1297 0.0861

For example, the coefficient for X1 in the FullTerm row describes the change in the log-odds of being full term rather than preterm for a one-unit increase in nutritional status, holding the other predictors fixed.

Example 25.4  

ExamplePredicted probabilities and classes from a multinomial model

A multinomial model gives one predicted probability for each response category. A common classification rule is to predict the category with the largest fitted probability.

pregnancy_predictions <- pregnancy_data |>
  select(Y) |>
  bind_cols(
    predict(pregnancy_fit, new_data = pregnancy_data, type = "prob"),
    predict(pregnancy_fit, new_data = pregnancy_data, type = "class")
  )

pregnancy_predictions |>
  slice_head(n = 10) |>
  knitr::kable(digits = 3)
Y .pred_Preterm .pred_Intermediate .pred_FullTerm .pred_class
Preterm 0.094 0.285 0.621 FullTerm
Preterm 0.254 0.561 0.185 Intermediate
Preterm 0.220 0.437 0.343 Intermediate
Preterm 0.335 0.638 0.027 Intermediate
Preterm 0.475 0.392 0.133 Preterm
Preterm 0.497 0.388 0.115 Preterm
Preterm 0.010 0.039 0.951 FullTerm
Preterm 0.335 0.567 0.097 Intermediate
Preterm 0.659 0.314 0.028 Preterm
Preterm 0.187 0.409 0.404 Intermediate

We can compare predicted classes to observed classes with a confusion matrix.

pregnancy_confusion <- conf_mat(
  pregnancy_predictions,
  truth = Y,
  estimate = .pred_class
)

pregnancy_confusion
              Truth
Prediction     Preterm Intermediate FullTerm
  Preterm           12            4        4
  Intermediate      10           23        5
  FullTerm           4            8       32
bind_rows(
  accuracy(pregnancy_predictions, truth = Y, estimate = .pred_class),
  mn_log_loss(
    pregnancy_predictions,
    truth = Y,
    .pred_Preterm,
    .pred_Intermediate,
    .pred_FullTerm
  )
) |>
  knitr::kable(digits = 3)
.metric .estimator .estimate
accuracy multiclass 0.657
mn_log_loss multiclass 0.827

Accuracy uses only the predicted class. Multinomial log loss uses the full set of predicted probabilities, so it rewards confident correct predictions and penalizes confident wrong predictions.

25.3 Ordinal Regression

25.3.1 Ordinal Response

When the response categories can be put into a meaningful order, the response is ordinal. Pregnancy duration can be treated as ordinal because

\[ \text{Preterm} < \text{Intermediate} < \text{FullTerm}. \]

Multinomial logistic regression can still be used, but it ignores the ordering. Ordinal logistic regression uses the ordering directly. A common ordinal logistic model is the proportional odds model, which models cumulative probabilities such as

\[ P(Y_i \le j) \]

rather than modeling only the separate category probabilities \(P(Y_i=j)\).

NoteWhy use an ordinal model?

An ordinal model can be more parsimonious because it uses the category ordering. Instead of fitting a completely separate coefficient vector for each non-baseline category, the proportional odds model uses one coefficient for each predictor and separate cutpoints between adjacent response categories.

25.3.2 Proportional Odds Assumption

The proportional odds model relies on an important assumption: each predictor has the same effect across the cumulative splits of the ordered response.

For the pregnancy duration categories, the cumulative splits are:

Split Comparison
1 Preterm versus Intermediate or FullTerm
2 Preterm or Intermediate versus FullTerm

The proportional odds assumption says that a predictor’s slope is the same for these cumulative comparisons, up to the sign convention used by the software. The cutpoints may differ, but the predictor effects are shared.

NoteParallel slopes idea

Another way to say this is that the model assumes parallel slopes on the cumulative logit scale.

For example, if nutritional status is associated with longer pregnancy duration, the proportional odds model assumes that this association works similarly when separating Preterm from higher categories and when separating FullTerm from lower categories.

WarningWhen the assumption may be questionable

The proportional odds assumption may be questionable if a predictor has a very different relationship with different category cutpoints.

For example, if smoking history strongly separates Preterm from the other categories but has little relationship to whether a pregnancy is FullTerm, then one shared slope may be too restrictive. In that case, a multinomial model or a more flexible ordinal model may be more appropriate.

Example 25.5  

ExampleFitting a proportional odds model

We can fit a proportional odds model using MASS::polr(). The response must be stored as an ordered factor.

ordinal_data <- pregnancy_data |>
  mutate(
    Y = ordered(
      Y,
      levels = c("Preterm", "Intermediate", "FullTerm")
    )
  )

ordinal_fit <- MASS::polr(
  Y ~ X1 + X2 + X3 + X4 + X5,
  data = ordinal_data,
  Hess = TRUE
)

broom::tidy(
  ordinal_fit,
  exponentiate = TRUE,
  conf.int = TRUE
) |>
  filter(coef.type == "coefficient") |>
  select(term, estimate, conf.low, conf.high, statistic) |>
  knitr::kable(digits = 4)
term estimate conf.low conf.high statistic
X1 1.0501 1.0274 1.0764 4.1332
X2 0.1386 0.0430 0.4164 -3.4296
X3 0.2558 0.0850 0.7348 -2.4950
X4 0.1883 0.0718 0.4669 -3.5128
X5 0.2036 0.0813 0.4815 -3.5239

The exponentiated coefficients are cumulative odds ratios under the proportional odds model. Their interpretation depends on the direction of the cumulative logits used by the software, so always check the model convention before writing a final applied interpretation.

The model also estimates cutpoints between adjacent ordered categories.

broom::tidy(ordinal_fit) |>
  filter(coef.type == "scale") |>
  knitr::kable(digits = 4)
term estimate std.error statistic coef.type
Preterm|Intermediate 2.9301 1.4929 1.9627 scale
Intermediate|FullTerm 5.0249 1.5445 3.2535 scale

Example 25.6  

ExamplePredictions from an ordinal logistic regression model

Like the multinomial model, the ordinal model can return one predicted probability for each category.

ordinal_probabilities <- predict(
  ordinal_fit,
  newdata = ordinal_data,
  type = "probs"
) |>
  as_tibble()

ordinal_predictions <- ordinal_data |>
  select(Y) |>
  bind_cols(
    ordinal_probabilities,
    tibble(
      .pred_class = predict(
        ordinal_fit,
        newdata = ordinal_data,
        type = "class"
      )
    )
  )

ordinal_predictions |>
  slice_head(n = 10) |>
  knitr::kable(digits = 3)
Y Preterm Intermediate FullTerm .pred_class
Preterm 0.057 0.272 0.671 FullTerm
Preterm 0.240 0.480 0.281 Intermediate
Preterm 0.150 0.439 0.411 Intermediate
Preterm 0.560 0.352 0.088 Preterm
Preterm 0.423 0.433 0.144 Intermediate
Preterm 0.460 0.414 0.126 Preterm
Preterm 0.010 0.066 0.924 FullTerm
Preterm 0.385 0.451 0.165 Intermediate
Preterm 0.752 0.209 0.039 Preterm
Preterm 0.121 0.408 0.471 FullTerm 
ordinal_confusion <- conf_mat(
  ordinal_predictions,
  truth = Y,
  estimate = .pred_class
)

ordinal_confusion
              Truth
Prediction     Preterm Intermediate FullTerm
  Preterm           13            7        1
  Intermediate       8           21        7
  FullTerm           5            7       33
accuracy(ordinal_predictions, truth = Y, estimate = .pred_class) |>
  knitr::kable(digits = 3)
.metric .estimator .estimate
accuracy multiclass 0.657

The ordinal model and multinomial model have the same in-sample accuracy here, but they use different assumptions. The ordinal model uses the response ordering; the multinomial model treats the categories as unordered.

25.4 Poisson Regression

Poisson regression is used when the response variable is a count:

\[ Y_i = 0,1,2,\ldots \]

Examples include the number of store visits, number of hospitalizations, number of defects, or number of events in a fixed time period.

25.4.1 The Poisson Distribution

The Poisson probability distribution is

\[ f(Y) = \frac{\mu^Y \exp(-\mu)}{Y!}. \]

For a Poisson random variable,

\[ \begin{align*} E(Y) &= \mu,\\ Var(Y) &= \mu. \end{align*} \]

The mean and variance are the same. This is an important assumption to remember when using Poisson regression.

NoteCounts are not just another continuous response

A count response is nonnegative and discrete. A linear model can predict negative fitted values, but a Poisson model with a log link keeps the fitted mean positive.

25.4.2 Poisson Regression Model

In Poisson regression, we usually assume

\[ Y_i \sim \text{Poisson}(\mu_i) \]

and use the log link

\[ \log(\mu_i)=\mathbf{x}_i^\prime\boldsymbol{\beta}. \]

Equivalently,

\[ \mu_i=\exp(\mathbf{x}_i^\prime\boldsymbol{\beta}). \]

This guarantees that the fitted mean count is positive.

Example 25.7  

ExamplePoisson regression for store customer counts

The Miller Lumber Company is a large retailer of lumber, paint, plumbing, electrical supplies, and other household supplies.

During a representative two-week period, in-store surveys were conducted and addresses of customers were obtained. The addresses were used to identify the metropolitan-area census tracts in which the customers lived. At the end of the survey period, the total number of customers who visited the store from each census tract was recorded.

The response is:

  • \(Y\): number of customers who visited the store from the census tract.

The predictors are:

Variable Meaning
\(X_1\) Number of housing units
\(X_2\) Average income, in dollars
\(X_3\) Average housing-unit age, in years
\(X_4\) Distance to nearest competitor, in miles
\(X_5\) Distance to store, in miles 
lumber_data <- read.table("CH14TA14.txt")

names(lumber_data) <- c("Y", "X1", "X2", "X3", "X4", "X5")

lumber_data |>
  slice_head(n = 10) |>
  knitr::kable(digits = 3)
Y X1 X2 X3 X4 X5
9 606 41393 3 3.04 6.32
6 641 23635 18 1.95 8.89
28 505 55475 27 6.54 2.05
11 866 64646 31 1.67 5.81
4 599 31972 7 0.72 8.11
4 520 41755 23 2.24 6.81
0 354 46014 26 0.77 9.27
14 483 34626 1 3.51 7.92
16 1034 85207 13 4.23 4.40
13 456 33021 32 3.07 6.03

Example 25.8  

ExampleFitting and interpreting a Poisson regression model 
lumber_recipe <- recipe(Y ~ ., data = lumber_data)

lumber_model <- poisson_reg() |>
  set_engine("glm")

lumber_workflow <- workflow() |>
  add_recipe(lumber_recipe) |>
  add_model(lumber_model)

lumber_fit <- lumber_workflow |>
  fit(data = lumber_data)

lumber_fit |>
  tidy() |>
  knitr::kable(digits = 4)
term estimate std.error statistic p.value
(Intercept) 2.9424 0.2072 14.1977 0.0000
X1 0.0006 0.0001 4.2623 0.0000
X2 0.0000 0.0000 -5.5340 0.0000
X3 -0.0037 0.0018 -2.0913 0.0365
X4 0.1684 0.0258 6.5343 0.0000
X5 -0.1288 0.0162 -7.9481 0.0000

The coefficients are on the log-mean scale. Exponentiating them gives multiplicative effects on the expected count.

lumber_fit |>
  tidy(exponentiate = TRUE) |>
  knitr::kable(digits = 4)
term estimate std.error statistic p.value
(Intercept) 18.9620 0.2072 14.1977 0.0000
X1 1.0006 0.0001 4.2623 0.0000
X2 1.0000 0.0000 -5.5340 0.0000
X3 0.9963 0.0018 -2.0913 0.0365
X4 1.1834 0.0258 6.5343 0.0000
X5 0.8792 0.0162 -7.9481 0.0000

For example, if an exponentiated coefficient is 1.18, then a one-unit increase in that predictor is associated with multiplying the expected count by 1.18, holding the other predictors fixed.

Example 25.9  

ExamplePredicted counts from a Poisson regression model 
lumber_predictions <- lumber_data |>
  select(Y) |>
  bind_cols(
    predict(lumber_fit, new_data = lumber_data, type = "numeric")
  )

lumber_predictions |>
  slice_head(n = 10) |>
  knitr::kable(digits = 3)
Y .pred
9 12.338
6 8.767
28 28.126
11 8.407
4 7.261
4 8.882
0 4.298
14 10.999
16 14.444
13 11.634 
lumber_predictions |>
  metrics(truth = Y, estimate = .pred) |>
  knitr::kable(digits = 3)
.metric .estimator .estimate
rmse standard 3.236
rsq standard 0.762
mae standard 2.482 
lumber_predictions |>
  ggplot(aes(x = .pred, y = Y)) +
  geom_point(alpha = 0.75) +
  geom_abline(slope = 1, intercept = 0, linetype = "dashed") +
  labs(
    x = "Predicted count",
    y = "Observed count",
    title = "Observed versus predicted customer counts"
  ) +
  theme_minimal()

Observed customer counts plotted against predicted customer counts from a Poisson regression model.

If the model predicted every count perfectly, all points would fall on the dashed line. In practice, count data are noisy, so we look for whether the predictions are reasonably aligned with the observed counts.

25.5 Recap

In this chapter, we introduced three regression models for non-continuous response variables.

Idea Meaning
Nominal response A categorical response whose categories have no natural order.
Multinomial logistic regression A model for nominal responses with more than two categories.
Baseline category The category used as the comparison group in a multinomial logistic model.
Category-specific coefficients Separate coefficients for each non-baseline category comparison.
Multinomial predicted probabilities One fitted probability for each response category.
Multinomial log loss A metric that evaluates the full set of predicted category probabilities.
Ordinal response A categorical response whose categories have a meaningful order.
Proportional odds model An ordinal logistic model based on cumulative probabilities.
Proportional odds assumption The assumption that predictor effects are shared across cumulative category splits.
Parallel slopes A way to describe the proportional odds assumption on the cumulative logit scale.
Cutpoints Thresholds that separate adjacent ordered response categories.
Targeted exploratory plot A focused plot designed to show one important predictor-response relationship clearly.
Count response A nonnegative integer response variable.
Poisson distribution A count distribution where the mean and variance are both \(\mu\).
Poisson regression A regression model for count responses.
Log link A link function that models \(\log(\mu_i)\) as a linear predictor.
Mean-count ratio The exponentiated Poisson coefficient, interpreted as a multiplicative effect on the expected count.
Observed versus predicted plot A plot used to compare observed counts to fitted counts.

25.6 Check your understanding

NoteProblems

  1. What is the difference between a nominal response and an ordinal response?

  2. Why is multinomial logistic regression appropriate for a response with more than two unordered categories?

  3. What is the baseline category in a multinomial logistic regression model?

  4. If a response has three categories, how many non-baseline log-odds equations are fit?

  5. In the pregnancy duration example, what does it mean that Preterm is the baseline category?

  6. Why does a multinomial model return several predicted probabilities for each observation?

  7. How is the predicted class usually chosen from multinomial predicted probabilities?

  8. Why can accuracy miss information that multinomial log loss uses?

  9. Why might an ordinal model be preferred over a multinomial model for ordered categories?

  10. What are cumulative probabilities in the proportional odds model?

  11. What are cutpoints in an ordinal logistic regression model?

  12. What does the proportional odds assumption say?

  13. What are the two cumulative splits in the pregnancy duration example?

  14. What does the phrase parallel slopes mean in the proportional odds model?

  15. Why should we be careful when interpreting exponentiated coefficients from polr()?

  16. Why can targeted plots be better than one large scatterplot matrix?

  17. What kinds of response variables are appropriate for Poisson regression?

  18. Why does Poisson regression often use a log link?

  19. What does it mean that the Poisson mean and variance are both \(\mu\)?

  20. How do we interpret an exponentiated coefficient in Poisson regression?

  21. Why is a plot of observed versus predicted counts useful?

  22. Why should the multinomial, ordinal, and Poisson models not be treated as interchangeable?

  1. Nominal categories have no natural order; ordinal categories do. Product choice is nominal, while severity levels such as mild, moderate, and severe are ordinal.

  2. It models multiple category probabilities. Multinomial logistic regression compares each non-baseline category to a baseline category.

  3. It is the reference category. The model expresses log-odds for each other category relative to the baseline.

  4. Two equations are fit. With three categories and one baseline category, there are two non-baseline comparisons.

  5. The other categories are compared to preterm. The Intermediate and FullTerm coefficient rows describe log-odds relative to Preterm.

  6. There is one probability for each possible category. The probabilities should sum to 1 for each observation.

  7. Choose the category with the largest predicted probability. This is the maximum-probability classification rule.

  8. Accuracy uses only the final class label. Log loss also uses how confident the predicted probabilities were.

  9. It uses the ordering. An ordinal model can be more parsimonious and more appropriate when the response categories have a meaningful order.

  10. They are probabilities of being at or below a category. For example, \(P(Y \le \text{Intermediate})\) is cumulative.

  11. They separate adjacent ordered categories. Cutpoints locate the boundaries between response levels on the model scale.

  12. It says predictor effects are shared across cumulative splits. The model uses the same slope for each predictor when comparing lower versus higher ordered response categories.

  13. The splits are Preterm versus higher categories and lower categories versus FullTerm. More specifically: Preterm versus Intermediate or FullTerm, and Preterm or Intermediate versus FullTerm.

  14. The predictor slopes are the same across cutpoints. The cutpoints can shift, but the model does not estimate separate predictor effects for each cumulative split.

  15. Software conventions differ. The sign and direction of the cumulative logit determine how an exponentiated coefficient should be described.

  16. They focus attention. A large matrix can be useful for scanning, but smaller targeted plots make important relationships easier to see and explain.

  17. Nonnegative integer counts. Examples include number of visits, number of defects, or number of events.

  18. It keeps fitted means positive. Modeling \(\log(\mu_i)\) linearly implies \(\mu_i=\exp(\mathbf{x}_i^\prime\boldsymbol{\beta})>0\).

  19. The variance increases with the mean. Larger expected counts also imply larger variability under the Poisson model.

  20. It is a multiplicative effect on the expected count. Values above 1 increase the expected count; values below 1 decrease it.

  21. It shows how well the fitted counts align with the observed counts. Large departures from the diagonal can reveal poor fit or unusual observations.

  22. They are designed for different response structures. Nominal categories, ordered categories, and counts require different model assumptions and interpretations.