Contrasts and comparisons for zero-inflation models

This vignette is the last in a 4-part series:

Contrasts and Pairwise Comparisons
Comparisons of Slopes, Floodlight and Spotlight Analysis (Johnson-Neyman Intervals)
Contrasts and Comparisons for Generalized Linear Models
Contrasts and Comparisons for Zero-Inflation Models

Lastly, we show an example for models with zero-inflation component.

What is a zero-inflated model?

A zero-inflated model is a statistical approach used when dealing with count data that has an excessive number of zero values. Imagine counting something that can be zero, like the number of customers a store gets in a day, and it happens that there are a lot more zeros in the data than a typical count model (e.g., Poisson regression) would expect. That’s where we need zero-inflated regression models. These models consider two ways zeros can happen:

True Zeros: These are days the store is naturally closed, or maybe there’s just no demand for the product.
Counting Zeros: These are days the store is open but just happens to get no customers. Maybe it’s bad luck, or a random fluctuation.

The model treats these differently. It uses one part (the zero-inflation component, a logistic regression) to predict the probability of a true zero, based on things that make the store less likely to be open at all. Then it uses another part (the conditional, or count component, a count regression) to predict the number of customers on days the store is actually open, considering other factors like weather or discounts.

Consequently, such regression models usually have two parts in their formula, or (depending on the package) separate formulas for the count and the zero-inflation components. Adjusted predictions can be calculated for both parts, and contrasts or comparisons can be calculated for both parts, too.

How to choose predictors for zero-inflation models?

The two model parts do not necessarily need to use the same predictors. Therefore, it is not always straightforward to find predictors that can be used in the zero-inflation model. Think about why you have excess zeros in your data. Are they true zeros (inherently no counts) or due to limitations (measurement limitations, biological process, …)? Choose variables that explain why some data points have zero counts even when conditions might allow for some count. For instance, if modeling customer complaints, store location in a remote area might predict zero complaints due to fewer customers.

Zero-inflation models using the glmmTMB package

In the following example, we use the Salamanders dataset from the glmmTMB package.We fit a zero-inflated Poisson regression model to the data, with mined as the predictor variable.

Adjusted predictions using estimate_means() can be made for the different model components:

The conditional, or count component, which predicts the average count of salamanders. This is the default and uses type = "fixed". This would return the predicted mean from the count component only, which is the conditional mean (average counts) of the response only for “counting zeros”. It does not take into account the probability of “true zeros”.
The full model, which predicts the average count of the response, including the zero-inflation component. This would return the expected value of the response for an average observation, which can be a “true zero” or a “count zero”. Use type = "zero_inflation" to calculate adjusted predictions that also take the zero-inflation component into account. For the above example, use this option if you want to predict the average number of customers per week, including days the store is closed.
The zero-inflation probabilities, which predicts the probabilities whether an observation is a “true zero” or not. Use type = "zi_prob" for this. It only related to the zero-inflation component of the model.

library(modelbased)
library(glmmTMB)

data(Salamanders)
m <- glmmTMB(count ~ mined + (1 | site),
  ziformula = ~mined,
  family = poisson(),
  data = Salamanders
)

Contrasts and comparisons for the conditional model

We will start with the conditional mean. For zero-inflated models, the conditional mean is predicted using predict = "conditional". This is the average count of the response, excluding the zero-inflation component.

# predicting the conditional mean
estimate_means(m, "mined", predict = "conditional")
#> Estimated Marginal Means
#> 
#> mined | Mean |   SE |       95% CI |     z
#> ------------------------------------------
#> yes   | 1.12 | 0.26 | [0.61, 1.63] |  4.29
#> no    | 3.51 | 0.32 | [2.89, 4.14] | 11.00
#> 
#> Variable predicted: count
#> Predictors modulated: mined
#> Predictors averaged: site
#> Predictions are on the conditional-scale.

estimate_contrasts(m, "mined", predict = "conditional")
#> Marginal Contrasts Analysis
#> 
#> Level1 | Level2 | Difference |   SE |       95% CI |    z |      p
#> ------------------------------------------------------------------
#> no     | yes    |       2.39 | 0.40 | [1.60, 3.18] | 5.93 | < .001
#> 
#> Variable predicted: count
#> Predictors contrasted: mined
#> Predictors averaged: site
#> p-values are uncorrected.
#> Contrasts are on the conditional-scale.

Contrasts and comparisons for the full model

By default, adjusted predictions are returned for the full model, i.e. the average expected count of the response, including the zero-inflation component.

# predicting the expected value of the response
estimate_means(m, "mined")
#> Estimated Marginal Means
#> 
#> mined | Mean |   SE |       95% CI |     z
#> ------------------------------------------
#> yes   | 0.27 | 0.05 | [0.17, 0.37] |  5.47
#> no    | 2.27 | 0.22 | [1.83, 2.70] | 10.19
#> 
#> Variable predicted: count
#> Predictors modulated: mined
#> Predictors averaged: site
#> Predictions are on the response-scale.

estimate_contrasts(m, "mined")
#> Marginal Contrasts Analysis
#> 
#> Level1 | Level2 | Difference |   SE |       95% CI |    z |      p
#> ------------------------------------------------------------------
#> no     | yes    |       1.99 | 0.23 | [1.55, 2.44] | 8.78 | < .001
#> 
#> Variable predicted: count
#> Predictors contrasted: mined
#> Predictors averaged: site
#> p-values are uncorrected.
#> Contrasts are on the response-scale.

Contrasts and comparisons for the zero-inflation probabilities

If you’re interested in the probabilities of being a “true zero” or not, use predict = "zprob".

# predicting the zero-inflation probabilities
estimate_means(m, "mined", predict = "zprob")
#> Estimated Marginal Means
#> 
#> mined | Probability |   SE |       95% CI |     z
#> -------------------------------------------------
#> yes   |        0.76 | 0.04 | [0.67, 0.84] | 17.54
#> no    |        0.36 | 0.03 | [0.30, 0.41] | 12.75
#> 
#> Variable predicted: count
#> Predictors modulated: mined
#> Predictors averaged: site
#> Predictions are on the zprob-scale.

estimate_contrasts(m, "mined", predict = "zprob")
#> Marginal Contrasts Analysis
#> 
#> Level1 | Level2 | Difference |   SE |         95% CI |     z |      p
#> ---------------------------------------------------------------------
#> no     | yes    |      -0.40 | 0.05 | [-0.50, -0.30] | -7.92 | < .001
#> 
#> Variable predicted: count
#> Predictors contrasted: mined
#> Predictors averaged: site
#> p-values are uncorrected.
#> Contrasts are on the zprob-scale.