# Zero-inflated negative binomial regression

Zero-inflated negative binomial regression is for modeling count variables with excessive zeros and it is usually for overdispersed count outcome variables. Furthermore, theory suggests that the excess zeros are generated by a separate process from the count values and that the excess zeros can be modeled independently.

require(pscl)

## Loading required package: pscl
## Classes and Methods for R developed in the
##
## Political Science Computational Laboratory
##
## Department of Political Science
##
## Stanford University
##
## Simon Jackman
##
## hurdle and zeroinfl functions by Achim Zeileis

library(MASS)
library(boot)
library(ggplot2)


## Examples of Zero-Inflated negative binomial regression

• Example 1. School administrators study the attendance behavior of high school juniors at two schools. Predictors of the number of days of absence include gender of the student and standardized test scores in math and language arts.

• Example 2. The state wildlife biologists want to model how many fish are being caught by fishermen at a state park. Visitors are asked how long they stayed, how many people were in the group, were there children in the group and how many fish were caught. Some visitors do not fish, but there is no data on whether a person fished or not. Some visitors who did fish did not catch any fish so there are excess zeros in the data because of the people that did not fish.

## Theoretical background

### ZINB Regression

A zero-inflated model assumes that zero outcome is due to two different processes. For instance, in the example of fishing presented here, the two processes are that a subject has gone fishing vs. not gone fishing. If not gone fishing, the only outcome possible is zero. If gone fishing, it is then a count process. The two parts of the a zero-inflated model are a binary model, usually a logit model to model which of the two processes the zero outcome is associated with and a count model, in this case, a negative binomial model, to model the count process.

The zero-inflated negative binomial (ZINB) model is based on the negative binomial model with quadratic variance function (p=2), i.e., NB2. The ZINB model is obtained by specifying a negative binomial distribution for the data generation process referred to earlier as Process 2: $g(y_i)=\frac{\Gamma(y_i+\alpha^{-1})}{y_i!\Gamma(\alpha^{-1})}\left(\frac{\alpha^{-1}}{\alpha^{-1}+\mu_i}\right)^{\alpha^{-1}}\left(\frac{\mu_i}{\alpha^{-1}+\mu_i}\right)^{y_i},\quad y_i=0,1,2,\ldots$ Thus the ZINB model is defined to be \begin{align*} \mathsf{P}[Y_i=0|{\bf x}_i,{\bf z}_i]&=F({\bf z}_i^{\top}{\boldsymbol\gamma})+(1-F({\bf z}_i^{\top}{\boldsymbol\gamma}))(1+\alpha\mu_i)^{-\alpha^{-1}},\\ \mathsf{P}[Y_i|{\bf x}_i,{\bf z}_i]&=(1-F({\bf z}_i^{\top}{\boldsymbol\gamma}))\frac{\Gamma(y_i+\alpha^{-1})}{y_i!\Gamma(\alpha^{-1})}\left(\frac{\alpha^{-1}}{\alpha^{-1}+\mu_i}\right)^{\alpha^{-1}}\left(\frac{\mu_i}{\alpha^{-1}+\mu_i}\right)^{y_i},\quad y_i>0. \end{align*} In this case, the conditional expectation and conditional variance of $$Y_i$$ are \begin{align*} \mathsf{E}[Y_i|{\bf x}_i,{\bf z}_i]&=\mu_i(1-F({\bf z}_i^{\top}{\boldsymbol\gamma})),\\ \mathsf{Var}[Y_i|{\bf x}_i,{\bf z}_i]&=\mathsf{E}[Y_i|{\bf x}_i,{\bf z}_i](1+\mu_i(F({\bf z}_i^{\top}{\boldsymbol\gamma}+\alpha))). \end{align*} As with the ZIP model, the ZINB model exhibits overdispersion because the conditional variance exceeds the conditional mean.

Finally, note that R does not estimate $$\alpha$$ but $$\theta$$, the inverse of $$\alpha$$.

# Wildlife fish data revised

Let's pursue Example 2 from above.

We have data on 250 groups that went to a park. Each group was questioned about how many fish they caught (count), how many children were in the group (child), how many people were in the group (persons), and whether or not they brought a camper to the park (camper).

In addition to predicting the number of fish caught, there is interest in predicting the existence of excess zeros, i.e., the probability that a group caught zero fish. We will use the variables child, persons, and camper in our model. Let's look at the data.

zinb <- read.csv("http://www.karlin.mff.cuni.cz/~pesta/prednasky/NMFM404/Data/fish.csv")
zinb <- within(zinb, {
nofish <- factor(nofish)
livebait <- factor(livebait)
camper <- factor(camper)
})

summary(zinb)

##  nofish  livebait camper     persons          child
##  0:176   0: 34    0:103   Min.   :1.000   Min.   :0.000
##  1: 74   1:216    1:147   1st Qu.:2.000   1st Qu.:0.000
##                           Median :2.000   Median :0.000
##                           Mean   :2.528   Mean   :0.684
##                           3rd Qu.:4.000   3rd Qu.:1.000
##                           Max.   :4.000   Max.   :3.000
##        xb                  zg              count
##  Min.   :-3.275050   Min.   :-5.6259   Min.   :  0.000
##  1st Qu.: 0.008267   1st Qu.:-1.2527   1st Qu.:  0.000
##  Median : 0.954550   Median : 0.6051   Median :  0.000
##  Mean   : 0.973796   Mean   : 0.2523   Mean   :  3.296
##  3rd Qu.: 1.963855   3rd Qu.: 1.9932   3rd Qu.:  2.000
##  Max.   : 5.352674   Max.   : 4.2632   Max.   :149.000


## histogram with x axis in log10 scale
ggplot(zinb, aes(count, fill = camper)) +
geom_histogram() +
scale_x_log10() +
facet_grid(camper ~ ., margins=TRUE, scales="free_y")

## stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
## stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
## stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.


## Analysis methods you might consider

Before we show how you can analyze this with a zero-inflated negative binomial analysis, let's consider some other methods that you might use.

• OLS Regression - You could try to analyze these data using OLS regression. However, count data are highly non-normal and are not well estimated by OLS regression.
• Zero-inflated Poisson Regression - Zero-inflated Poisson regression does better when the data is not overdispersed, i.e. when variance is not much larger than the mean.
• Ordinary Count Models - Poisson or negative binomial models might be more appropriate if there are not excess zeros.

# Zero-inflated NB model

Now let's build up our model. We are going to use the variables child and camper to model the count in the part of negative binomial model and the variable persons in the logit part of the model. We use the pscl to run a zero-inflated negative binomial regression. We begin by estimating the model with the variables of interest.

m1 <- zeroinfl(count ~ child + camper | persons, data = zinb, dist = "negbin",
EM = TRUE)
summary(m1)

##
## Call:
## zeroinfl(formula = count ~ child + camper | persons, data = zinb,
##     dist = "negbin", EM = TRUE)
##
## Pearson residuals:
##     Min      1Q  Median      3Q     Max
## -0.5861 -0.4617 -0.3886 -0.1974 18.0129
##
## Count model coefficients (negbin with log link):
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept)   1.3711     0.2561   5.353 8.63e-08 ***
## child        -1.5152     0.1956  -7.747 9.42e-15 ***
## camper1       0.8790     0.2693   3.264   0.0011 **
## Log(theta)   -0.9854     0.1759  -5.600 2.14e-08 ***
##
## Zero-inflation model coefficients (binomial with logit link):
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept)   1.6028     0.8363   1.916   0.0553 .
## persons      -1.6663     0.6790  -2.454   0.0141 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Theta = 0.3733
## Number of iterations in BFGS optimization: 2
## Log-likelihood: -432.9 on 6 Df


The output looks very much like the output from two OLS regressions in R.

Below the model call, you will find a block of output containing negative binomial regression coefficients for each of the variables along with standard errors, z-scores, and p-values for the coefficients. A second block follows that corresponds to the inflation model. This includes logit coefficients for predicting excess zeros along with their standard errors, z-scores, and p-values.

All of the predictors in both the count and inflation portions of the model are statistically significant. This model fits the data significantly better than the null model, i.e., the intercept-only model. To show that this is the case, we can compare with the current model to a null model without predictors using chi-squared test on the difference of log likelihoods.

m0 <- update(m1, . ~ 1)

pchisq(2 * (logLik(m1) - logLik(m0)), df = 3, lower.tail = FALSE)

## 'log Lik.' 1.280471e-13 (df=6)


From the output above, we can see that our overall model is statistically significant.

Note that the model output above does not indicate in any way if our zero-inflated model is an improvement over a standard negative binomial regression. We can determine this by running the corresponding standard negative binomial model and then performing a Vuong test of the two models. We use the MASS package to run the standard negative binomial regression.

summary(m2 <- glm.nb(count ~ child + camper, data = zinb))

##
## Call:
## glm.nb(formula = count ~ child + camper, data = zinb, init.theta = 0.2552931119,
##
## Deviance Residuals:
##     Min       1Q   Median       3Q      Max
## -1.3141  -1.0361  -0.7266  -0.1720   4.0163
##
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept)   1.0727     0.2425   4.424 9.69e-06 ***
## child        -1.3753     0.1958  -7.025 2.14e-12 ***
## camper1       0.9094     0.2836   3.206  0.00135 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Negative Binomial(0.2553) family taken to be 1)
##
##     Null deviance: 258.93  on 249  degrees of freedom
## Residual deviance: 201.89  on 247  degrees of freedom
## AIC: 887.42
##
## Number of Fisher Scoring iterations: 1
##
##
##               Theta:  0.2553
##           Std. Err.:  0.0329
##
##  2 x log-likelihood:  -879.4210

vuong(m1, m2)

## Vuong Non-Nested Hypothesis Test-Statistic: -3.819662
## (test-statistic is asymptotically distributed N(0,1) under the
##  null that the models are indistinguishible)
## in this case:
## model2 > model1, with p-value 6.6817e-05


• The predictors child and camper in the part of the negative binomial regression model predicting number of fish caught (count) are both significant predictors.
• The predictor person in the part of the logit model predicting excessive zeros is statistically significant.
• For these data, the expected change in log(count) for a one-unit increase in child is -1.515255 holding other variables constant.
• A camper (camper = 1) has an expected log(count) of 0.879051 higher than that of a non-camper (camper = 0) holding other variables constant.
• The log odds of being an excessive zero would decrease by 1.67 for every additional person in the group. In other words, the more people in the group the less likely that the zero would be due to not gone fishing. Put plainly, the larger the group the person was in, the more likely that the person went fishing.
• The Vuong test suggests that the zero-inflated negative binomial model is a significant improvement over a standard negative binomial model.

We can get confidence intervals for the parameters and the exponentiated parameters using bootstrapping. For the negative binomial model, these would be incident risk ratios, for the zero inflation model, odds ratios. We use the boot package. First, we get the coefficients from our original model to use as start values for the model to speed up the time it takes to estimate. Then we write a short function that takes data and indices as input and returns the parameters we are interested in. Finally, we pass that to the boot function and do 1200 replicates, using snow to distribute across four cores. Note that you should adjust the number of cores to whatever your machine has. Also, for final results, one may wish to increase the number of replications to help ensure stable results.

dput(round(coef(m1, "count"), 4))

## structure(c(1.3711, -1.5152, 0.879), .Names = c("(Intercept)",
## "child", "camper1"))

dput(round(coef(m1, "zero"), 4))

## structure(c(1.6028, -1.6663), .Names = c("(Intercept)", "persons"
## ))

f <- function(data, i) {
require(pscl)
m <- zeroinfl(count ~ child + camper | persons,
data = data[i, ], dist = "negbin",
start = list(count = c(1.3711, -1.5152, 0.879), zero = c(1.6028, -1.6663)))
as.vector(t(do.call(rbind, coef(summary(m)))[, 1:2]))
}

set.seed(10)
(res <- boot(zinb, f, R = 1200, parallel = "snow", ncpus = 4))

##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot(data = zinb, statistic = f, R = 1200, parallel = "snow",
##     ncpus = 4)
##
##
## Bootstrap Statistics :
##        original       bias    std. error
## t1*   1.3710504 -0.083293503  0.39439368
## t2*   0.2561136 -0.002616572  0.03190918
## t3*  -1.5152609 -0.061393126  0.26877799
## t4*   0.1955916  0.006027950  0.02026508
## t5*   0.8790522  0.091642830  0.47150032
## t6*   0.2692734  0.001873325  0.01998125
## t7*  -0.9853566  0.080190346  0.21907572
## t8*   0.1759504  0.002580684  0.01689353
## t9*   1.6031354  0.473483335  1.59330502
## t10*  0.8365225  3.767327930 15.65780310
## t11* -1.6665917 -0.462324216  1.56790036
## t12*  0.6793077  3.771998568 15.69674577


The results are alternating parameter estimates and standard errors. That is, the first row has the first parameter estimate from our model. The second has the standard error for the first parameter. The third column contains the bootstrapped standard errors, which are considerably larger than those estimated by zeroinfl.

Now we can get the confidence intervals for all the parameters. We start on the original scale with percentile and bias adjusted CIs. We also compare these results with the regular confidence intervals based on the standard errors.

## basic parameter estimates with percentile and bias adjusted CIs
parms <- t(sapply(c(1, 3, 5, 7, 9), function(i) {
out <- boot.ci(res, index = c(i, i + 1), type = c("perc", "bca"))
with(out, c(Est = t0, pLL = percent[4], pUL = percent[5],
bcaLL = bca[4], bcaLL = bca[5]))
}))

row.names(parms) <- names(coef(m1))
## print results
parms

##                          Est        pLL        pUL      bcaLL      bcaLL
## count_(Intercept)  1.3710504  0.5132187  2.0539932  0.7563477  2.3268123
## count_child       -1.5152609 -2.1495185 -1.0807496 -1.9866681 -0.9827549
## count_camper1      0.8790522  0.1119459  1.8628376 -0.2267101  1.6676446
## zero_(Intercept)  -0.9853566 -1.3142565 -0.4465283 -1.4471843 -0.6407219
## zero_persons       1.6031354  0.3519777  8.0795923 -0.1857838  3.6867011

## compare with normal based approximation
confint(m1)

##                         2.5 %     97.5 %
## count_(Intercept)  0.86910572  1.8730573
## count_child       -1.89860155 -1.1318879
## count_camper1      0.35126797  1.4068178
## zero_(Intercept)  -0.03635521  3.2418992
## zero_persons      -2.99701357 -0.3354987


The bootstrapped confidence intervals are considerably wider than the normal based approximation. The bootstrapped CIs are more consistent with the CIs from Stata when using robust standard errors.

Now we can estimate the incident risk ratio (IRR) for the negative binomial model and odds ratio (OR) for the logistic (zero inflation) model. This is done using almost identical code as before, but passing a transformation function to the h argument of boot.ci, in this case, exp to exponentiate.

## exponentiated parameter estimates with percentile and bias adjusted CIs
expparms <- t(sapply(c(1, 3, 5, 7, 9), function(i) {
out <- boot.ci(res, index = c(i, i + 1), type = c("perc", "bca"), h = exp)
with(out, c(Est = t0, pLL = percent[4], pUL = percent[5],
bcaLL = bca[4], bcaLL = bca[5]))
}))

row.names(expparms) <- names(coef(m1))
## print results
expparms

##                         Est       pLL          pUL     bcaLL      bcaLL
## count_(Intercept) 3.9394864 1.6706640    7.7989834 2.1304808 10.2452309
## count_child       0.2197509 0.1165403    0.3393411 0.1371516  0.3742786
## count_camper1     2.4086158 1.1184525    6.4420052 0.7971518  5.2996704
## zero_(Intercept)  0.3733061 0.2686740    0.6398457 0.2352317  0.5269119
## zero_persons      4.9685866 1.4218768 3227.9195721 0.8304531 39.9129615


To better understand our model, we can compute the expected number of fish caught for different combinations of our predictors. In fact, since we are working with essentially categorical predictors, we can compute the expected values for all combinations using the expand.grid function to create all combinations and then the predict function to do it. Finally we create a graph.

newdata1 <- expand.grid(0:3, factor(0:1), 1:4)
colnames(newdata1) <- c("child", "camper", "persons")
newdata1\$phat <- predict(m1, newdata1)

ggplot(newdata1, aes(x = child, y = phat, colour = factor(persons))) +
geom_point() +
geom_line() +
facet_wrap(~camper) +
labs(x = "Number of Children", y = "Predicted Fish Caught")


## Things to consider

Here are some issues that you may want to consider in the course of your research analysis.

• Question about the over-dispersion parameter is in general a tricky one. A large over-dispersion parameter could be due to a miss-specified model or could be due to a real process with over-dispersion. Adding an over-dispersion problem does not necessarily improve a miss-specified model.
• The zero inflated negative binomial model has two parts, a negative binomial count model and the logit model for predicting excess zeros, so you might want to review the Negative Binomial Regression and Logit Regression.
• Since zero inflated negative binomial has both a count model and a logit model, each of the two models should have good predictors. The two models do not necessarily need to use the same predictors.
• Problems of perfect prediction, separation or partial separation can occur in the logistic part of the zero-inflated model.
• Count data often use exposure variable to indicate the number of times the event could have happened. You can incorporate exposure (also called an offset) into your model by using the offset() function.
• It is not recommended that zero-inflated negative binomial models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.
• Pseudo-R-squared values differ from OLS R-squareds.

## References

• UCLA: IDRE (Institute for Digital Research and Education). Data Analysis Examples. from http://www.ats.ucla.edu/stat/dae/ (accessed January 31, 2014)
• SAS 9.3 (2013). PROC COUNTREG help page. SAS Institute, Cary NC.
• Long, J. S. (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage Publications.