# Interpret zero-inflated negative binomial regression

I am trying to estimate a zero-inflated negative binomial model with 11 predictor variables and the number of reported crimes as a response variable. The model seems to work OK, but I'm uncertain on how to interpret the results. Below is my model and the results:

#estimate zero-inflated NB model
zinf.nbi <- zeroinfl(CRIME ~ VAR1 + VAR2 + VAR3 + VAR4
+ VAR5 + VAR6 + VAR7 + VAR8 + VAR9 + VAR10
+ VAR 11, data = mydata, dist = "negbin")
summary(zinf.nbi)

> summary(zinf.nbi)

Call:
zeroinfl(formula = CRIME ~ VAR1 + VAR2 + VAR3 + VAR4 + VAR5
+ VAR6 + VAR7 + VAR8 + VAR9 + VAR10 + VAR 11,
data = mydata, dist = "negbin")

Pearson residuals:
Min       1Q   Median       3Q      Max
-0.47719 -0.17583 -0.08080 -0.02709 26.99868

Count model coefficients (negbin with log link):
Estimate Std. Error z value Pr(>|z|)
(Intercept)  -2.682578   0.269317  -9.961  < 2e-16 ***
VAR1          1.436770   0.249026   5.770 7.95e-09 ***
VAR2         -0.648535   0.268608  -2.414 0.015760 *
VAR3         -0.130107   0.239543  -0.543 0.587029
VAR4         -0.008985   0.267949  -0.034 0.973249
VAR5         -0.807941   0.269470  -2.998 0.002715 **
VAR6         -1.396990   0.396299  -3.525 0.000423 ***
VAR7          0.314514   0.113696   2.766 0.005670 **
VAR8         -1.959792   0.207233  -9.457  < 2e-16 ***
VAR9          0.711452   0.338171   2.104 0.035394 *
VAR10        -0.013628   0.132889  -0.103 0.918316
VAR11         0.092719   0.034799   2.664 0.007712 **
Log(theta)   -1.429807   0.103981 -13.751  < 2e-16 ***

Zero-inflation model coefficients (binomial with logit link):
Estimate Std. Error z value Pr(>|z|)
(Intercept)   1.14267    0.46786   2.442 0.014593 *
VAR1          1.13108    0.51718   2.187 0.028742 *
VAR2         -0.68871    0.33832  -2.036 0.041781 *
VAR3          0.16412    0.37019   0.443 0.657527
VAR4          0.57907    0.42818   1.352 0.176241
VAR5          0.83822    0.40451   2.072 0.038247 *
VAR6          0.02991    0.73117   0.041 0.967368
VAR7          0.01186    0.19025   0.062 0.950282
VAR8         -1.33618    0.39677  -3.368 0.000758 ***
VAR9          1.40246    0.39349   3.564 0.000365 ***
VAR10        -0.14713    0.22707  -0.648 0.517000
VAR11        -2.71317    0.64939  -4.178 2.94e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Theta = 0.2394
Number of iterations in BFGS optimization: 42
Log-likelihood: -2649 on 25 Df


As far as I understand, the first block (the count component) is a summary of the full model and can be interpreted as a standard negative binomial model. The second block (the zero component), on the other hand, predicts whether or not the outcome is a certain zero. Now, what I would like to know is:

a) How do I interpret the second block of the model in relation to the first block? As you can see in the results, some variables are significant in both the first and the second block.

b) Which block should I present in my final results? The first block or the second block?

a) Here https://rpubs.com/kaz_yos/pscl-2 is a nice example of how to interpret the results of a ZINB model.

b) Obviusly you have to present both blocks.

Note: ZINB regression model two separate processes so they produce two sets of coefficients: one for the count part of the model and the other for the logistic part of the model.

A common way of interpreting logistic regression models is to exponentiate the coefficients, which places the coefficients in an odds-ratio scale. With zero-inflated models the logistic part of the model predicts non-occurrence of the outcome.

Here you can fins another example https://stats.idre.ucla.edu/other/dae/.

• Answers are better when they don't require links & are standalone (since links can change or be removed over time). I think this is particularly important for your part (a)--would you mind explaining in (a) a little bit so as to not rely on the link? Jan 4, 2018 at 4:08
• Thanks for the links, they are helpful. But as @PatrickCoulombe notes above, would you mind elaborate a bit on how to interpret the second block of the model in relation to the first block? E.g. how do I interpret VAR8 which have a negative coefficient in both blocks and are both significant? Jan 4, 2018 at 16:49
• I have regression results I am tying to interpret that are similar to these. It seems to me that the literature suggests that variables with the same coefficient signs (+/-) in both blocks of the regression output are "dissonant" effects. Essentially they suggest there is some threshold that must be achieved to get to the count portion of the model. I find that interpreting this model in prose is difficult and the literature is unhelpful. Any further answers would be greatly appreciated. Mar 23, 2020 at 14:06