Interpreting a zero-inflation negative binomial model

Question

I am currently running a series of zero-inflated negative binomial models on the impact of the magnitude and direction of change in various weather parameters on a number of insect behaviours (represented as counts of the number of times the behaviour is recorded in a set period of time following exposure). My current model is examining fecundity as below. The random effects are both categorical and correspond to the day on which the behaviour was recorded and the rearing conditions of the males in the lab.

model2<-glmmTMB(No.eggs.laid~Change.6hrs*Direction.6hrs + (1|Day) + (1|Sex.ratio.line.male), family = "nbinom1",ziformula = ~Change.6hrs*Direction.6hrs + (1|Day) + (1|Sex.ratio.line.male), data = charlotte.egg)

Here is the summary output of my model:

Family: nbinom1  ( log )
Formula:          
No.eggs.laid ~ Change.6hrs * Direction.6hrs + (1 | Day) + (1 |  
    Sex.ratio.line.male)
Zero inflation:                
~Change.6hrs * Direction.6hrs + (1 | Day) + (1 | Sex.ratio.line.male)
Data: charlotte.egg

     AIC      BIC   logLik deviance df.resid 
  2980.0   3033.3  -1477.0   2954.0      430 

Random effects:

Conditional model:
 Groups              Name        Variance  Std.Dev. 
 Day                 (Intercept) 1.316e-02 1.147e-01
 Sex.ratio.line.male (Intercept) 8.032e-10 2.834e-05
Number of obs: 443, groups:  Day, 7; Sex.ratio.line.male, 10

Zero-inflation model:
 Groups              Name        Variance Std.Dev.
 Day                 (Intercept) 0.58920  0.7676  
 Sex.ratio.line.male (Intercept) 0.01893  0.1376  
Number of obs: 443, groups:  Day, 7; Sex.ratio.line.male, 10

Overdispersion parameter for nbinom1 family (): 5.42 

Conditional model:
                                   Estimate Std. Error z value
(Intercept)                         2.92753    0.11709  25.003
Change.6hrs                        -0.26768    0.12984  -2.062
Direction.6hrsIncrease             -0.06377    0.13621  -0.468
Change.6hrs:Direction.6hrsIncrease  0.23991    0.15250   1.573
                                   Pr(>|z|)    
(Intercept)                          <2e-16 ***
Change.6hrs                          0.0392 *  
Direction.6hrsIncrease               0.6397    
Change.6hrs:Direction.6hrsIncrease   0.1157    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Zero-inflation model:
                                   Estimate Std. Error z value
(Intercept)                         -0.9118     0.6278  -1.452
Change.6hrs                         -2.9058     1.3911  -2.089
Direction.6hrsIncrease              -0.8555     0.6355  -1.346
Change.6hrs:Direction.6hrsIncrease   3.3083     1.4543   2.275
                                   Pr(>|z|)  
(Intercept)                          0.1464  
Change.6hrs                          0.0367 *
Direction.6hrsIncrease               0.1783  
Change.6hrs:Direction.6hrsIncrease   0.0229 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

I have a few questions with regards to the output of this model:

1. What does the zero-inflation model actually represent?
2. Are these p-values sufficient in interpreting the model or do further statistical tests of significance need to be completed in order to infer relationships between variables?
3. With this type of model how would you go about determining the significance of the random effects? With non-zero inflated models I am able to do this by using the anova() function to compare a model with and without a particular random effect however when I tried to do this only one p-value is generated. As such I am not sure if this pertains to either the conditional or the zero-inflation model.

Answer 1

1. What does the zero-inflation model actually represent?

this is a model for the occurance of non zeros vs zeros. It can be interpreted in the same was as a logistic regression model where success means a non-zero count and you are modelling the probability of obtaining a non-zero count.

2. Are these p-values sufficient in interpreting the model or do further statistical tests of significance need to be completed in order to infer relationships between variables?

Try to interpret the coefficient estimates, not the p values, but yes, the p values can be interpreted, as the probability of obvserving these data, or data more extreme, if the null hypothesis is true. That is, each p value relates to a specific test of a specific null hypothesis and that is the only context in which you can interpret p values.

3. With this type of model how would you go about determining the significance of the random effects? With non-zero inflated models I am able to do this by using the anova() function to compare a model with and without a particular random effect however when I tried to do this only one p-value is generated. As such I am not sure if this pertains to either the conditional or the zero-inflation model.

Again, don't worry too much about p values from these tests. You have repeated measures and therefore you are accounting for this using random intercepts. It is sufficient to report the variance of these random intercepts. In your case you can note that the variance of one of these variance components in both parts of the model is small in comparison to the other. Having said that, it is good to seek a parsimoneous model, so if you have reason to believe that there should not be any correlation within either of your grouping variables for either part of the model, then you can remove the corresponding random term from the model and perform a likelihood ratio test in the same way as you do with a model without zero inflation - note that you have 2 parts to the model that include random effects: the main part and the ziformula part.