# How to interpret rate ratios (RR) and odds ratio (OR) from a zero altered/hurdle negative binomial model

I am having a really hard time interpreting results from a zero altered negative binomial model. This is related to my question but some things have changed. I am doing this in R.

Quick background on my data...I am trying to model bythotrephes prey found in fish stomachs. My data is zero inflated and overdispersed. After doing model diagnostics/selection, decided that ZANB/hurdle model was best.

Based on AICc, the most parsimonious model is:

hurdle(Byths ~ depth * Season,
data = dietGLM, dist = "negbin")


where Byths == bythotrephes count; depth is the max. depth of where the fish was collected (max. trawl depth), and Season is a categorical variable (Pre Hypoxia, Peak Hypoxia, and Post Hypoxia). I didn't scale nor transform anything for the model.

    Call:
hurdle(formula = Byths ~ depth * Season, data = dietGLM, dist = "negbin")

Pearson residuals:
Min      1Q  Median      3Q     Max
-0.8326 -0.5676 -0.2256  0.1534  4.7221

Count model coefficients (truncated negbin with log link):
Estimate Std. Error z value Pr(>|z|)
(Intercept)       5.88537    0.62991   9.343  < 2e-16 ***
depth            -0.01528    0.01699  -0.899 0.368510
SeasonPeak       -3.03182    0.86296  -3.513 0.000443 ***
SeasonPost       -2.74276    1.95642  -1.402 0.160937
depth:SeasonPeak  0.05199    0.02165   2.401 0.016339 *
depth:SeasonPost  0.06410    0.03995   1.604 0.108609
Log(theta)        0.13513    0.19772   0.683 0.494335
Zero hurdle model coefficients (binomial with logit link):
Estimate Std. Error z value Pr(>|z|)
(Intercept)       4.02258    1.10901   3.627 0.000287 ***
depth            -0.09761    0.02534  -3.851 0.000117 ***
SeasonPeak       -2.55477    1.65554  -1.543 0.122790
SeasonPost       -3.94742    3.35201  -1.178 0.238944
depth:SeasonPeak  0.08342    0.03684   2.264 0.023569 *
depth:SeasonPost  0.09163    0.06704   1.367 0.171698
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Theta: count = 1.1447
Number of iterations in BFGS optimization: 17
Log-likelihood: -344.6 on 13 Df


Then I did:

exp(cbind(Exponentiated_Odds_Ratio=coef(h5), confint(h5)))


which gives..

                           Exponentiated_Odds_Ratio        2.5 %       97.5 %
count_(Intercept)                  359.73629541 1.046652e+02 1236.4199806
count_depth                          0.98483495 9.525747e-01    1.0181878
count_SeasonPeak                     0.04822789 8.886849e-03    0.2617271
count_SeasonPost                     0.06439212 1.391624e-03    2.9795012
count_depth:SeasonPeak               1.05336831 1.009601e+00    1.0990332
count_depth:SeasonPost               1.06619778 9.858986e-01    1.1530371
zero_(Intercept)                    55.84473421 6.353222e+00  490.8744754
zero_depth                           0.90700275 8.630488e-01    0.9531952
zero_SeasonPeak                      0.07770974 3.028865e-03    1.9937515
zero_SeasonPost                      0.01930438 2.706583e-05   13.7686161
zero_depth:SeasonPeak                1.08699475 1.011268e+00    1.1683924
zero_depth:SeasonPost                1.09595488 9.610133e-01    1.2498444


Putting everything together (for me it is easier to look at this, so I hope this is allowed...):

I've read sooo many papers and book chapters on hurdle models and OR and I am still not understanding how to clearly explain my results. I have a limited stats background so that doesn't help.

This is what I have so far but honestly it doesn't make sense to me....

• For depth exp(-0.10) = 0.091 (OR), does this mean that for one unit increase in depth, the odds of my fish of interest consuming bythotrephes increases by a factor of 0.91 ? What exactly does this mean?
• For the count part of model: Fish are RR = 0.05; exp(-3.03) less likely to consume bythotrephes during 'SeasonPeak' compared to 'SeasonPre'?

I am not sure how to interpret the categorical variables, how to put in words the results between RR and OR, and how to interpret interaction terms.

I have looked at other questions: this, this, this, and many others on similar topics.

I am happy to include more information/data if it helps understand my question. The simpler you can explain this the better. Try to explain this as if you were explaining it to a kid :)