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In the negative binomial component of the model, a significant group-by-pre-release UPI interaction was found, indicating that the effect of pre-release UPI on post-release UPI was different in the comparison and intervention groups.
The coefficients from the negative binomial component of the model can be exponentiated to obtain an incidence rate ratio (IRR) representing the factor change in the mean count of the outcome per unit change in the explanatory factor [ 46].
The exponentiated coefficient of the negative binomial component of the model corresponds to a Prevalence Rate Ratio, which in this analysis we interpret as ratio of reported impacts (problems), or "Problem Rate Ratio" (PRR) as in a previous study [ 40].
For the binomial component of the zero-inflated model, visitor taxon was a statistically significant predictor of the excess number of visits (relative to the Poisson distribution expectation) that resulted in zero removals (χ 1 2, N = 382 = 14.7, P = 6.2 × 10−4).
For the binomial component of the zero-inflated model, visitor taxon was a statistically significant predictor of the excess number of visits (relative to the Poisson-distribution expectation) that resulted in zero insertions (χ 1 2, N = 382 = 49.2, P = 2.0 × 10−11).
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In this paper we consider a case, where the random variables in the ratio are joint binomial components of a multinomial distribution.
"Hurdle" models are mixture models that include two processes: (1) a binomial component modelling the probability of a count having a zero value, used to determine the factors influencing breeding probability; (2) a truncated Poisson model for the positive litter size counts only (excluding females that did not breed).
For the negative binomial component, we used the covariates which resulted in a p-value less than 0.1 from a multivariable negative binomial model.
Regression coefficients were presented as incidence rate ratios for the binomial component and odds ratios for the excess zero component.
In fact, the spatial component of the binomial generalized linear mixed model, the spatial autoregressive coefficient, is similar to the local and interaction effects in network models [ 36] in what Anselin [ 37] called the family of autocorrelation models.
For (ii), we used a maximum likelihood approach to estimate both the alpha and beta parameters in the beta component of the beta-binomial model; we have clarified this point in the same subsection of the Materials and methods of the revised manuscript.
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