Semi-parametric approach for modelling overdispersed count data with application to Industry 4.0

The paper deals with a test for the goodness-of-fit of a model for count data, in the framework of Generalized Linear Models. The motivating example concerns the study on the effectiveness of policy incentives for the adoption of 4.0 technologies by Small and Medium Enterprises. According to the literature, openness to Industry 4.0 should be measured in terms of the number of 4.0 technologies adopted, represented by a count variable. To investigate the effectiveness of public policy interventions to encourage the adoption of 4.0 technologies, we propose the application of a model for count data with a permutation ANOVA to test the goodness-of-fit and for the model selection. When the distribution of the response is neither Poisson nor Negative Binomial, and in the quite common situation in which the response variance is much greater than the mean, the classic Poisson regression and Negative Binomial regression are not valid. The proposed testing method is based on the combination of permutation tests on the significance of the regression coefficient estimates. The power behaviour of the proposed semi-parametric solution is investigated through a comparative Monte Carlo simulation study. The performance of such a method is compared to those of the two main parametric competitors. The application of the permutation test to an interesting case study is presented. The dataset is original, and related to a sample survey carried out in Italy, about the adoption of Industry 4.0 technologies by Italian enterprises.