Modelling Negative Binomial as a substitute model to Poisson for raters agreement on ordinal scales with sparse data

The Poisson distribution has been widely used for modelling rater agreement using loglinear models. Mostly in all life or social science researches, subjects are being classified into categories by rater, interviewers or observers and most of these tables indicate that the cell counts are mixtures of either too big values and two small values or zeroes which are sparse data. We refer to sparse as a situation when a large number of cell frequencies are very small. For these kinds of tables, there are tendencies for overdispersion in which the variance of the outcome or response exceeds the nominal variance, that is, when the response is greater than it should be under the given model or the true variance is bigger than the mean. In these types of situations assuming Poisson models means we are imposing the mean-variance equality restriction on the estimation. This implies that we will effectively be requiring the variance to be less than it really is, and also, as a result, we will underestimate the true variability in the data. Lastly, this will lead us to underestimating the standard errors, and so to overestimating the degree of precision in the coefficients. The Negative Binomial, which has a variance function, would be better for modelling rater agreement with sparse data in the table in order to allow the spread of the observations or counts. We observed that assuming Negative Binomial as the underline sampling plan is better for modelling rater agreement when there are sparse data in a limited number of example.