Pearson's goodness of fit statistic as a score test statistic

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R 2 STATISTICS FOR MIXED MODELS (2005, background): “One could also base them on score statistics (see Jaffrézic, et al., 2003, and Smyth, 2003, for using the score test to construct a goodness-of-fit measure) and possibly other functions of the New Prairie Press https://newprairiepress.org/agstatconference/2005/proceedings/12 data.”
Small-sample estimation of negative binomial dispersion, with applications to SAGE data. (2007, methodology): “Models used to capture this additional variation include overdispersed generalized linear models (GLMs), with the counts as either binomial (Baggerley and others, 2004) or Poisson (Lu and others, 2005), the latter proving to be more powerful in most situations.”