In this study, we extend earlier work of Freeland (1998) and Jung and Tremayne (2003), and develop a general formula for a score statistic to test for dependence in an integer autoregressive process with an arbitrary arrivals distribution. We give two statistics that cater for arrivals processes that may be under-, equi- or overdispersed. The first is based on the Katz family which includes Poisson, binomial and negative binomial distributions as special cases. The second uses the generalized Poisson which includes the Poisson distribution as a special case and can also cater for under- and over- dispersion. The null distribution of the tests is provided and consistency is discussed. Size and power properties are investigated under different model assumptions by Monte Carlo simulations. The autocorrelation coefficient is also investigated as a benchmark for comparison.