The purpose of this paper is to determine an appropriate variance function (mean-variance relationship) which can be used in the semi-parametric analysis of over-dispersed count data (for example, for analysis of count data by extended quasi-likelihood and double extended quasi-likelihood). We use hypothesis testing approach through a broader class of models and data analytic approach. The models considered are the three parameter negative binomial distribution and the extended quasi-likelihood. Wide analysis involving tests, data analysis and simulations indicate that the three parameter generalized negative binomial distribution does not improve in fit to count data over the simpler negative binomial distribution. Further data analysis and simulations using the extended quasi-likelihood indicate that the negative binomial variance function is preferable over a simpler variance function for data with small mean and small over-dispersion. Otherwise is a preferable variance function over the negative binomial variance function.