Effect of Missing Responses on the $$C(\alpha )$$ C ( α ) or Score Tests in One-way Layout of Count Data

One-way layout of count data having over/under dispersion arises in many practical situations. For example, in the mice toxicology data, Barnwal and Paul (1988, Biometrika, 75(2), 215–222) sought to assess as to whether the means of several groups of count data in the presence of such over/under dispersion are equal. Specifically, they developed and studied five statistics, two of which are score tests and the other three statistics are based on data transformed to normality. After extensive simulation study they recommended the score tests. Saha (2008, J. Stat. Plan. Inference, 138(7), 2067–2081) developed two similar test statistics for the homogeneity of the means in over/under dispersed count data situations in which no likelihood exists. Again through extensive simulations, Saha recommended a score-type statistic using a double extended quasi-likelihood (Lee and Nelder 2001, Biometrika, 88(4), 987–1006). However, as in the continuous and some other discrete data situations, some observations might be missing in the one way layout of count data. The purpose of this paper is to (i) develop estimation procedures for the parameters involved in the one way layout of count data under different missing data scenarios, (ii) study the comparative behaviour of the score tests developed by Barnwal and Paul (1988, Biometrika, 75(2), 215–222) and score type statistic developed by Saha (2008, J. Stat. Plan. Inference, 138(7), 2067–2081) for complete data, and (iii) study the comparative effect of missing data on the score and score-type statistic under different missing data scenarios. Extensive Monte Carlo simulations and real life data analysis show that for complete data as well as for data under different missing data scenarios, the score-type statistic (Saha 2008, J. Stat. Plan. Inference, 138(7), 2067–2081) has some edge in terms of power over the score test statistic (Barnwal and Paul 1988, Biometrika, 75(2), 215–222) showing that the estimation under missing data methodology works well. The effect of missing data is as follows: If missing data are MCAR (missing completely at random) power increases as the difference in means and sample size increases. However, under MAR/ MNAR (missing at random/missing not at random) power remains stable in spite of percentage missing as the estimation procedure replaces the missing data by their estimates resulting in the sample size remaining nearly as the original sample size.