The generalized estimating equations (GEE) is one of the statistical approaches for the analysis of longitudinal data with correlated response. A working correlation structure for the repeated measures of the outcome variable of a subject needs to be specified by this method and the GEE estimator for the regression parameter will be the most efficient if the working correlation matrix is correctly specified. Hence it is desirable to choose a working correlation matrix that is the closest to the underlying structure among a set of working correlation structures. The quasi-likelihood Information criteria (QIC) was proposed for the selection of the working correlation structure and the best subset of explanatory variables in GEE. However, its success rate in selecting the true correlation structure has been established to be about 29.4%. Likewise, past studies have shown that its bias increases with the number of parameters. By considering longitudinal data with binary response, we establish numerically through simulations the consistency property of QIC in selecting the true working correlation structure and the conditions for its consistency. Further, we propose a modified QIC that penalizes for the number of parameter estimates in the original QIC and numerically establish that the penalization enhances the consistency of QIC in selecting the true working correlation structure. The results indicate that QIC selects the true correlation structure with probability approaching one if only parsimonious structures are considered otherwise the selection rates are less than 50% regardless of the increase in the sample size, measurements per subject and level of correlation. Further, we established that the probability of selecting the true correlation structure R 0 almost surely converges to one when we penalize for the number of correlation parameters estimated.

This study sought to analyse trend in the monthly water demand data series in Kisumu city at both seasonal and non-seasonal levels using the parametric method of Ordinary Least Squares (OLS) and non-parametric methods of Mann-Kendall tau and Sen's T test. Sen's test was applied to validate the Mann Kendall trend test and to estimate the magnitude of the trend and its direction. The significance of the slope of the OLS equation was tested using the F-Test based on the Analysis of Variance (ANOVA). Secondary monthly water consumption data obtained from KIWASCO for the period January 2004 to December 2013 were used. Using logarithmically transformed data, the study established by OLS that residential water demand in Kisumu City had a significant increasing trend (F Calc =(105.13) > F (1;119)(α=0:05) =(5.15)). Kendall's tau test corroborated the OLS results of a significant increasing trend. The Sens T test indicated that most of the months registered significant upward trend with Sen's slope estimates showing positive rates of change in residential water demand for these months.

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