Zero-Inflated Regression Methods for Insecticides

The numerical abundance of many species sharing the same ecosystem very different levels of the organism and are in constant change, depending on many factors. Due to the heterogeneous strucspeciese of the life cycles of organisms and abiotic resources in the environment based on census population densities derived from overdispersion (variance is higher than means in Poisson distribution) (Cox, 1983; Cameron and Trivedi, 1998) and a large number of zero values (zero-inflated data) is observed (Yesilova et al, 2011). In such a case, zero-inflated Poisson (ZIP) regression model is a appropriate approach for analyzing a dependent variable having excess zero observations (Lambert, 1992; Bohning, 1998; Bohning et al, 1999; Yau and Lee, 2001; Lee et al, 2001; Khoshgoftaar et al, 2005; Yesilova et al, 2010). Zero-inflation is also likely in data sets, excess zero observations. In such cases, a zeroinflated negative binominal (ZINB) regression model is an alternative method (Ridout et al, 2001; Yau, 2001; Cheung, 2002; Jansakul, 2005; Long and Frese, 2006; Hilbe, 2007; Yesilova et al, 2009; Yesilova et al, 2010). Morever, The Poisson hurdle model and negative binomial hurdle model (Rose and Martin, 2006; Long and Frese, 2006; Hilbe, 2007; Yesilova et al, 2009; Yesilova et al, 2010), and zero-inflated generalized Poisson (ZIGP) model (Consul, 1989, Consul and Famoye, 1992; Czado et al., 2007) are widely used in the analysis of zero-inflated data. In this part, the analysis of data with many zeros for Notonecta viridis Delcourt (Heteroptera: Notonectidae) and Chironomidae species (Diptera) were carried out by means of using the models of Poisson Regression (PR), negative binomial (NB) regression, zero-inflated Poisson (ZIP) regression, zero-inflated negative binomial (ZINB) regression and negative binomial hurdle (NBH) model.


Introduction
The numerical abundance of many species sharing the same ecosystem very different levels of the organism and are in constant change, depending on many factors. Due to the heterogeneous strucspeciese of the life cycles of organisms and abiotic resources in the environment based on census population densities derived from overdispersion (variance is higher than means in Poisson distribution) (Cox, 1983;Cameron and Trivedi, 1998) and a large number of zero values (zero-inflated data) is observed (Yeşilova et al, 2011). In such a case, zero-inflated Poisson (ZIP) regression model is a appropriate approach for analyzing a dependent variable having excess zero observations (Lambert, 1992;Böhning, 1998 (Consul, 1989, Consul andFamoye, 1992;Czado et al., 2007) are widely used in the analysis of zero-inflated data. In this part, the analysis of data with many zeros for Notonecta viridis Delcourt (Heteroptera: Notonectidae) and Chironomidae species (Diptera) were carried out by means of using the models of Poisson Regression (PR), negative binomial (NB) regression, zero-inflated Poisson (ZIP) regression, zero-inflated negative binomial (ZINB) regression and negative binomial hurdle (NBH) model.

Samplings
The study was based on periodical samplings of the coastal band of Van Lake, conducted between July-September 2005 and May-September 2006. Samples were taken at totally twenty sampling points as streams entrance (6 points), settlement coastlines (7 points) and naspeciesal coastlines (7 points). Samples were taken according to Hansen et al. (2000). The

Poisson regression
The logarithm of mean of Poisson distribution (  ) is supposed to be a linear function of independent variables ( i x ) is, Poisson Regression Model can be written as In equation 1, i y denotes dependent variable having Poisson distribution. Likelihood function for PR model is, (Böhning, 1998) In equation 2,  are unknown parameters.  can be estimated by maximizing log likelihood function according to ML (Khoshgoftaar et al, 2005;Yau, 2006).

Zero inflated poisson regression
ZIP regression is [13], In equation (4), i  represents the possibility of extra zeros' existence. Log likelihood function for ZIP model is (Yau, 2006),  .
I , given in equation (5) is the indicator function for the specified event. Then i  and i  parameters can be obtained following link functions, and In equations 6 and 7, B(nxp) and G(nxq) are covariate matrixes.  and  are respectively unknown parameter vectors with px1 and qx1 dimension (Yau, 2006).
I , given in equation 9 is the indicator function for the specified event. The model descripted by Lambert (1992) can be given as, Here, X(nxp) and G(nxq) covariate matrixes,  and  are respectively unknown parameter vectors with px1 and qx1 dimension. Maximum likelihood estimations for  ,  and  can be obtained by using EM algorithm.

Negative binomial hurdle model
Log-likelihood for negative binomial hurdle model (Hilbe, 2007), In equation (10),   In equations, LL indicates log likelihood, r indicates parameter number and n indicates sample size.

Results
In this study, R statistical software program was used. Insect densities were included to the model as dependent variable. Besides years, months, species and station are included as independent variables to the model. The 66 (20.63%) of the 320 dependent variable were zero valued. The distribution of the insect densities was skewed to right because of excess zeros.  Table 3. Parameter estimations and standard errors for negative binomial regression.
ML parameter estimations and standard errors for zero-inflated Poisson regression both count model and logit model were given in Table 4 and  Table 5. Parameter estimations and standard errors for ZIP logit model.
ML parameter estimations and standard errors for zero-inflated negative binomial regression both count model and logit model were given in Table 6 and Table 7 Table 7. Parameter estimations and standard errors for ZINB logit model.
ML parameter estimations and standard errors for Poisson hurdle both count model and logit model were given in Table 8 and ML parameter estimations and standard errors obtained for the NBH count model was given in Table 8. While stations and species were significant on the insect densities, the effect of years and the effect of months were not significant on the insect densities. ML parameter estimations and standard errors obtained for the NBH logit model was given in Table 9. The effects months, years and species were not significant on the insect densities. However, the effect of station was significant on the insect densities.  Table 9. Parameter estimations and standard errors for PH logit model.
ML parameter estimations and standard errors obtained for negative binomial hurdle both count model and logit model were given in Table 10 and Table 11, respectively. ML parameter estimations and standard errors obtained for the NBH count model was given in Table 10. While stations and species were significant on the insect densities, the effect of years and the effect of months were not significant on the insect densities. Insect densities observed at monthly sampling ranges depending on water temperaspeciese were increased with the rise of temperaspeciese, but specifically after the month of July such intensity was decreased at the rate of 16% ( -0.19128 ~0.8434961 e ) towards the month of September within the both years. It has been determined that insect intensities observed at different stations have shown differentiation at the rate of 5%. Chironomid larvae which are included in prey of notonectidae fed by different sources of food at aquatic environment have been found at rather lower density in reference to notonectid density. However, it is hard to guess that such decrement has been formed under the impact of notonectidae. Nevertheless notonectidae do not depend on a single host, their sources of food are rather wide range of variety. Small arthropods on the water surface, small crustaceans living in water, larvae of aquatic insects, snails, small fish or larvae of frog are among their preys (Bruce et al., 1990).