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Teresa Bago d'Uva. On the use of zero-inflated and hurdle models for modeling vaccine adverse event count data.
For a better fit, an overdispersed model that incorporates excess zeros should serve as an alternative[ 6 ]. Zero modified models such as ZI and hurdle count models are capable of incorporating excess zeros. They are applied to count data when over-dispersion exists and excess zeros are indicated[ 5 , 10 ]. ZIP model can thus be seen as a mixture of two component distributions, a zero part and no-zero components, given by:.
From Equation 6 , zero observations arise from both zero-component distribution and Poisson distribution. The most natural choice to model probability of excess zeros is to use a logistic regression model:. The effect of covariates on count data excluding excess zeros can be modeled through Poisson regression:. The ZINB distribution is given by:. PLH model is a two-component model comprising of a hurdle component models zero versus non-zero counts, and a truncated Poisson count component is employed for the non-zero counts:. For PLH model, the most natural choice to model probability of zeros is to use a logistic regression model:.
Similarly, for the hurdle models, the NBLH can be used instead of Poisson distribution above in case of over-dispersion:. While both ZI and hurdle models need distributional assumptions for their count component, both classes differ with respect to their dependencies of estimation of parameters of "zero" component on these assumptions[ 1 ]. Much as the hurdle model will be consistent in the absence of a good model for the non-zero counts, however, one of its weaknesses is that it assumes all zeros to come from a single population.
As a measure of the relative goodness of fit of a statistical model, AIC not only rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters. Since the log-likelihood is expected to increase as parameters are added to a model, the AIC criteria penalize models with larger q. This penalty function may also be a function of n , the number of observations[ 13 ]. This penalty discourages over-fitting. The AIC is specified as:.
A model with lowest AIC is preferred[ 22 ]. AIC is optimal in selecting the model with the least mean squared error while BIC is not asymptotically optimal[ 24 ]. Two datasets were used as a case study in the analysis of count data on human helminths.
Table 3 from Functional Form and Heterogeneity in Models for Count Data - Semantic Scholar
The first data set came from a study carried out in in Chikwawa district, in the Lower Shire Valley-southern Malawi. The Chikwawa data were collected from a rural population mainly engaged in subsistence farming. The area lies between and m above sea level. The rainy season extends from December to March. The design and methods for data collection are described in[ 25 , 26 ]. Briefly, data were collected in eighteen 18 villages, purposively selected from the control and intervention arms of a cluster randomized study design. There was only one round of treatment following community based and house to house approaches for mass drug administration MDA.
Over 90 percent of the eligible population were treated. All infected participants in non-intervention arm received appropriate treatment. After the follow-up assessment, both arms had mass treatment. In the study, polyparasitism was considered basing on the number of species an individual was hosting.
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The focus was on S. Polyparasitism is the epidemiology of multiple species parasite infections[ 26 ]. Ten percent of the households were randomly selected from the villages for baseline survey using random number tables[ 25 ]. Subjects for geo-helminth survey were selected using a two stage-design. In short, at first stage villages were selected, then at second stage, sample of households was listed and chosen.
In the selected households, all members aged one year and above were invited to participate. Consenting individuals had their demographic details completed and were given full body clinical examinations except genitals for females for chronic manifestations of human helminths. In addition they had anthropometric measurements taken and were asked to provide a single fresh stool and urine sample. All body samples were analyzed in the laboratory using Katz test to determine geohelminth infection and intensity. Further details are provided in[ 25 ]. The second data, was school-based, collected in a cross-sectional study that was carried out in Kafue and Luangwa Districts of Lusaka province, Zambia in The two districts were selected on the basis of their ecological representativeness of the country in general[ 27 ].
In each of these districts, ten 10 primary schools were selected. Approximately schoolchildren, aged 6 to 15 years, were recruited from every school. The altitude and geographical location longitude, latitude of the surveyed schools were obtained from the archives of the Zambian Survey Department Further details of the study design are given elsewhere in[ 27 ]. Data on S. Two laboratory technicians prepared and read specimen filters.
Both technicians read each specimen independently. This was useful in increasing sensitivity of the technique, particularly where egg intensity was low. Individual data sheets were used to collect ancillary information on each child examined. In addition, data on intermediate host snails were also obtained through field collections and laboratory-based species identification.
Sampling of potential schistosomiasis transmission sites was done based on water body proximity to respective primary school, that is, the nearest likely infection source.
Zero adjusted models with applications to analysing helminths count data
The consent process was a four stage process. First stage, oral informed consent was obtained at the traditional authority TA level. Registers were kept for documentation whereby, for each individual in the selected household, a column was kept to indicate whether an individual had orally consented to participate in the study or not. Similarly, in the Zambia study, ethical approval for data collection on urinary Schistosomiasis in school children was received from University of Zambia Ethics Committee[ 27 ]. Table 1 gives summary statistics for study participants who had outcomes of interest.
The study had participants of which The mean age years of Of these, had hookworm representing Figure 1 shows that a large proportion of individuals i.
Indeed, there was overwhelming evidence of overdispersion. This was confirmed by the presence of excess zeros Figure 1. Distribution of egg counts for S. With regards to zero capturing, the Poisson model was again not appropriate as it could only capture of the zeros whereas the zero adjusted based models were much better in capturing the zero counts.
NBLH was used for fitting a final model to model helminth infection intensity and determination of factors that foster infections due to its ease of implementation and its direct link with the observed data. Those in the intervention area were at a reduced chance of infection relative to those in control area. A positive association was also observed between those who did fishing in Shire river relative to those who did not fish Table 4. Table 5 gives characteristics of study population.
A total of school children aged 6 to 15 years were enrolled into the study from 20 selected primary schools in the two districts, Kafue and Luangwa, The remaining children 6.
nitdoconstacon.ml Overall S. Infection intensity had a mean of However there was a significant difference in the mean intensity of infection, with For Schistosomiasis, a large proportion of individuals were "non egg excretors" With regards to zero capturing, the Poisson model was again not appropriate as it could only capture of the zeros whereas the zero adjusted models were much better in capturing the zero counts. This paper considered several count data models to quantify factors associated with prevalence and intensity of S.
Related Functional Form and Heterogeneity in Models for Count Data
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