Vaccine - (Page 8) M. Oviedo et al. / Vaccine 26 (2008) 6157–6164 6163 Appendix A. Statistical details A.1. Negative binomial model The negative binomial model is a particular case of generalized linear models (GLM). Negative binomial regression is appropriate for modelling overdispersed count data. For this model we have: E(casesi ) = log i i A.3. Measures of quality of adjustment The statistics used to measure the fit of the adjustment are similar to those of any generalized linear model, i.e., the deviance scale and/or Pearson’s 2 -statistic is calculated. Deviance is defined as: n = exp{xi ˇ} + offset taking logarithms, - D= i=1 casesi log casesi − (casesi − ˆ i ) ˆi = xi ˇ + log(offset) + εi = ˇ0 + ˇ1 xi1 + ˇ2 xi2 + . . . + ˇk xik + log(offset) + εi 1. The number of reported cases of hepatitis A, casesi follows a Poisson distribution of parameter i , conditioned to εi. 2. exp{εi } follows a standard Gamma distribution. Assuming 1 and 2, casesi follows a negative binomial distribution. 1. The offset means a term of the form log (offset) included in the model but with coefficient constrained to be equal to 1. Thus an offset is a term for which a coefficient is not estimated. In this study offset is the Catalonia population by age group and year. Thus by including an offset we end up fitting a model for the rate of occurrence, as was desired. The use of an offset then is just a trick that allows us to use Poisson or negative binomial regression, which are only appropriate for count data, to fit a rate model. A.2. Overdispersion of the model The Poisson regression has the disadvantage of being sensitive to overdispersion (also named extra-Poisson variance); this is the violation of the property of equidispersion in the Poisson regression, i.e., the observed variance exceeds the observed mean. This causes an underestimation of the real variance, meaning that the estimate of the confidence interval is narrower than it should be and gives unbiased but inefficient estimates. The system function glm( ) is modified to include estimation of the additional parameter – phi ( ) – to adjust for a negative binomial generalized linear model. This model is based on the negative binomial probability distribution which maintains the asymmetry of count data but is more flexible in the form of distribution than the Poisson model. This adaptability is provided by the term , named dispersion parameter, which is added to the expression of variance. When overdispersion is present, the value of will be positive. When the value is 0, the negative binomial distribution is equivalent to the Poisson distribution, showing that the latter is a particular case of the former. Var(casesi ) = E(casesi ) = E(casesi ) = ; Var(casesi ) = (equidispersion)Poisson + 2 Deviance always decreases as the number of variables in the model increases and, therefore, criteria that penalize the models with a greater number of variables are applied. The criterion of selection of the best model was to select that which presented the minimum Akaike Information Criterion (AIC) and had correct residuals. AIC = D + 2×p, where p is the degrees of freedom of the model. The second measure of quality of adjustment used was Pearson’s 2 -statistic which is defined as: n 2 = i=1 (casesi − ˆ i ) V ( ˆ i) 2 Validation of the residuals was made using the Wilcoxon signed rank test to verify that the residuals followed a Normal distribution centred on zero and with constant variance. The lack of correlation between residuals was proven using the Ljung-Box statistic and the plots of the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the residuals. 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Changing travel-related global epidemiology of hepatitis A. Am J Med 2005;118(Suppl. 10A):S46–9. [9] Innis BL, Snitbhan R, Kunasol P, Laorakpongse T, Poopatanakool W, Kozik CA, et al. Protection against hepatitis A by an inactivated vaccine. JAMA 1994;271:1328–34. [10] Werzberger A, Mensch B, Kuter B, Brown L, Lewis J, Sitrin R, et al. A controlled trial of a formalin-inactivated hepatitis A vaccine in healthy children. N Engl J Med 1992;327:453–7. [11] Dominguez A, Salleras L, Carmona G, Batalla J. Effectiveness of a mass hepatitis A vaccination program in preadolescents. Vaccine 2003;21:698–701. [12] Last JM. A dictionary of epidemiology. Oxford: Oxford University Press; 2001. [13] Hardin JW, Hilbe JM. Generalized linear models and extensions. Second ed. Stata Press; 2007. [14] Armstrong GL, Bell BP. Hepatitis A virus infections in the United States: model-based estimates and implications for childhood immunization. Pediatrics 2002;109:839–45. [15] Rao ASRS, Chen MH, Pham BZ, Tricco AC, Gilca V, Dubal B, et al. Cohort effects in dynamic models and their impact on vaccination programmes: an example from Hepatitis A BMC Infectious Diseases 2006;6:174. (http://www.biomedcentral.com/1471-2334/6/174). [16] Bauch CT, Rao ASRS, Pham BZ, Krahn M, Gilca V, Duval B, et al. A dynamic model for assessing universal hepatitis A vaccination in Canada. Vaccine 2007;25:1719–26. [17] Domínguez A, Oviedo M, Carmona G, Batalla J, Salleras L, Plasència A. Impact and effectiveness of a hepatitis A vaccination programme of preadolescents seven years after introduction. Vaccine 2008;26:1737–41. Negative binomial Another form of correction of overdispersion in count data is correction of standard errors. This consists of dividing the coefficients of the parameters estimated by the Poisson regression by the respective standard errors. The interpretation of the estimated coefficients in the negative binomial model is identical to that of the Poisson model, since the structures of the mean are the same. http://www.biomedcentral.com/1471-2334/6/174
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