SUPPLEMENTARY MATERIAL: General mixed Poisson regression models with varying dispersion

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1 SUPPLEMENTARY MATERIAL: General mixed Poisson regression models with varying dispersion Wagner Barreto-Souza and Alexandre B. Simas Departamento de Estatística, Universidade Federal de Minas Gerais Pampulha, Belo Horizonte, MG - Brazil s: wagnerbs85@gmail.com; wagnerbs@est.ufmg.br Departamento de Matemática, Universidade Federal da Paraíba Cidade Universitária, João Pessoa, PB - Brazil alexandre@mat.ufpb.br September 3,

2 1 Additional simulated results We here present additional simulation results by considering more two different scenarios. We set the following regression structures for the mean and dispersion parameters: logµ i = β 0 +β 1 x 1i +β 2 x 2i and logφ i = α 0 +α 1 x 1i +α 2 x 3i, for i = 1,...,n, with x 1i, x 2i and x 3i generated independently from standard uniform distributions and kept fixed along the simulation. We keep here the set (β 0,β 1,β 2 ) = (1,1.5,2) and n = 25,50,100,200, as in the article. With respect to the dispersion parameter, we consider here different values for (α 0,α 1,α 2 ). In the first scenario, we take (α 0,α 1,α 2 ) = (1,2, 3) and the second one we get (α 0,α 1,α 2 ) = (0.5,1,4). Therangeofφis(0.135,20.085)forthefirstscenarioand(1.649, )forthesecondscenario. We compute the mean and the RMSE (root of the mean square error) of the estimates of the parameters. We use 5000 replicas of Monte Carlo. The results for the first scenario of the PIG case are given in Tables 1 and 2, while the results for the second scenario are given in Tables 5 and 6. The results related to the NB case are given in Tables 3 and 4 for the first scenario and in Tables 7 and 8 for the second scenario. By looking at Table 1 (PIG case for the first scenario), Table 5 (PIG case for the second scenario), Table 3 (NB case for the first scenario) and Table 7 (NB case for the second scenario), we observe that the estimates obtained from the EM algorithm are the best in both bias and mean squared error criteria. We also observe that the Quasi-Poisson model has the poorest behavior even for large sample size. The empirical standard deviation of the parameter estimates of the model and the average of the standard errors obtained from the observed information matrix for the PIG case are given in Tables 2 and 6 for the first and second scenario, respectively. The results for the NB case are presented in Tables 2 (first scenario) and 6 (second scenario). The results agree even for small sample sizes, suggesting that the observed information matrix is suitable for estimating the standard errors of the parameter estimators. Basically, all observations made for the scenario considered in the paper hold for both additional scenarios considered in this Supplementary Material. 2 Plots of the total local influence In this Suplementary Material we also present the graphics concerning the total local influence analysis under all the five perturbation schemes introduced on the data analysis presented in the article. In Figure 1, we give the plots related to the NB case. The plots related to the PIG case are given in Figure 2. 2

3 Table 1: Mean and root of the mean square error in parentheses of the estimates of the parameters for the PIG, quasi-poisson and GAMLSS models under the first scenario n = 25 PIG β (0.508) (0.501) (0.510) β (0.667) (0.643) (0.668) β (0.492) (0.505) (0.495) α (2.552) (2.090) (2.895) α (2.821) (2.332) (2.947) α (3.362) (1.983) (3.771) n = 50 PIG β (0.389) (0.421) (0.390) β (0.441) (0.469) (0.445) β (0.394) (0.446) (0.394) α (1.184) (2.016) (1.206) α (1.244) (2.314) (1.255) α (1.389) (1.332) (1.420) n = 100 PIG β (0.213) (0.231) (0.213) β (0.247) (0.266) (0.248) β (0.218) (0.243) (0.218) α (0.621) (2.272) (0.628) α (0.849) (1.980) (0.854) α (0.768) (1.281) (0.773) n = 200 PIG β (0.164) (0.184) (0.164) β (0.195) (0.223) (0.194) β (0.174) (0.199) (0.174) α (0.417) (2.282) (0.419) α (0.520) (2.032) (0.518) α (0.554) (1.255) (0.556) 3

4 Table 2: Empirical and theoretical standard errors of the estimates of the parameters for the PIG model under the first scenario n = 25 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 50 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 100 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 200 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical

5 Table 3: Mean and root of the mean square error in parentheses of the estimates of the parameters for the NB, quasi-poisson and GAMLSS models under the first scenario n = 25 NB β (0.492) (0.526) (0.492) β (0.620) (0.672) (0.622) β (0.501) (0.514) (0.501) α (2.134) (2.135) (2.642) α (2.138) (2.047) (2.407) α (3.014) (1.658) (3.754) n = 50 NB β (0.338) (0.370) (0.338) β (0.441) (0.487) (0.442) β (0.403) (0.429) (0.403) α (0.895) (2.195) (0.906) α (1.091) (2.108) (1.097) α (1.125) (1.108) (1.136) n = 100 NB β (0.219) (0.251) (0.219) β (0.248) (0.280) (0.248) β (0.233) (0.261) (0.233) α (0.470) (2.274) (0.471) α (0.642) (1.674) (0.643) α (0.653) (0.798) (0.655) n = 200 NB β (0.180) (0.200) (0.180) β (0.200) (0.222) (0.200) β (0.182) (0.199) (0.182) α (0.336) (2.502) (0.337) α (0.428) (1.640) (0.427) α (0.459) (0.913) (0.458) 5

6 Table 4: Empirical and theoretical standard errors of the estimates of the parameters for the NB model under the first scenario n = 25 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 50 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 100 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 200 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical

7 Table 5: Mean and root of the mean square error in parentheses of the estimates of the parameters for the PIG, quasi-poisson and GAMLSS models under the second scenario n = 25 PIG β (0.251) (0.268) (0.251) β (0.305) (0.316) (0.309) β (0.287) (0.304) (0.285) α (2.617) (2.743) (3.211) α (3.391) (2.113) (4.177) α (3.578) (2.558) (4.449) n = 50 PIG β (0.158) (0.166) (0.159) β (0.199) (0.207) (0.200) β (0.202) (0.211) (0.203) α (1.437) (2.074) (1.945) α (1.984) (1.569) (2.530) α (2.578) (2.203) (3.826) n = 100 PIG β (0.117) (0.124) (0.117) β (0.132) (0.138) (0.132) β (0.119) (0.125) (0.119) α (0.875) (2.164) (0.930) α (1.223) (1.304) (1.318) α (1.483) (2.183) (1.949) n = 200 PIG β (0.078) (0.080) (0.078) β (0.090) (0.092) (0.090) β (0.083) (0.087) (0.083) α (0.615) (2.062) (0.648) α (0.850) (1.236) (0.907) α (1.047) (2.287) (1.238) 7

8 Table 6: Empirical and theoretical standard errors of the estimates of the parameters for the PIG model under the second scenario n = 25 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 50 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 100 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 200 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical

9 Table 7: Mean and root of the mean square error in parentheses of the estimates of the parameters for the NB, quasi-poisson and GAMLSS models under the second scenario n = 25 NB β (0.233) (0.257) (0.233) β (0.262) (0.280) (0.263) β (0.288) (0.318) (0.290) α (2.620) (3.060) (6.417) α (4.492) (2.080) (6.810) α (3.682) (1.874) (6.875) n = 50 NB β (0.140) (0.146) (0.140) β (0.157) (0.163) (0.156) β (0.175) (0.187) (0.175) α (1.233) (2.153) (2.355) α (1.936) (1.400) (2.646) α (2.507) (2.255) (4.068) n = 100 NB β (0.108) (0.111) (0.108) β (0.127) (0.132) (0.127) β (0.119) (0.123) (0.119) α (0.820) (2.048) (0.893) α (1.103) (1.442) (1.213) α (1.530) (2.222) (1.984) n = 200 NB β (0.077) (0.079) (0.077) β (0.091) (0.094) (0.091) β (0.089) (0.093) (0.089) α (0.531) (2.174) (0.548) α (0.743) (1.219) (0.780) α (0.959) (2.078) (1.139) 9

10 Table 8: Empirical and theoretical standard errors of the estimates of the parameters for the NB model under the second scenario n = 25 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 50 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 100 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical n = 200 β 0 β 1 β 2 α 0 α 1 α 2 Empirical Theoretical

11 Plot of Case weights conformal Plot of mean pert conformal Plot of disp pert conformal Plot of simul pert conformal Plot of hidden pert conformal Figure 1: Total local influence plots based on the conformal curvature for NB regression model under the perturbation schemes considered in the article 11

12 Plot of Case weights conformal Plot of mean pert conformal Plot of disp pert conformal Plot of simul pert conformal Plot of hidden pert conformal Figure 2: Total local influence plots based on the conformal curvature for PIG regression model under the perturbation schemes considered in the article 12