Robust Estimation for Zero-Inflated Poisson Regression - Franklin ...

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12 D. B. Hall and J. Shen Scand J StatistTable 5. Frequency distribution for use of forcecountsCumulativeValue Frequency Per cent per cent0 12,715 94.07 94.071 573 4.24 98.312 151 1.12 99.423 40 0.30 99.724 12 0.09 99.815 13 0.10 99.906 5 0.04 99.947 1 0.01 99.958 1 0.01 99.969 1 0.01 99.9610 2 0.01 99.9812 1 0.01 99.9914 1 0.01 99.9922 1 0.01 100.00Table 6. Parameter estimates and Wald p-values (in parentheses) for zero inflated Poissonmodel for use of force countsPoisson componentMixing probabilityVariable ML RES ML RESsite – (< 0.001) – (< 0.001) – (0.006) – (0.023)sex – (0.970) – (0.644) – (0.014) – (0.135)race – (0.022) – (0.303) – (0.043) – (0.283)dai – (< 0.001) – (0.076) – (0.001) – (0.110)age 2.16 (0.010) 1.46 (0.261) 3.27 (0.047) 2.39 (0.328)age 2 −2.64 (0.002) −1.91 (0.144) −3.53 (0.036) −2.67 (0.283)priors 0.328 (< 0.001) 0.409 (< 0.001) −0.417 (0.009) −0.293 (0.137)priors 2 −0.0879 (0.188) −0.141 (0.061) 0.240 (0.149) 0.138 (0.406)los 0.248 (0.002) 0.201 (0.038) −1.42 (< 0.001) −1.88 (< 0.001)los 2 −0.0814 (0.051) −0.0442 (0.392) 0.558 (< 0.001) 0.741 (< 0.001)avgutil −0.370 (0.323) −0.281 (0.659) 0.332 (0.600) 0.491 (0.682)avgutil 2 0.175 (0.632) 0.0694 (0.915) −0.472 (0.445) −0.699 (0.573)utildiff −0.0177 (0.908) −0.224 (0.294) −1.52 (< 0.001) −1.72 (< 0.001)utildiff 2 0.122 (0.172) 0.226 (0.061) 1.18 (< 0.001) 1.36 (< 0.001)ML, maximum likelihood; RES, robust expectation-solution.The significant predictors identified in the RES-fitted model largely reproduce those foundby Jackson (2004). In particular, the variable priors is positively associated with the PoissonUOF incident rate as is los, which, as should be expected, has a positive influence on thePoisson rate of incidents and a significant negative effect on the probability of the zero state.utildiff, which is a measure of the variability in crowding at the detention facility, was alsofound to have a positive influence on the probability of being involved in UOF incidents.The RES results in Table 6 were obtained using tuning constant c = 0.01. In addition, werefit this model using the RES method with c = 0.025 to investigate sensitivity to this choice.Generally speaking, effects of using this larger tuning constant were negligible on parameterestimates and p-values associated with γ. Forβ, the larger c value had slightly greater effects,although mainly on the Wald p-values rather than the parameter estimates. Because the largerc value effectively reduces the estimated variability in the data, p-values tended to drop somewhat,although not drastically; qualitative conclusions about the effects of the explanatoryvariables remain mostly the same, with the only change being to priors 2 and utildiff 2 whosep-values dropped just below the traditional 0.05 significance level.© 2009 Board of the Foundation of the Scandinavian Journal of Statistics.
Scand J Statist Robust ZIP regression 13A30B302525Chi−square residuals201510Chi−square residuals20151055000.0 1.0 2.0 3.0Chi−square quantiles (RES)0.0 1.0 2.0 3.0Chi−square quantiles (MLE)Fig. 1. Chi-square residual plots for model fit to the use of force data: (A) chi-square quantiles (robustexpectation solution); (B) chi-square quantiles (maximum likelihood estimation).To further compare the RES and ML fits of this model, we produced plots of chi-squareresiduals. These plots are similar in concept to half normal plots (Atkinson, 1981; Vieira et al.,2000), and were constructed by first binning the data into b bins according to the observedvalues of the response. In this case, we used b = 6 bins corresponding to 0, 1, 2, 3, 4, [5, ∞).Then we calculated residuals defined as the contribution to the chi-square goodness-of-fitstatistic r j = n{f n (y) − f θ,n (y)} 2 /f θ,n (y), j = 1, ..., 6, for each bin based on the fitted model (usingRES or ML). The idea behind this plot is if the model is correctly specified, any collectionof b − 1 residuals should be approximately i.i.d. χ 2 (1) random variables. The residual plot isa plot of ordered r (j) s against χ 2 (1) quantiles. A simulated envelope for this chi-square plotwas constructed in the same way as is typically done in half normal plots (e.g. Vieira et al.,2000). The plots show that most of the residuals fall within the boundaries of the envelopeusing RES (Fig. 1A), but not with ML (Fig. 1B), indicating that the former method is moreappropriate for these data. The one large point in Fig. 1(A) is a desirable feature. It correspondsto the outlying values in the largest bin, which should have large residuals if theestimation downweights these outliers as intended. Although our choice of binning schemeis somewhat arbitrary, we examined chi-square residual plots for other choices of the binsand obtained similar results (omitted for the sake of brevity).Figure 2 displays the downweighting scheme for the UOF model using the RES method. Inparticular, we plot the quantity ω(B i ){ψ c (y i ) − o i (β, c)}/(y i − μ i ) versus y i , i = 1, ..., n. Whilemost observations receive weights near 1, Fig. 2 shows the expected behaviour, with downweightingtending to occur for the larger values of y, and the most extreme downweightingfor the one observation for which y = 22.Finally, note that a negative binomial regression model does not fit these data as well asthe ZIP model, even when fit with ML estimation. For example, Akaike information criterionvalues (smaller is better form) for these two models are 5929.3 (negative binomial)and 5908.4 (ZIP). In addition, the negative binomial model has the disadvantage of not© 2009 Board of the Foundation of the Scandinavian Journal of Statistics.
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