Generalized additive models for location, scale and shape
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Cites methods from "Generalized additive models for loc..."
...This method is included in a broader methodology, the GAMLSS [29], which offers a general framework that includes a wide range of known methods for constructing growth curves....
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...The first four distributions were fitted using the GAMLSS (Generalized Additive Models for Location, Scale and Shape) software [27] and the last using the ‘‘xriml’’ module in the STATA software [28]....
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...Using GAMLSS, comparisons were carried out for length/height-for-age, weight-for-age and weight-forlength/height....
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...The GAMLSS allows for modelling the mean (or location) of the growth variable under consideration as well as other parameters of its distribution that determine scale and shape....
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"Generalized additive models for loc..." refers methods in this paper
...(a) Procedure 1: estimate the hyperparameters λ by one of the methods (i) minimizing a profile generalized Akaike information criterion GAIC over λ, (ii) minimizing a profile generalized cross-validation criterion over λ, (iii) maximizing the approximate marginal density (or profile marginal likelihood) for λ by using a Laplace approximation or (iv) approximately maximizing the marginal likelihood for λ by using an (approximate) EM algorithm. (b) Procedure 2: for fixed current hyperparameters λ, use the GAMLSS (RS or CG) algorithm to obtain posterior mode (MAP) estimates of .β, γ/. Procedure 2 is nested within procedure 1 and a numerical algorithm is used to estimate λ. We now consider the methods in more detail. A.2.1. Minimizing a profile generalized Akaike information criterion over λ GAIC (Akaike, 1983) was considered by Hastie and Tibshirani (1990), pages 160 and 261, for hyperparameter estimation in GAMs. In GAMs a cubic smoothing spline function h.x/ is used to model the dependence of a predictor on explanatory variable x. For a single smoothing spline term, since λ is related to the smoothing degrees of freedom df = tr.S/ through equation (6), selection (or estimation) of λ may be achieved by minimizing GAIC.#/, which is defined in Section 6.2, over λ. When the model contains p cubic smoothing spline functions in different explanatory variables, then the corresponding p smoothing hyperparameters λ= .λ1, λ2, . . . , λp/ can be jointly estimated by minimizing GAIC.#/ over λ. However, with multiple smoothing splines Σpj=1 tr.Sj/ is only an approximation to the full model complexity degrees of freedom. The GAIC.#/ criterion can be applied more generally to estimate hyperparameters λ in the distribution of random-effects terms. The (model complexity) degrees of freedom df need to be obtained for models with random-effects terms. This has been considered by Hodges and Sargent (2001). The degrees of freedom of a model with a single random-effects term can be defined as the trace of the random-effect (shrinkage) smoother S, i.e. df= tr.S/, where S is given by equation (6). As with smoothing terms, when there are other terms in the model Σpj=1tr.Sj/ is only an approximation to the full model complexity degrees of freedom. The full model complexity degrees of freedom for model (1) are given by df= tr.A−1B/ where A is defined in Appendix C and B is obtained from A by omitting the matrices Gjk for j =1, 2, . . . , Jk and k =1, 2, . . . , p. A.2.2. Minimizing a generalized cross-validation over λ The generalized cross-validation criterion was considered by Hastie and Tibshirani (1990), pages 259– 263, for hyperparameter estimation in GAMs....
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...Using an Akaike information criterion, i.e. GAIC.2/, for hyperparameter selection, as discussed in Section 6.2 and Appendix A.2.1, led to the conclusion that the random-effect parameters for ν and τ are not needed, i.e. σ3 =σ4 =0. The remaining random-effect parameters were estimated by using the approximate marginal likelihood approach, which is described in Appendix A.2.3, giving fitted parameter values σ̂1 = 13:14 and σ̂2 = 0:0848 with corresponding fixed effects parameter values β̂1 =164:8, β̂2 =−2:213, β̂3 =−0:0697 and β̂4 =2:148 and an approximate marginal deviance of 3118.62 obtained from equation (14) in Appendix A.2.3. This was the chosen fitted model. Since ν̂ = β̂3 =−0:0697 is close to 0, the fitted conditional distribution of yij is approximately defined by σ̂−1 ij log.yij=μ̂ij/∼ tτ̂ , a t-distribution with τ̂ =exp.β̂4/=8:57 degrees of freedom, for i=1, 2, . . . , nj and j =1, 2, . . . , J . Fig. 5 plots the sample and fitted medians (μ) of prind against state (ordered by the sample median). The fitted values of σ (which are not shown here) vary very little. The heterogeneity in the sample variances of prind between the states (in Fig. 4) seems to be primarily due to sampling variation caused by the high skewness and kurtosis in the conditional distribution of y (rather than either the variance–mean relationship or the random effect in σ). Fig. 6 provides marginal (Laplace-approximated) profile deviance plots, as described in Section 6.2, for each of ν and τ , for fixed hyperparameters, giving 95% intervals .−0:866, 0:788/ for ν and .4:6, 196:9/ for τ , indicating considerable uncertainty about these parameters. (The fitted model suggests a log-transformation for y, whereas the added variable plot that was used by Hodges (1998) suggested a Box–Cox transformation parameter ν =0:67 which, although rather different, still lies within the 95% interval for ν....
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...The Akaike information criterion AIC (Akaike, 1974) and the Schwarz Bayesian criterion SBC (Schwarz, 1978) are special cases of the GAIC....
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...(a) Procedure 1: estimate the hyperparameters λ by one of the methods (i) minimizing a profile generalized Akaike information criterion GAIC over λ, (ii) minimizing a profile generalized cross-validation criterion over λ, (iii) maximizing the approximate marginal density (or profile marginal likelihood) for λ by using a Laplace approximation or (iv) approximately maximizing the marginal likelihood for λ by using an (approximate) EM algorithm. (b) Procedure 2: for fixed current hyperparameters λ, use the GAMLSS (RS or CG) algorithm to obtain posterior mode (MAP) estimates of .β, γ/. Procedure 2 is nested within procedure 1 and a numerical algorithm is used to estimate λ. We now consider the methods in more detail. A.2.1. Minimizing a profile generalized Akaike information criterion over λ GAIC (Akaike, 1983) was considered by Hastie and Tibshirani (1990), pages 160 and 261, for hyperparameter estimation in GAMs....
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...Using an Akaike information criterion, i.e. GAIC.2/, for hyperparameter selection, as discussed in Section 6.2 and Appendix A.2.1, led to the conclusion that the random-effect parameters for ν and τ are not needed, i.e. σ3 =σ4 =0. The remaining random-effect parameters were estimated by using the approximate marginal likelihood approach, which is described in Appendix A.2.3, giving fitted parameter values σ̂1 = 13:14 and σ̂2 = 0:0848 with corresponding fixed effects parameter values β̂1 =164:8, β̂2 =−2:213, β̂3 =−0:0697 and β̂4 =2:148 and an approximate marginal deviance of 3118.62 obtained from equation (14) in Appendix A.2.3. This was the chosen fitted model. Since ν̂ = β̂3 =−0:0697 is close to 0, the fitted conditional distribution of yij is approximately defined by σ̂−1 ij log.yij=μ̂ij/∼ tτ̂ , a t-distribution with τ̂ =exp.β̂4/=8:57 degrees of freedom, for i=1, 2, . . . , nj and j =1, 2, . . . , J . Fig. 5 plots the sample and fitted medians (μ) of prind against state (ordered by the sample median). The fitted values of σ (which are not shown here) vary very little. The heterogeneity in the sample variances of prind between the states (in Fig. 4) seems to be primarily due to sampling variation caused by the high skewness and kurtosis in the conditional distribution of y (rather than either the variance–mean relationship or the random effect in σ). Fig. 6 provides marginal (Laplace-approximated) profile deviance plots, as described in Section 6.2, for each of ν and τ , for fixed hyperparameters, giving 95% intervals .−0:866, 0:788/ for ν and .4:6, 196:9/ for τ , indicating considerable uncertainty about these parameters. (The fitted model suggests a log-transformation for y, whereas the added variable plot that was used by Hodges (1998) suggested a Box–Cox transformation parameter ν =0:67 which, although rather different, still lies within the 95% interval for ν. Furthermore the wide interval for τ suggests that a conditional distribution model for yij defined by σ −1 ij log.yij=μij/∼N.0, 1/ may provide a reasonable model. This model has σ̂1 =13:07 and σ̂2 =0:105.) Fig. 7(a) provides a normal QQ-plot for the (normalized quantile) residuals, which were defined in Section 6.2, for the chosen model. Fig. 7(a) indicates an adequate model for the conditional distribution of y. The outlier case for Washington state, identified by Hodges (1998), does not appear to be an outlier in this analysis....
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...In our opinion the Schwarz Bayesian criterion (SBC) is too conservative (i.e. restrictive) in its model selection, leading to bias in the selected functions for µ, σ, ν and τ (particularly at turning-points), whereas the AIC is too liberal, leading to rough (or erratic) selected functions....
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...The conclusion, by looking at the SBC values, is that the Box–Cox t-distribution family fits best, indicating that the distribution of the river flow is both skew and leptokurtic....
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...There is strong support for including a smoothing term for loglos as indicated by the reduction in AIC and SBC for model III compared with model II....
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...This model resulted in an SBC of 3325....
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...Column I of Table 3 shows the global deviance GD and SBC for the resulting model I given by {µ= poly.r1, 2/ + r2 + r3 + .tw + p + p1 Å t90/ Å tp − .tw + p + p1 Å t90/}, for a variety of distribution families, where any scale and shape parameters (e.g. σ, ν and τ ) were modelled as constants....
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