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b/docs/_setup.R @@ -34,6 +34,7 @@ opts_chunk$set( echo = FALSE, tidy = TRUE, cache = TRUE, + results = "hide", ## dev = "tikz", fig.width = 7, fig.height = 5, diff --git a/docs/cap04_resultados-e-discussao.Rnw b/docs/cap04_resultados-e-discussao.Rnw index c3ea5af..18e2b6b 100644 --- a/docs/cap04_resultados-e-discussao.Rnw +++ b/docs/cap04_resultados-e-discussao.Rnw @@ -1631,10 +1631,20 @@ seção \ref{sec:fish}). O estudo tem por objetivo a modelagem do número de peixes capturados por grupos de visitantes em um Parque Estadual. As covariáveis mensuradas foram \texttt{np}, o número de pessoas no grupo, \texttt{nc}, o número de crianças e \texttt{ca} variável binária que -indica a presença ou não de um campista no grupo. Com estrutura dos -dados vamos modelar o número de peixes capturados em duas partes, as -contagens nulas e as não nulas, seção \ref{cap02:zeros}. Abaixo -definimos os preditores considerados para as duas partes +indica a presença ou não de um campista no grupo. + +Como já antecipado pela visualização e apresentação dos dados, modelos +estruturados de forma convencional, que pressupõe apenas um processo +estocástico na geração de dados, não se ajustaram adequadamente. A +seguir apresentamos a alternativa de inclusão de um efeito de barreira +para acomodar a quantidade excessiva de valores zero. Os modelos +Poisson, Binomial Negativo e COm-Poisson sob esta estruturação são +ajustados e comparados. + +Com a estrutura dos dados vamos modelar o número de peixes capturados em +duas partes, as contagens nulas e as não nulas, seção +\ref{cap02:zeros}. Abaixo definimos os preditores considerados para as +duas partes \noindent Preditor 1: \quad $ @@ -1649,13 +1659,23 @@ Preditor 1: \quad $ Preditor 2: \quad $ \begin{aligned} g(\mu) &= \beta_0 + \beta_1 \textrm{ca} + - \beta_2 \textrm{np} + \beta_4 \textrm{nc} + - \beta_5 (\textrm{np} \cdot \textrm{nc}) \\ + \beta_2 \textrm{np} + \beta_3 \textrm{nc} + + \beta_4 (\textrm{np} \cdot \textrm{nc}) \\ logit(\pi) &= \gamma_0 + \gamma_1 \textrm{ca} + \gamma_2 \textrm{np} + \gamma_3 \textrm{nc} + - \gamma_5 (\textrm{np} \cdot \textrm{nc}) + \gamma_4 (\textrm{np} \cdot \textrm{nc}) \end{aligned}$ \\ +\noindent +sendo $g(\mu)$ e $logit(\pi)$ as funções de ligação que relacionam os +preditores lineares com as médias dos modelos para contagens não nulas e +contagens zero respectivamente. Os preditores lineares foram propostos +de forma aninhada. No primeiro temos os efeitos aditivos de todas as +covariáveis mensuradas para a parte das contagens nulas e efeitos +aditivos do número de pessoas e de crianças para a parte das contagens +não nulas. No segundo temos os efeitos aditividos de todas as +covariáveis acréscido do efeito de interação entre o número de pessoas e +de crianças para ambas as partes do modelo. <<logLik-fish, include=FALSE>>= @@ -1667,22 +1687,22 @@ anHB <- myanova(m1HB, m2HB) ## Obtem as estimativas dos parametros de dispersao dispHC <- sapply(list(m1HC, m2HC), function(m) m@coef[1]) dispHB <- sapply(list(m1HB, m2HB), function(m) m$theta) -dispersions <- c(NA, NA, dispHC, dispHB) +dispersions <- c(NA, NA, dispHB, dispHC) ## Empilhando tab <- data.frame(pred = rep(paste("Preditor", 1:2), 3)) -tab <- cbind(tab, rbind(anHP, anHC, anHB), dispersions) +tab <- cbind(tab, rbind(anHP, anHB, anHC), dispersions) ## ## Gerando o código latex ## digits <- c(1, 1, 0, 2, 2, 2, 0, -2, 2) -xtable(tab, digits = digits) +## xtable(tab, digits = digits) @ \begin{table}[ht] \centering \small -\caption{Medidas de ajuste para avaliação e comparação entre preditores +\caption{Medidas de ajuste para avaliação e comparação de preditores e modelos com componente de barreira ajustados} \label{tab:ajuste-fish} \begin{tabular}{lcccccrc} @@ -1691,14 +1711,14 @@ Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & \\ \midrule Preditor 1 & 7 & -857,48 & 1728,96 & & & & \\ Preditor 2 & 10 & -744,58 & 1509,17 & 225,79 & 3 & 1,12E-48 & \\[0.3cm] -COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$ \\ - \midrule - Preditor 1 & 8 & -409,85 & 835,71 & & & & -8,77 \\ - Preditor 2 & 11 & -402,30 & 826,59 & 15,12 & 3 & 1,72E-03 & -3,77 \\[0.3cm] Binomial Negativo & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\theta}$ \\ \midrule Preditor 1 & 8 & -399,79 & 815,58 & & & & 0,20 \\ - Preditor 2 & 11 & -393,72 & 809,44 & 12,14 & 3 & 6,91E-03 & 0,37 \\ + Preditor 2 & 11 & -393,72 & 809,44 & 12,14 & 3 & 6,91E-03 & 0,37 \\[0.3cm] +COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$ \\ + \midrule + Preditor 1 & 8 & -409,85 & 835,71 & & & & -8,77 \\ + Preditor 2 & 11 & -402,30 & 826,59 & 15,12 & 3 & 1,72E-03 & -3,77 \\ \bottomrule \end{tabular} \begin{tablenotes} @@ -1709,3 +1729,211 @@ Binomial Negativo & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) \item Fonte: Elaborado pelo autor. \end{tablenotes} \end{table} + +Na tabela \ref{tab:ajuste-fish} as medidas de ajuste dos modelos +Poisson, Binomial Negativo e COM-Poisson são apresentadas para +comparação dos resultados. Observa-se pelas log-verossimilhanças +maximimizadas que o modelo Poisson não se ajustou adequadamente quando +comparado aos demais. Isso se deve ao fato discutido na seção +\ref{cap02:zeros}, que mesmo modelando os zeros podemos ter diferentes +nÃveis de dispersão para as contagens nulas. Nesse exemplo as contagens +não nulas são superdispersas, conforme visto pelas estimativas dos +parâmetros extras do modelo Binomial Negativo e COM-Poisson. Conforme +indicado pelos nÃveis descritivos dos TRV's aplicados nos modelos +encaixados temos a indicação de que o modelo com efeitos de interação é +distinto do modelo com efeitos aditivos definidos no preditor 1. + + +<<coef-fish, results="asis">>= + +##====================================================================== +## Estimativas dos parâmetros + +coHP <- rbind( + c(NA, NA), + do.call(rbind, summary(m2HP)$coef)[, c(1, 3)] +) +## +coHB <- with(summary(m2HB)$coef, { + last <- nrow(count) + rbind( + c(exp(count[last, 1]), count[last, 3]), + rbind(count[-last, c(1, 3)], zero[, c(1, 3)]) + ) +}) +## +coHC <- summary(m2HC)@coef[, c(1, 3)] + +## Empilha +pnames <- c("phi", paste("beta", 0:4, sep = "_"), + paste("gamma", 0:4, sep = "_")) +tab <- data.frame(pnames, cbind(coHP, coHB, coHC)) + +## xtable(tab) + +@ + +As estimativas dos parâmetros para cada especificação de modelos são +exibidas na tabela \ref{tab:coef-fish}. Observe primeiramente que as +estimativas dos parâmetros $\gamma_i$, $i = 0, 1, 2, 3, 4$ são +idênticas, independentemente do modelo adotado. Esse resultado é +esperado, pois na construção dos modelos com componente de barreira, a +modelagem da parte que contempla os valores zero é realizada via +distribuição Bernoulli com parâmetro $\pi = logit(Z\gamma)$. As +diferenças entre os modelos ocorre na distribuição considerada para a +parte das contagens não nulas. + +\begin{table}[ht] +\centering +\caption{Estimativas dos parâmetros e razões entre as estimativa e erro + padrão para os três modelos em estudo} +\label{tab:coef-fish} +\begin{tabular}{lcccccc} + \toprule + & \multicolumn{2}{c}{Poisson} & \multicolumn{2}{c}{Binomial Negativo} & \multicolumn{2}{c}{COM-Poisson} \\ + \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} + Parâmetro & Estimativa & Est/EP & Estimativa & Est/EP & Estimativa & Est/EP \\ + \midrule + Extra ($\phi$, $\theta$) & & & 0,37 & -2,08 & -3,77 & -9,52 \\ + $\beta_0$ & -1,01 & -5,44 & -1,75 & -2,90 & -0,62 & -29,74 \\ + $\beta_1$ & 0,74 & 7,88 & 0,41 & 1,23 & 0,10 & 29,20 \\ + $\beta_2$ & 0,89 & 18,55 & 1,05 & 6,41 & 0,14 & 21,86 \\ + $\beta_3$ & 0,49 & 1,11 & -0,06 & -0,05 & -0,33 & -17,53 \\ + $\beta_4$ & -0,45 & -3,69 & -0,32 & -0,90 & 0,04 & 33,41 \\ + $\gamma_0$ & -2,58 & -5,08 & -2,58 & -5,08 & -2,59 & -5,09 \\ + $\gamma_1$ & 0,98 & 3,00 & 0,98 & 3,00 & 1,00 & 3,04 \\ + $\gamma_2$ & 1,25 & 5,60 & 1,25 & 5,60 & 1,26 & 5,61 \\ + $\gamma_3$ & -0,93 & -1,05 & -0,93 & -1,05 & -0,93 & -1,06 \\ + $\gamma_4$ & -0,41 & -1,41 & -0,41 & -1,41 & -0,41 & -1,41 \\ + \bottomrule +\end{tabular} +\begin{tablenotes} + \small +\item Fonte: Elaborado pelo autor. +\end{tablenotes} +\end{table} + +Nos efeitos estimados para a parte da modelagem dos valores não nulos +têm-se algumas diferenças consideráveis. Destaca-se que o valor das +estimativas dos modelos Poisson e Binomial Negativo são comparáveis +entre si, pois modelam a média da distribuição, mas não comparáveis com +as estimativas do modelo COM-Poisson, pois este modelo um parâmetro que +não representa, unicamente, a média. Contudo, independente da +distribuição o sinal dos efeitos deve ser o mesmo. Isso não ocorre nas +estimativas dos parâmetros $\beta_3$, positiva no modelo Poisson e +negativa nos demais e $\beta_4$, positiva no modelo COM-Poisson e +negativa nos demais. Porém esses efeitos não tem impacto significativo +para definição dos parâmetros das distribuições, conforme podemos ver na +figura \ref{fig:pref-fish} que mostra as médias calculadas com base nas +três formulações. A seguir discutimos sobre os valores apresentados dos +erros padrão dessas estimativas. + +Calculando a magnitude desses efeitos quando escalonados pelo seu erro +padrão, calculado pelo negativo do inverso da matriz hessiana, temos +diferenças substanciais. O modelo COM-Poisson indica erros padrões das +estimativas muito menores que os apresentados no modelo Binomial +Negativo. Sob investigações do problema, encontramos que este resultdo +se deve por inconsistências no procedimento numérico para determinação +da matriz hessiana por diferenças finitas no modelo +COM-Poisson. Portanto, os erros padrão sob o modelo COM-Poisson +apresentados não são confiáveis. + +<<pred-fish, fig.height=4.5, fig.width=7, fig.cap="Valores preditos do número de peixes capturados considerando o número de crianças e pessoas no grupo e a presença de um campista">>= + +##====================================================================== +## Preditos + +## Região para predição +pred <- expand.grid(campista = c("Não", "Sim"), + ncriancas = 0:3, npessoas = 1:4) + +##------------------------------------------- +## Estimando as médias + +## Pelo Hurdle Poisson +aux <- predict(m2HP, newdata = pred, type = "response") +predHP <- cbind(pred, fit = aux, model = "HP") + +## Pelo Hurdle Binomial Negativo +aux <- predict(m2HB, newdata = pred, type = "response") +predHB <- cbind(pred, fit = aux, model = "HB") + +## Pelo Hurdle COM-Poisson +aux <- predict(m2HC, newdata = pred, type = "response") +predHC <- cbind(pred, fit = aux, model = "HC") + +## +pred.all <- rbind(predHP, predHB, predHC) + +##------------------------------------------- +## Gráfico final + +key <- list( + lines = list(lty = 1:3), + text = list( + paste("Hurdle", c("Poisson", "COM-Poisson", + "Binomial Negativo"))) +) + +xyplot(npeixes ~ npessoas | campista, + groups = ncriancas, data = subset(fish, npeixes < 50), + jitter.x = TRUE, + jitter.y = TRUE, + type = c("p", "g"), + xlab = "Número de pessoas no grupo", + ylab = "Número de peixes capturados", + key = key, + alpha = 0.3, + strip = strip.custom( + strip.names = TRUE, var.name = "campista" + )) + + as.layer( + xyplot(fit ~ npessoas | campista, + groups = ncriancas, data = predHP, + type = "l", + lty = 1) + ) + + as.layer( + xyplot(fit ~ npessoas | campista, + groups = ncriancas, data = predHC, + type = "l", + lty = 2) + ) + + as.layer( + xyplot(fit ~ npessoas | campista, + groups = ncriancas, data = predHB, + type = "l", + lty = 3) + ) +##------------------------------------------- +## Legenda +cols <- trellis.par.get("superpose.line")$col[1:4] +draw.key( + key = list( + cex = 0.9, + columns = 1, + lines = list( + lty = 1, lwd = 2, col = cols), + text = list( + as.character(paste(unique(fish$ncriancas), "crianças"))) + ), draw = TRUE, + vp = grid::viewport( + x = grid::unit(0.22, "npc"), y = grid::unit(0.6, "npc"))) + +@ + +Embora tenhamos constatado problemas nos algoritmos numéricos para +determinar a curvatura da log-verossimilhança, temos que as estimativas +pontuais são coerentes com os demais modelos, conforme visto no figura +\ref{fig:pred-fish} onde temos apresentadas as médias calculadas com +base nos três modelos estudados. Observa-se em todos os modelos a +mesma tendência. + +Com esse ilustramos a extensão do modelo COM-Poisson para acomodar +excesso de zeros e ressaltamos que nesse exemplo tivemos contagens não +nulas superdispersas e para esses casos temos a distribuição Binomial +Negativa sendo a principal alternativa. Porém em casos que as contagens +não nulas se apresentam de forma subdispersa não temos opções +prontamente disponÃveis para análise e o modelo COM-Poisson com +componente de barreira, conforme apresentado, se torna uma abordagem +atrativa. -- GitLab