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$\chi^2$ de Pearson. \section{Modelo Binomial Negativo} \label{cap02:binomneg} +Uma das principais alternativas paramétricas para dados de contagem +superdispersos é a adoção da distribuição Binomial Negativa. A função +massa de probabilidade da distribuição Binomial Negativa pode ser +deduzida de um processo hierárquico de efeitos aleatórios onde se assume +que + +\begin{equation} + \label{eqn:proc-binomneg} + \begin{split} + Y \mid & b \sim Poisson(b) \\ + & b \sim Gama(\mu, \phi) + \end{split} +\end{equation} + +\noindent +A função massa de probabilidade decorrente da estrutura descrita em +\ref{eqn:proc-binomneg} é deduzida integrando os efeitos aleatórios, +considere $f(y \mid b)$ como a função massa de probablidade da +distribuição Poisson (vide expressão em \ref{eqn:pmf-poisson}) e $g(b +\mid \mu, \phi)$ a função densidade da distribuição Gama \footnote{O + desenvolvimento detalhado da integral pode ser visto em + \citeonline[pág. 303-305]{Paula2013}. Obs.: A função densidade do + modelo Gama está parametrizada para que $\mu$ represente a média da + distribuição.} + +\begin{equation} + \label{eqn:proc-binomneg} + \begin{split} + Pr(Y = y \mid \mu,\phi) &= \int_0^\infty f(y \mid b) + g(b \mid \mu,\phi) db\\ + &= \frac{\phi^\phi}{y!\mu^\phi\Gamma(\phi)} + \int_0^\infty e^{-b(1 + \phi/\mu)} b^{y+\phi-1}db \\ + &= \frac{\Gamma(\phi + y)}{\Gamma(y+1)\Gamma(\phi)} + \left ( \frac{\mu}{\mu + \phi} \right )^y + \left ( \frac{\phi}{\mu + \phi} \right )^\phi + \qquad y = 0, 1, 2, \cdots + \end{split} +\end{equation} + +\noindent +com $\mu >0$ e $\phi > 0$. Ressaltamos que esse é um caso particular de +um modelo de efeito aleatório cuja a integral tem solução analítica e +por consequência o modelo marginal tem forma fechada. Outro caso que se +baseia no mesmo princípio é o modelo \textit{Inverse Gaussian Poisson}, +que como o nome sugere adota a distribuição Inversa Gaussiana para os +efeitos aleatórios. Na figura \ref{fig:distr-binomneg} são apresentadas +as distribuições Binomial Negativa para diferentes parâmetros $\phi$ em +comparação com a distribuição Poisson equivalente em locação. Note que +quanto menor o parâmetro $\phi$, maior a dispersão da distribuição. Isso +introduz uma propriedade importante desse modelo, para $\phi \rightarrow +\infty$ a distribuição reduz-se a Poisson. + +<<distr-binomneg, fig.cap="Probabilidades pela distribuição Binomial Negativa para diferentes valores de $\\phi$ com $\\mu = 5$", fig.height=3.5, fig.width=7>>= + +##------------------------------------------- +## Parametros da distribuição +mu <- 5 +phis <- c("p1" = 1, "p2" = 5, "p3" = 30) +vars <- mu + (1/phis) * mu^2 + +##------------------------------------------- +## Calculando as probabilidades +y <- 0:15 + +## Binomial Negativa +py.bn <- sapply(phis, function(p) dnbinom(y, size = p, mu = mu)) +da.bn <- as.data.frame(py.bn) +da.bn <- cbind(y, stack(da.bn)) + +## Poisson +py.po <- sapply(phis, function(p) dpois(y, lambda = mu)) +da.po <- as.data.frame(py.po) +da.po <- cbind(y, stack(da.po)) + +##------------------------------------------- +## Objetos para grafico da lattice +fl <- substitute( + expression(phi == p1, phi == p2, phi == p3), + list(p1 = phis[1], p2 = phis[2], p3 = phis[3])) +cols <- trellis.par.get("superpose.line")$col[1:2] +yaxis <- pretty(da.po$values, n = 2) +ylim <- c(-0.07, max(da.po$values)*1.2) +key <- list( + columns = 2, + lines = list(lty = 1, col = cols), + text = list(c("Poisson", "Binomial Negativa"))) + + +##------------------------------------------- +## Grafico +xyplot(values ~ c(y - 0.15) | ind, data = da.po, + type = c("h", "g"), + xlab = "y", ylab = expression(P(Y == y)), + ylim = ylim, xlim = extendrange(y), + scales = list(y = list(at = yaxis)), + layout = c(NA, 1), + key = key, + strip = strip.custom(factor.levels = fl)) + + as.layer(xyplot( + values ~ c(y + 0.15) | ind, data = da.bn, + type = "h", col = cols[2])) +for(i in 1:3){ + trellis.focus("panel", i, 1, highlight=FALSE) + grid::grid.text(label = sprintf("E[Y]: %.1f\nV[Y]: %.1f", + mu, mu), + x = .62, y = 0.03, + default.units = "npc", + gp = grid::gpar(col = cols[1]), + just = c("left", "bottom")) + grid::grid.text(label = sprintf("E[Y]: %.1f\nV[Y]: %.1f", + mu, vars[i]), + x = .08, y = 0.03, + default.units = "npc", + gp = grid::gpar(col = cols[2]), + just = c("left", "bottom")) +} +trellis.unfocus() + +@ + +Os momentos média e variância da distribuição Binomial Negativa são +expressos como $E(Y) = \mu$ e $V(Y) = \mu + \mu^2/\phi$. Note que pelas +expressões fica evidente a característica da Binomial Negativa de +acomodar somente superdispersão, pois $E(Y)$ é menor que $V(Y)$ para +qualquer $\phi$. Percebemos também quanto maior o parâmetro $\phi$ mais +$E(Y)$ se aproxima de $V(Y)$, e no limite $\phi \rightarrow \infty$, +$E(Y) = V(Y)$ fazendo com que a distribuição Binomial Negativa se reduza +a Poisson. A relação funcional entre média e variância é ilustrada na +figura \ref{fig:mv-binombeg} onde apesentamos as médias e variâncias +para $\mu$ entre 0 e 10 e $\phi$ entre 0 e 50. O comportamento dessa +relação proporciona um mairo flexibilidade à distribuição em acomodar +superdispersão, uma característica importante exibida nesta figura é que +para a Binomial Negativa se aproximar a Poisson em contagens altas o +$\phi$ deve ser extremamente grande. + +<<mv-binomneg, fig.cap="Relação Média e Variância na distribuição Binomial Negativa", fig.height=4, fig.width=4, fig.show="hide", results="asis">>= + +##------------------------------------------- +## Parâmetros considerados +phi <- seq(0.5, 50, length.out = 50) +col <- rev(brewer.pal(n = 8, name = "RdBu")) +col <- colorRampPalette(colors = col)(length(phi)) + +##------------------------------------------- +## Etiquetas da legenda +labels <- substitute( + expression(phi == p1, phi == p2, phi == p3), + list(p1 = min(phi), p2 = median(phi), p3 = max(phi))) + +##------------------------------------------- +## Gráfico + +## Curva identidade representando a Poisson +par(mar = c(4, 4, 3, 3)) +curve(mu + 1*0, + from = 0, to = 10, xname = "mu", + ylab = expression(V(Y) == mu %.% (mu + mu^2~"/"~phi)), + xlab = expression(E(Y) == mu)) +grid() +## Curvas da relação média e variância da Binomial Negativa +for (a in seq_along(phi)) { + curve(mu + (mu^2)/phi[a], + add = TRUE, xname = "mu", col = col[a], lwd = 2) +} +plotrix::color.legend( + xl = 11, yb = 2.5, xr = 12, yt = 6.5, + gradient = "y", align = "rb", + legend = round(fivenum(phi)[c(1, 3, 5)]), + rect.col = col) +mtext(text = expression(phi), side = 3, cex = 1.5, + line = -4.5, at = 11.5) + +wrapfigure() +@ + +O emprego do modelo Binomial Negativo em problemas se regressão ocorre +de maneira similar aos MLG's, com excessão de que a distribuição só +pertence a família exponencial de distribuições se o parâmetro $\phi$ +for conhecido e assim o processo sofre algumas +alterações. Primeiramente, assim como na Poisson, definimos $g(\mu_i) = +X\beta$, comumente utiliza-se a função $g(\mu_i) = +\log(\mu_i)$. Desenvolvendo a log-verossimilhança e suas funções +derivadas, função escore e matriz de informação de Fisher chegamos que a +matriz de informação é bloco diagonal caracterizando a ortogonalidade +dos parâmetros $\beta$ de locação e $\phi$ de dispersão. Deste fato +decorre que a estimação dos parâmetros pode ser realizada em paralelo, +ou seja, estima-se o vetor $beta$ pelo método de \textit{IWLS} e +posteriormente o parâmetro $\phi$ pelo método de Newton-Raphson, faz-se +osdois procedimentos simultaneamente até a convengência dos parâmetros. + \section{Modelo COM-Poisson} \label{cap02:compoisson} \lipsum[1] -- GitLab