---
title: "Modelagem Estat\\(\\'{i}\\)stica da incid\\(\\^{e}\\)ncia de
        c\\(\\^{a}\\)nceres\\newline via um GLMM Multinomial"
# short-title: ""
author: "Henrique Laureano \\newline \\url{http://leg.ufpr.br/~henrique}"
short-author: "leg.ufpr.br/~henrique"
# email: "laureano\\@ufpr.br"
date: "\\href{https://cidamo.github.io/CiDWeek/}{CiDWeek I}, 03-07/02/2020"
short-date: "leg.ufpr.br/~henrique"
department: ""
institute: ""
# short-institute: "LEG"
section-titles: true
safe-columns: true # enables special latex macros for columns
output:
  legtheme::beamer_leg
---

```{r, include=FALSE}
library(knitr)

knit_hooks$set(rmdsize = function(before, options, envir) {
    if (before) "\\footnotesize"
    else "\\normalsize"
})

opts_chunk$set(rmdsize = TRUE,
               warning = FALSE,
               cache = TRUE,
               cache_path = "20min_cache/")
```

###

\vspace{.5cm}

\begin{columns}
 \begin{column}{.4\linewidth}
  \centering
   \includegraphics[width=.75\linewidth]{what.png}
 \end{column}
 \begin{column}{.6\linewidth}
  O qu\(\^{e}\)?
 \end{column}
\end{columns}

\begin{columns}
 \begin{column}{.4\linewidth}
  \centering
   \includegraphics[width=.75\linewidth]{why.jpg}
 \end{column}
 \begin{column}{.6\linewidth}
  Por qu\(\^{e}\)?
 \end{column}
\end{columns}

\begin{columns}
 \begin{column}{.4\linewidth}
  \centering
   \includegraphics[width=.75\linewidth]{roll_safe.png}
 \end{column}
 \begin{column}{.6\linewidth}
  Como?
 \end{column}
\end{columns}

### \textcolor{beamer@UIUCorange}{O qu\(\^{e}\)?}\newline\hspace*{.5cm}\rotatebox[origin=c]{180}{$\Lsh$} Modelagem Estat\(\'{i}\)stica da incid\(\^{e}\)ncia de c\(\^{a}\)nceres

Considere uma certa estrutura familiar,
\vspace{.25cm}
\begin{columns}
 \begin{column}{.45\linewidth}
  \centering
   \includegraphics[width=\linewidth]{twins.jpg}
 \end{column}
 \begin{column}{.55\linewidth}
  g\(\^{e}\)meos, por exemplo.\newline
  Agora imagine que um deles teve c\(\^{a}\)ncer.
 \end{column}
\end{columns}
\vspace{.2cm}
Como fica a probabilidade do outro:

+ Ter o mesmo tipo de c\(\^{a}\)ncer?

+ Ter de um tipo diferente? Mas similar.

+ Ter de um tipo completamente diferente?

### \textcolor{beamer@UIUCorange}{Por qu\(\^{e}\)?}

Em rela\(\c{c}\~{a}\)o ao problema em si,

+ Estudar a probabilidade de ocorr\(\^{e}\)ncia dos eventos\newline
  (neste caso, c\(\^{a}\)nceres)
  \textcolor{beamer@UIUCblue}{ao longo do tempo};

+ Verificar como as doen\(\c{c}\)as s\(\~{a}\)o conectadas,
  \textcolor{beamer@UIUCblue}{ao longo do tempo}.

\vspace{-.25cm}
\noindent{\color{beamer@UIUCorange}\rule{\linewidth}{0.25mm}}

Em rela\(\c{c}\~{a}\)o ao "como fazer",
\vspace{.3cm}
\begin{columns}
 \begin{column}{.45\linewidth}
  \centering
   \includegraphics[width=\linewidth]{nazare.jpeg}
 \end{column}
 \begin{column}{.55\linewidth}
  Porque \(\'{e}\) complicado.
 \end{column}
\end{columns}

### \textcolor{beamer@UIUCorange}{Como?}\newline\hspace*{.5cm}\rotatebox[origin=c]{180}{$\Lsh$} Via um GLMM Multinomial

\noindent{\color{beamer@UIUCblue}Indo por partes,~\rule{.45\linewidth}{0.25mm}}

\begin{columns}
 \begin{column}{.54\linewidth}

  GLMM: Generalized Linear Mixed Model
  \newline
  \hspace*{.5cm}\rotatebox[origin=c]{180}{$\Lsh$} GLM: Generalized Linear Model
  \newline
  \hspace*{1.15cm}\rotatebox[origin=c]{180}{$\Lsh$} LM: Linear Model

 \end{column}
 \begin{column}{.46\linewidth}

  i.e.,

  \begin{itemize}
   \item[\textcolor{beamer@UIUCblue}{LM}]: \(\mathbb{E}[Y | X] = X\beta\);

   \item[\textcolor{beamer@UIUCblue}{GLM}]:
    \(\mathbb{E}[Y | X] = g^{-1}(X\beta)\);

   \item[\textcolor{beamer@UIUCblue}{GLMM}]:
    \(\mathbb{E}[Y | X, u] = g^{-1}(X\beta + Zu)\).
  \end{itemize}
 \end{column}
\end{columns}

\noindent{\color{beamer@UIUCorange}Multinomial?~\rule{.8\linewidth}{0.25mm}}
\[
 \begin{aligned}
  Y | \{X, u\} &\sim \text{Multinomial}(p_{1}, p_{2}, \dots, p_{k}),\\
             u &\sim \text{Normal Multivariada}(0, \Sigma).
 \end{aligned}
\]
\noindent
{\color{beamer@UIUCblue}Ponto importante:~\rule{.525\linewidth}{0.25mm}}

Fun\(\c{c}\~{a}\)o de incid\(\^{e}\)ncia acumulada:
\[ F_{k}(t | X, u) = \text{Multinom}(Y | X, u) \times \Phi(t, X, u). \]

###

\begin{center}
 \includegraphics[width=.9\linewidth]{thank_you.jpg}
\end{center}
