population.Rnw

\section{Population averages}

\begin{frame}
  {\Large Population averages}
\end{frame}

\begin{frame}{Issue}
  \begin{itemize}
   \item Natural summaries
    \begin{itemize}
      \item transition rate $\lambda_{jk}$ from state $j$ to state $k$
      \item $p(t)$, the probability-in-state vector
      \item $E_j(t)$, expected amount of time in state $j$
      \item $v_j(t)$, expected number of visits to state $j$ (lifetime risk)
    \end{itemize}
    \item Hazard models for $\lambda$ are natural
    \item Coefficients from the hazard models do not translate in a simple
      way to the other summaries.
 \end{itemize}
\end{frame}

\begin{frame}{Fundamental Issue}
  We are infatuated with simplicity.
  
 \begin{equation*}
   \log(\lambda(t)) = \beta_0(t) + \beta_1x_1 + \beta_2 x_2 + \ldots
 \end{equation*}
 
 \begin{itemize}
   \item This is the proportional hazards model
     \begin{itemize}
       \item The only time-varying coefficient is $\beta_0$, the 
              ``baseline hazard''
       \item All terms are linear, no interactions
     \end{itemize}
     
   \item If it holds, then the effect of any given covariate is 
     captured by the one number summary $\exp(\beta)$ =
     hazard ratio.
     
   \item What is remarkable is how well this model fits the data for 
     acute endpoints such as time to death for subjects with advanced cancer,
     or waiting time on an organ transplant list.

   \item The model can be stretched to cover repeated events of the same type,
     but not always.
 \end{itemize}
\end{frame}


\begin{frame}{Why focus on simplicity}
  \begin{itemize}
    \item Terse summaries for our papers
    \item Too many projects on our plate
    \item Thoughtful simplicity: models which over-summarize are fit in order to 
      better understand the data, but with the larger context always in mind.
  \end{itemize}
  \pause
  ``All models are wrong, the question is whether they are wrong enough to 
  not be useful.''  GEP Box
  
  ``A model is a lie that helps you see the truth.'' Howard Skipper
\pause  
  ``For every complex question there is a simple and wrong solution.'' 
  A Einstien
\end{frame}

\begin{frame}{Marginal estimates}
  \begin{itemize}
    \item model with $x_1$, $x_2$, $x_3$, \ldots
    \item $PMM_{x1=c} = E_X (\hat y(x) | x_1=c)$ 
    \item Population Marginal Mean
    \item Idea
      \begin{itemize}
        \item Compare treatment A to treatment B
        \item Pretend we have a population of subjects = the other covariates
        \item For each of those subjects we can compute the predicted response
          for their covariates, under treatment A and then under treatment B
        \item Take an average; $PMM_{A} - PMM_B$
      \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Implement}
  \begin{itemize}
    \item Which $\hat y$  
    \item What population for the other covariates $X$
      \begin{itemize}
        \item data set as a whole
        \item fixed data set, e.g., US 2000 age/sex distribution
        \item external data set (calibration)
        \item balanced factorial design (Yates, 1934)
      \end{itemize}
    \item Computation. 
      \begin{itemize}
        \item simple approach: brute force
        \item \code{yates} function
      \end{itemize}
    \item Standard error
      \begin{itemize}
        \item simple in a few cases
        \item parametric simulation
        \item other?
        \item open issue
      \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Old idea}
  \begin{itemize}
    \item $\hat y = S(t)$, population=data: direct adjusted survival
    \item linear model, $\hat y= X\hat\beta$, population=data: closely
      related to survey sampling estimates
    \item g-estimates of causal modeling --- sort of
    \item first instinct of a statistician is to change $Z$ to $E(Z)$
      \pause
    \item linear model, $\hat y= X\hat\beta$, population= factorial for the
      categoricals, data for continuous: SAS GLM type III (SGTT)
      \pause
      \begin{enumerate}
        \item clever and efficient formulas, but forgot what is being computed
        \item horrible documentation (document an algorithm)
        \item factorial population is rarely appropriate
        \item other SAS (and R) procedures do something different (NSTT)
      \end{enumerate}
  \end{itemize}
\end{frame}

\begin{frame}{Issues}
  \begin{itemize}
    \item The model has to be correct
    \item $\hat y$ should be unbiased
      \begin{itemize}
        \item since many values will be averaged, 
          bias is more worrisome than variance
        \item model will often be ``rich'' wrt $x_1$ shape and/or interactions
          with other variables
        \item similar to the thinking used in propensity scoring
      \end{itemize}
    \item Convince our clients that not everything is a hazard ratio
      \begin{itemize}
        \item Expected time in state (RMST)
        \item Expected number of visits
      \end{itemize}
    \item More computation
    \item Variance and power need exploration
  \end{itemize}
\end{frame}