Doc/papers/CLS-philosophy.tex

   1 \documentclass{article}
   2
   3 \title{CLS: an approach for a new statistical system}
   4 \author{AJ Rossini}
   5 \date{\today}
   6
   7 \begin{document}
   8
   9 \maketitle
  10
  11 \section{Introduction}
  12 \label{sec:intro}
  13
  14 Statisticians who use a computer for data analysis invariably take one
  15 of two approaches (considered in the extremes here for illustration):
  16 \begin{enumerate}
  17 \item the \emph{FORTRAN} approach of coding numerical and algorithmic
  18   information into the computer program code used for the data
  19   analysis, or
  20 \item the \emph{GUI} approach, via Microsoft Excel, SPSS, Minitab, and
  21   similar approaches, where tasks are facilitated, sometimes with
  22   accompanying workflow support.
  23 \end{enumerate}
  24 Both approaches have co-existed since the early 80s, with the FORTRAN
  25 approach dating back to the dawn of the computing era.
  26
  27 \section{Components of a procedure}
  28 \label{sec:components}
  29
  30 define a statistical procedure as a decision-making approach which
  31 entails the intertwining of formal and informal structure.
  32
  33 Components:
  34 \begin{enumerate}
  35 \item \label{statproc-decision} Decision to make
  36 \item \label{statproc-assessment} Assessment approach to use
  37   (some are inherently different, others just look different)
  38 \item \label{statproc-normalization} Normalization of the problem for
  39   assessment/comparison with other reference behaviours
  40 \item \label{conclusion} Type of conclusion desired, and instance of
  41   that conclusion (when data is present)
  42 \end{enumerate}
  43
  44 This forms an \textit{abstract class} of a procedure, which can be
  45 represented by a real class, which can then be instantiated through
  46 the application of data.
  47
  48 \subsection{Decision}
  49 \label{sec:components:decision}
  50
  51 By example, consider the t-test as an instance of a procedure,
  52 representing the general class of testing hypotheses surrounding 2
  53 means.  Related would be formal likelihood tests with distributions,
  54 the superspace/classes from regression and ANOVA.
  55 Questions could be:
  56 \begin{itemize}
  57 \item are the 2 means the same?
  58 \item what is the difference?
  59 \item what is the strength of the difference?
  60 \end{itemize}
  61
  62 \subsection{Core Assessment}
  63 \label{sec:components:assessment}
  64
  65 This is the construction of the model and parameters that would be
  66 used to form the term used to make the assessment.  Here, we could
  67 consider
  68 \begin{equation}
  69   \label{eq:assess:ex:1}
  70   \hat{E}[Y|G=1] - \hat{E}[Y|G=0]
  71 \end{equation}
  72 as the fundamental quantity to compare.    This can arise from many
  73 sources such as regression models
  74 \begin{equation}
  75   \label{eq:assess:ex:2}
  76   Y = \mu + \beta G + \epsilon \\
  77   E[\epsilon] = 0
  78 \end{equation}
  79 or
  80 \begin{equation}
  81   \label{eq:assess:ex:2}
  82   E[Y|G] = \mu + \beta G
  83 \end{equation}
  84
  85 \subsection{Normalized Behavior}
  86 \label{sec:components:normbeh}
  87 Let $X=(Y,G)$ from above, the whole data.
  88
  89 empirical adjustment:
  90 \begin{equation}
  91   \label{eq:norm:ex:1}
  92   \frac{ \hat\mu_1 - \hat\mu_0}%
  93   {\hat{SE}(\hat\mu_1 - \hat\mu_0)}
  94 \end{equation}
  95 or regression-model-based:
  96 \begin{equation}
  97   \label{eq:norm:ex:2}
  98   \frac{ \hat\beta}%
  99   {\hat{SE}(\hat\beta)}
 100 \end{equation}
 101 or likelihood-model-based: (FIXME!)
 102 \begin{equation}
 103   \label{eq:norm:ex:3}
 104   -2 \log \frac{ L(\hat\beta|X)}%
 105   {L(0|X)}
 106 \end{equation}
 107 or score-model-based:
 108 \begin{equation}
 109   \label{eq:norm:ex:4}
 110   \cal{I}^{-1}(\beta=0,X) S(\beta=0,X)
 111 \end{equation}
 112
 113 \subsection{Conclusion Desired}
 114 \label{sec:component:conclusion}
 115
 116 Value or Range on the Target Scale (existing parameter describing
 117 data-oriented substantive model)
 118
 119 Translation of Value/Range on the Decision Scale (what to do, what to
 120 decide about the problem, i.e. in a testing framework).
 121
 122 \section{Class Implementation}
 123 \label{sec:class}
 124
 125
 126 \section{Discussion}
 127 \label{sec:disc}
 128
 129
 130
 131 \end{document}