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\documentclass[aspectratio=169]{beamer}
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\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{enumerate}

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\title{Mathematical Biostatistics Boot Camp: Lecture 1, Introduction}
\author{Brian Caffo}
\date{\today}
\institute[Department of Biostatistics]{
  Department of Biostatistics \\
  Johns Hopkins Bloomberg School of Public Health\\
  Johns Hopkins University
}


%\logo{\includegraphics[height=0.5cm]{Logo_PPT.pdf}}

\begin{document}
\frame{\titlepage}


%\section{Table of contents}
\frame{
  \frametitle{Table of contents}
  \tableofcontents
}



\section{Biostatistics}
\frame{
\frametitle{Biostatistics defined}
From the Johns Hopkins Department of Biostatistics 2007 self
  study:
  \begin{quote}
    Biostatistics is a theory and methodology for the acquisition
    and use of quantitative evidence in biomedical research.
    Biostatisticians develop innovative designs and analytic methods
    targeted at increasing available information, improving the
    relevance and validity of statistical analyses, making best use of
    available information and communicating relevant uncertainties.
  \end{quote}
}


\frame{
\frametitle{Example: hormone replacement therapy}
  A large clinical trial (the Women's Health Initiative)
  published results in 2002 that contradicted prior evidence on the
  efficacy of hormone replacement therapy for post menopausal women
  and suggested a negative impact of HRT for several key health
  outcomes. {\bf Based on a statistically based protocol, the study was
  stopped early due an excess number of negative events.} \\ \ \\
  {\tiny See WHI writing group paper JAMA 2002, Vol 288:321 - 333. for the paper and
  Steinkellner et al. Menopause 2012, Vol 19:616 – 621 for a recent discussion
  of the long term impacts}
 
  }

\frame{
\frametitle{Example: ECMO}
  In 1985 a group at a major neonatal intensive care center
  published the results of a trial comparing a standard treatment and
  a promising new extracorporeal membrane oxygenation treatment (ECMO)
  for newborn infants with severe respiratory failure. {\bf Ethical
  considerations lead to a statistical randomization scheme whereby
  one infant received the control therapy, thereby opening the study
  to sample-size based criticisms.} \\ \ \\
  {\tiny For a review of the statistical discussion and discussion, see Royall Statistical Science 1991, Vol 6, No. 1, 52-88}

}


\frame{
\frametitle{Summary}
\begin{itemize}
\item These examples illustrate the central role that biostatistics
  plays in public health and the importance of performing design,
  analysis and interpretation of statistical data correctly.
\item At the Johns Hopkins Bloomberg School of Public Health,
the prevailing philosophy for conducting biostatistics includes:
  \begin{itemize}
  \item A tight coupling of the statistical methods with the ethical and scientific goals.
  \item Emphasizing scientific interpretation of statistical evidence to impact policy.
  \item Acknowledging and assumptions and evaluating the robustness of conclusions to them.
  \end{itemize}
\end{itemize}
}


\section{Experiments}
\frame{
\frametitle{Experiments}
  Consider the outcome of an {\bf experiment} such as:
  \begin{itemize}
  \item a collection of measurements from a sampled population
  \item measurements from a laboratory experiment
  \item the result of a clinical trial
  \item the result from a simulated (computer) experiment
  \item values from hospital records sampled retrospectively
  \item ...
  \end{itemize}
}

\section{Set notation}
\frame{
\frametitle{Notation}
\begin{itemize}
\item The {\bf sample space}, $\Omega$, is the collection of possible
  outcomes of an experiment
  \begin{list}{}{}
  \item Example: die roll $\Omega = \{1,2,3,4,5,6\}$
  \end{list}
\item An {\bf event}, say $E$, is a subset of $\Omega$
  \begin{list}{}{}
  \item Example: die roll is even $E = \{2,4,6\}$
  \end{list}
\item An {\bf elementary} or {\bf simple} event is a particular result
  of an experiment
  \begin{list}{}{}
  \item Example: die roll is a four, $\omega = 4$
  \end{list}
\item $\emptyset$ is called the {\bf null event} or the {\bf empty set}
\end{itemize}
}


\frame{
\frametitle{Interpretation of set operations}
Normal set operations have particular interpretations in this setting
\begin{enumerate}
\item $\omega \in E$ implies that $E$ occurs when $\omega$ occurs
\item $\omega \not\in E$ implies that $E$ does not occur when $\omega$ occurs
\item $E \subset F$ implies that the occurrence of $E$ implies the occurrence of $F$
\item $E \cap F$ implies the event that both $E$ and $F$ occur
\item $E \cup F$ implies the event that at least one of $E$ or $F$ occur
\item $E \cap F=\emptyset$ means that $E$ and $F$ are {\bf mutually exclusive}, or cannot both occur\item $E^c$ or $\bar E$ is the event that $E$ does not occur
\end{enumerate}
}


\frame{
\frametitle{Set theory facts}
\begin{itemize}
\item DeMorgan's laws
  \begin{list}{}{}
  \item $(A \cap B)^c = A^c \cup B^c$
  \item $(A \cup B)^c = A^c \cap B^c$
  \item Example: If an alligator or a turtle you are not[$(A \cup B)^c$]
    then you are not an alligator and you are also not a turtle ($A^c \cap B^c$)
  \item Example: If your car is not both hybrid and diesel [$(A \cap B)^c$]
    then your car is either not hybrid or not diesel ($A^c \cup B^c$)
  \end{list}
\item $(A^c)^c = A$
\item $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$
\end{itemize}
}

\section{Probability}
\frame{
\frametitle{Probability: some discussion}
\begin{itemize}
\item Useful strategy used in much of science:\\
For a given experiment
  \begin{itemize}
  \item attribute all that is known or theorized to a systematic model (mathematical function)
  \item attribute everything else to randomness, {\em even if the
      process under study is known not the be ``random'' in any sense
      of the word}
  \item Use probability to quantify the uncertainty in your conclusions
  \item Evaluate the sensitivity of your conclusions to the assumptions of your model
  \end{itemize}
\end{itemize}
}


%\frame{
%\frametitle{Probability: some discussion}
%\begin{itemize}
%\item Probability has been found extraordinarily useful, even if true {\em
% randomness} is an elusive, undefined, quantity
%\item {\em frequentist} interpretation of probability
% \begin{itemize}
% \item A probability is the long proportion of times an event will occur in
% repeated identical repetitions of an experiment
% \end{itemize}
%\item Other definitions of probability exists
%\item There is not agreement, at all, in how probabilities should be interpreted
%\item There is (nearly) complete agreement on the mathematical rules probability must follow
%\end{itemize}
%}
%
%
%\frame{
%\frametitle{Probability: some discussion}
%\begin{itemize}
%\item An alternative interpretation of probability is so-called
% ``Bayesian''
%\item Named after the $18^{th}$ century Presbyterian Minister / mathematician
% Thomas Bayes
%\item A Bayesian interprets probability as a subjective degree of belief
% \begin{itemize}
% \item For the same event, two separate people could have differing
% probabilities
% \item Bayesian interpretations of probabilities avoid some of the philosophical
% difficulties of frequency interpretations
% \end{itemize}
%\end{itemize}
%}


\frame{
\frametitle{Probability: a useful exercise}
In the first few lectures, we'll largely talk about the mathematics and
basic uses of probability.
However, this is only a small starting point in the use of probability
in data analyses. Keep the following questions in the back of your mind
as we cover using probability for data analyses.
\begin{itemize}
\item What is being modeled as random?
\item Where does this attributed randomness arise from?
\item Where did the systematic model components arise from?
\item How did observational units come to be in the study and is there importance to the missing data points?
\item Do the results generalize beyond the study in question?
\item Were important variables unaccounted for in the model?
\item How drastically would inferences change depending on the answers to the previous questions?
\end{itemize}
}

\end{document}
 
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