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\title{A Big Simulation Study\footnote{See last slide for copyright information.}}
\subtitle{STA2053 Fall 2022}
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\frametitle{Design}
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A big simulation study(Brunner and Austin, 2009) with six factors: \pause
\begin{itemize}
    \item Sample size: $n$ = 50, 100, 250, 500, 1000 \pause
    \item $Corr(X_1,X_2)$: $\phi_{12}$ = 0.00, 0.25, 0.75, 0.80, 0.90 \pause
    \item Proportion of variance in $Y$ explained by $X_1$: 0.25, 0.50, 0.75 \pause
    \item Reliability of $W_1$: 0.50, 0.75, 0.80, 0.90, 0.95 \pause
    \item Reliability of $W_2$: 0.50, 0.75, 0.80, 0.90, 0.95 \pause
    \item Distribution  of latent variables and error terms: Normal, Uniform, $t$, Pareto. \pause
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There were $5\times 5\times 3\times 5\times5\times 4$ = 7,500 treatment combinations.
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\frametitle{Simulation study procedure}
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Within each of the $5\times 5\times 3\times 5\times5\times 4$ = 7,500 treatment combinations, 
\begin{itemize}
    \item 10,000 random data sets were generated 
    \item For a total of 75 million data sets 
    \item All generated according to the true model, with $\beta_2=0$. 
    \item Fit naive model, test $H_0:\beta_2=0$ at $\alpha= 0.05$. 
    \item Proportion of times $H_0$ is rejected is a Monte Carlo estimate of the Type I Error Probability.
    \item It should be around 0.05.
\end{itemize}
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\frametitle{Look at a small part of the results} 
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\begin{itemize}
    \item Both reliabilities = 0.90 
    \item Everything is normally distributed 
    \item $\beta_0=1$, $\beta_1=1$ and of course $\beta_2=0$.
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\begin{frame}
\frametitle{Table 1 of Brunner and Austin (2009, p.39)}
\framesubtitle{\emph{Canadian Journal of Statistics}, Vol. 37, Pages 33-46, Used without permission}  \pause
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\includegraphics[width=3in]{BrunnerAustinTable1}
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\frametitle{Marginal Mean Type I Error Probabilities}
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\includegraphics[width=3.5in]{MarginalMeans}
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\frametitle{Poison}
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\begin{itemize}
    \item The poison combination is measurement error in the variable for which you are ``controlling," and correlation between latent explanatory variables.
    \item As the sample size increases, the problem gets worse
    \item For a large enough sample size, no amount of measurement error in the explanatory variables is safe, assuming that the latent explanatory variables are correlated.
\end{itemize}
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\frametitle{Other kinds of regression, other kinds of measurement error} \pause
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\begin{itemize}
    \item Logistic regression 
    \item Proportional hazards regression in survival analysis 
    \item Log-linear models:  Test of conditional independence in the presence of classification error 
    \item Median splits 
    \item Even converting $X_1$ to ranks inflates Type I Error probability.
\end{itemize}
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\frametitle{Moral of the story}
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Use models that allow for measurement error in the explanatory variables. 
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\frametitle{Copyright Information}
This slide show was prepared by  \href{http://www.utstat.toronto.edu/brunner}{Jerry Brunner},
Department of Statistics, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/2053f22} {\footnotesize \texttt{http://www.utstat.toronto.edu/brunner/oldclass/2053f22}}
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