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\begin{document}
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\begin{center}
{\normal \scshape Humphreys, Sanchez de la Sierra, and van der Windt, 2012. \\ Fishing. Online Appendix B }
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\setcounter{figure}{1}
\section*{Quantities used to generate Figure 2}
\singlespace
Let $q$ denote the probability that a proposition is true, $\beta$ the probability that a researcher is prone to fishing, and $\gamma$ the probability that a non-fishing researcher adopts modified modules (for example due to learning). Let $p_T^o$ and $p_F^o$ denote the probability that a positive result is obtained under an ``orthodox'' plan  (set prior to data analysis) given that the statement is true or false respectively and let $p_T^f$ and $p_F^f$ and $p_T^m$ and $p_F^m$ denote the corresponding probabilities that a fisher will find positive results (given that positive results were not obtained under the orthodox model), and the probability that a nonfisher would receive positive results (conditional on adoption of a modified model). The ``compliers'' are those that do not submit results that deviate from the orthodox model, this group contains both unsuccessful fishers and nonfishers who do not implement modified models. To focus on the sorting function of partial registration, assume that the incentives of researchers and journals are not affected by the form of compliance required. In this case, the share of positive results, positive results that are false, and negative results that are false are given in the Tables below. 

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		% No Registration Case 				NEW
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{\renewcommand{\arraystretch}{3}
\begin{table}[h!]
\centering
\begin{tabular}{|m{3cm}|m{5cm}|}
		\hline

	Share of positives that are false& \multicolumn{1}{|>{\Large}c|}{$ \frac{(1-q)\left((\beta+(1-\beta)(1-\gamma))p_F^o+\beta(1-p_F^o)p_F^f +(1-\beta)\gamma p_F^m\right)} {(\beta+(1-\beta)(1-\gamma))(q p_T^o+(1-q)p_F^o)+\beta(q(1-p_T^o)p_T^f+(1-q)(1-p_F^o)p_F^f) +(1-\beta)\gamma ( q p_T^m+(1-q) p_F^m)} $} \\ \hline 

	Share of negatives that are false& \multicolumn{1}{|>{\Large}c|}{$\frac{q\left(\beta(1-p_T^o)(1-p_T^f)+(1-\beta)(1-\gamma)(1-p_T^o)+(1-\beta)\gamma (1-p_T^m)\right)}
	{\beta[q(1-p_T^f)(1-p_T^o)+(1-q)(1-p_F^f)(1-p_F^o)]+(1-\beta)(1-\gamma)[q(1-p_T^o)+(1-q)(1-p_F^o)]+(1-\beta)\gamma [q(1-p_T^m)+(1-q)(1-p_F^m)]}$} \\ \hline 

	Share of findings that are positive& \multicolumn{1}{|>{\large}c|}{{{ $(\beta+(1-\beta)(1-\gamma))(q p_T^o+(1-q)p_F^o)+(1-\beta)\gamma(q p_T^m+(1-q)p_F^m)+\beta(q (1-p_T^o) p_T^f+(1-q)(1-p_F^o) p_F^f)$}}}\\ \hline
	
\end{tabular}
\caption{Outcomes from \textit{No Registration}.  These quantities are illustrated in the first column of Figure 2.}
\label{noregistration}
\end{table}
}


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		% Full Registration Case 			NEW
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\begin{table}[h!]
\centering
\begin{tabular}{|m{6cm}|m{20cm}|}
	\hline
	Share of positives that are false & \multicolumn{1}{|>{\large}c|}{$\frac{(1-q)p_F^o}{ q p_T^o+(1-q) p_F^o }$}\\  \hline
	Share of negatives that are false & \multicolumn{1}{|>{\large}c|}{$	\frac{q(1-p_T^o)}{ q(1-p_T^o) + (1-q)(1-p_F^o)}$}\\  \hline
	Share of findings that are positive & \multicolumn{1}{|>{\large}c|}{$	qp_T^o +(1-q)p_F^o$}\\ \hline
\end{tabular}
\caption{Outcomes from \textit{Binding} registration.  These quantities are illustrated in the second column of Figure 2.}
\label{fullcompliance}
\end{table}
}



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	\multicolumn{2}{c}{\textbf{Share of positives that are false}:}\\ \hline
	\multicolumn{1}{|c|}{Compliers} & \multicolumn{1}{|>{\Large}c|}{$\frac{(1-q)p_F^o}{q p_T^o+(1-q)p_F^o }$}\\ \hline 
	\multicolumn{1}{|c|}{Noncompliers} & \multicolumn{1}{|>{\Large}c|}{$	\frac{(1-q)(\beta(1-p_F^o)p_F^f+(1-\beta)\gamma p_F^m)}
	{q(\beta(1-p_T^o)p_T^f+\gamma(1-\beta)p_T^m)+(1-q)(\beta(1-p_F^o)p_F^f+\gamma(1-\beta)p_F^m) }	$}\\ \hline

	\multicolumn{2}{c}{\textbf{Share of negatives that are false}:}\\ \hline
	\multicolumn{1}{|c|}{Compliers} & \multicolumn{1}{|>{\Large}c|}{$\frac{ q (1-p_T^o)((1-p_T^f)\beta+(1-\beta)(1-\gamma))}{q (1-p_T^o)((1-p_T^f)\beta+(1-\beta)(1-\gamma))+(1-q) (1-p_F^o)((1-p_F^f)\beta+(1-\beta)(1-\gamma))}$}\\ \hline
	\multicolumn{1}{|c|}{Noncompliers} &\multicolumn{1}{|>{\Large}c|}{$\frac{q(1-p_T^m)}{q(1-p_T^m)+(1-q)(1-p_F^m)}	$}\\ \hline

	\multicolumn{2}{c}{\textbf{Share of findings that are positive}:}\\ \hline
	\multicolumn{1}{|c|}{Compliers} & \multicolumn{1}{|>{\Large}c|}{$\frac{((1-q)p_F^o+q p_T^o)(\beta+(1-\beta)(1-\gamma))}
	{(1-\beta)(1-\gamma)+\beta[q(p_T^o+(1-p_T^o)(1-p_T^f))+(1-q)(p_F^o+(1-p_F^o)(1-p_F^f))]}$}\\ \hline
	\multicolumn{1}{|m{2cm}|}{Noncompliers} & \multicolumn{1}{|>{\Large}c|}{$	\frac{\beta(q(1-p_T^o)p_T^f +(1-q)(1-p_F^o)p_F^f)+(1-\beta)\gamma(q p_T^m+(1-q)p_F^m)}{\beta(q(1-p_T^o)p_T^f +(1-q)(1-p_F^o)p_F^f)+(1-\beta)\gamma}	$}\\\hline

\end{tabular}
\caption{Outcomes from \textit{Nonbinding} registration, by group. These quantities are illustrated in the final two columns of Figure 2.}
\label{partialcompliance}
\end{table}
}
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