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\title{Supporting Information for ``Assessing Threats to Inference with Simultaneous Sensitivity Analysis: The Case of U.S. Supreme Court Oral Arguments''}


\begin{document}
\citationstyle{apsr}

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\newcommand{\tit}{\bf Supporting Information for ``Assessing Threats to Inference with Simultaneous Sensitivity Analysis: The Case of U.S. Supreme Court Oral Arguments''}


{\title{\tit}}%\thanks{For helpful comments, discussion, and suggestions, we thank Larry Baum, Greg Caldeira, William Minozzi, and Dylan Small. We thank Timothy Johnson, James Spriggs, and Paul Wahlbeck for making their data available. A version of this paper was presented at the 2010 Conference on Empirical Legal Studies at Yale University.}}}

    \author{Jeffrey Budziak\and Daniel Lempert}

\date{\today}
\maketitle
\thispagestyle{empty}
%\begin{abstract}
%Political scientists relying on observational data face substantial challenges in drawing causal inferences.  A particularly problematic threat to inference is the unobserved confounder. As a means to assess this threat, we introduce simultaneous sensitivity analysis to the political science literature. As an application, we consider the potentially confounded relationship between Supreme Court justice voting and oral argument quality. We demonstrate that this relationship is sensitive to the presence of a confounder, to a degree that threatens inference, and explore the confounder both theoretically and empirically. More generally, we show how sensitivity analysis can guide inquiry related to a covariate that cannot be directly measured.
%\end{abstract}
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%\section{Supporting Information (Proposed Online-Only)}

\section*{Software for Rosenbaum-style Sensitivity Analysis}
%This Supplemental Information first gives references to software for conducting Rosenbaum-style sensitivity analyses in R and Stata.  Then, information about balance (Supplemental Tables \ref{ba.pbpd}-\ref{ba.rbld}) and additional sensitivity analyses (Supplemental Tables \ref{sa.s1pbpd}-\ref{sa.s2rbld}) is presented. Finally, Supplemental Table \ref{regcr} gives the regression results associated with Figure 2 in main text.

In R, \citeasnoun{ke14} implements primal sensitivity analysis after pair matching, for binary and ordinal/continuous responses. \citeasnoun{ke14} also allows primal sensitivity analysis after fixed-ratio matching with two or three controls, for ordinal/continuous responses. Primal sensitivity analysis for matched pairs is implemented in Stata by \citeasnoun{ga04} (for continuous responses) and by \citeasnoun{suov14} (for binary responses).  For Stata, \citeasnoun{le15} describes software implementing simultaneous sensitivity analysis after pair matching, for binary or continuous responses, and after matching with multiple controls and full matching, for ordinal/continuous responses.  

In all of the software above, inference is based on one of the commonly-used nonparametric tests: the McNemar Test, the Wilcoxon Signed-Rank Test, or the Hodges-Lehmann Aligned Rank Test.  Two R packages described in \citeasnoun{ro15} implement primal sensitivity analysis based on M-Tests for matched pairs and for matching with multiple controls. A two-parameter interpretation of the primal sensitivity analysis (which, roughly speaking, transforms a primal sensitivity analysis into a simultaneous sensitivity analysis) after pair matching is also available. Questions of design and analysis related to the power of a sensitivity analysis are addressed in \citeasnoun{ro12} and \citeasnoun{smet13}; the latter paper points to software that implements both papers' methods.
 
%\clearpage

\section*{Supplemental Tables}
Supplemental Tables \ref{ba.pbpd}-\ref{ba.rbld} give information about balance. Supplemental Tables \ref{sa.s1pbpd}-\ref{sa.s2rbld} present additional sensitivity analyses. Finally, Supplemental Table \ref{regcr} gives the regression results associated with Figure 2 in main text.

\begin{table}[th]\footnotesize
\label{bal.pbpd}
\begin{center}
\begin{tabular}{lrrrrrr}
\hline\\
\vspace{-6.38 mm}\\

& \multicolumn{2}{c}{Pre-Matching} & \multicolumn{2}{c}{Specification 1} & \multicolumn{2}{c}{Specification 2}\\
\hline\\
\vspace{-6.38 mm}\\
Covariate &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$}\\


\hline\\
\vspace{-6.38 mm}\\


U.S. Appellant & .1183 & .3329 & .0341 & .7370 & .0729 & .5119 \\
U.S. Appellee & -.1382 & .2584 & -.1315 & .2503 & -.0374 & .7460 \\
S.G. Appellant & .3404 & .0060 & .1425 & .0791 & .0128 & .8732 \\
S.G. Appellee & .0577 & .6364 & .0047 & .9683 & .0072 & .9524 \\
D.C. Elite Appellant & .0032 & .9790 & .0131 & .8879 & .0600 & .5359 \\
D.C. Elite Appellee & -.1231 & .3137 & -.0285 & .7815 & -.0366 & .6422 \\
Law Professor Appellant & .0468 & .7013 & .0100 & .9383 & .0110 & .9383 \\
Law Professor Appellee & -.0771 & .5276 & -.0515 & .6949 & -.0284 & .8348 \\
Clerk Appellant & .0083 & .9456 & 0 & 1 & -.0205 & .8479 \\
Clerk Appellee & -.0980 & .4223 & -.0159 & .8788 & -.0263 & .8185 \\
Elite Law School Appellant & .0156 & .8981 & .0084 & .9382 & .0196 & .8559 \\
Elite Law School Appellee & -.2187 & .0747 & -.0178 & .8592 & .0683 & .5265 \\
Liberal Decision Below & .1558 & .2029 & .0028 & .9769 & -.1288 & .1606 \\
Relative Experience & .2665 & .0304 & .1016 & .3710 & -.0065 & .9459 \\
Case Complexity & -.0881 & .4707 & -.0269 & .7850 & .0402 & .6852 \\
Court Median Ideology & .1626 & .1841 & .0448 & .5953 & .0199 & .8238 \\


\hline

\end{tabular}
\end{center}
\vspace{-6.38 mm}
\caption{Covariate balance for two matching specifications. The standardized difference of means (sdm) and a randomization inference-based $p$ value are presented for the unmatched sample and matching Specifications 1 and 2.  Petitioner-better, positive-difference cases are considered treated; the corresponding sensitivity analyses are presented in Table 1 and Supplemental Table \ref{sa.s1pbpd}.  See text for details.}
\label{ba.pbpd}
\end{table}


\begin{table}[th]\footnotesize
\label{bal.rbpd}
\begin{center}
\begin{tabular}{lrrrrrr}
\hline\\
\vspace{-6.38 mm}\\

& \multicolumn{2}{c}{Pre-Matching} & \multicolumn{2}{c}{Specification 1} & \multicolumn{2}{c}{Specification 2}\\
\hline\\
\vspace{-6.38 mm}\\
Covariate &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$}\\


\hline\\
\vspace{-6.38 mm}\\

%TAB2

U.S. Appellant & -.1040 & .3688 & -.0147 & .8689 & .0428 & .6394 \\
U.S. Appellee & .1763 & .1287 & .1808 & .0464 & .0007 & .9940 \\
S.G. Appellant & -.1384 & .2323 & -.0592 & .5178 & -.0133 & .8909 \\
S.G. Appellee & .2264 & .0516 & .2357 & .0091 & .0222 & .8040 \\
D.C. Elite Appellant & .0557 & .6301 & .1066 & .1721 & .0457 & .5118 \\
D.C. Elite Appellee & .0725 & .5307 & .1053 & .1546 & .0570 & .5305 \\
Law Professor Appellant & .0537 & .6427 & .0512 & .3173 & .0549 & .4416 \\
Law Professor Appellee & .0537 & .6427 & 0 & 1 & .0279 & .5637 \\
Clerk Appellant & -.0606 & .6003 & .0180 & .8399 & 0 & 1 \\
Clerk Appellee & .1311 & .2578 & .0984 & .2191 & .0203 & .8292 \\
Elite Law School Appellant & -.1284 & .2676 & -.0352 & .6537 & -.0398 & .5954 \\
Elite Law School Appellee & .0949 & .4124 & .1271 & .1372 & -.0293 & .7449 \\
Liberal Decision Below & -.0334 & .7724 & -.0904 & .2915 & -.0405 & .6495 \\
Relative Experience & -.3462 & .0032 & -.2204 & .0190 & .0713 & .3047 \\
Case Complexity & -.1563 & .1775 & -.0288 & .7082 & .0425 & .6008 \\
Court Median Ideology & .0358 & .7569 & .0324 & .7203 & .0390 & .6697 \\

\hline

\end{tabular}
\end{center}
\vspace{-6.38 mm}
\caption{Covariate balance for two matching specifications. The standardized difference of means (sdm) and a randomization inference-based $p$ value are presented for the unmatched sample and matching Specifications 1 and 2.  Respondent-better, positive-difference cases are considered treated; the corresponding sensitivity analyses are presented in Supplemental Tables \ref{sa.s1rbpd} and \ref{sa.s2rbpd}.  See text for details.}
\label{ba.rbpd}
\end{table}


\begin{table}[th]\footnotesize
\label{bal.pbmd}
\begin{center}
\begin{tabular}{lrrrrrr}
\hline\\
\vspace{-6.38 mm}\\

& \multicolumn{2}{c}{Pre-Matching} & \multicolumn{2}{c}{Specification 1} & \multicolumn{2}{c}{Specification 2}\\
\hline\\
\vspace{-6.38 mm}\\
Covariate &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$}\\


\hline\\
\vspace{-6.38 mm}\\

U.S. Appellant & .0894 & .4843 & -.0272 & .7843 & -.0662 & .5560 \\
U.S. Appellee & -.2242 & .0812 & -.1629 & .1955 & -.0108 & .9300 \\
S.G. Appellant & .3630 & .0052 & .1976 & .0401 & -.0288 & .7240 \\
S.G. Appellee & -.0231 & .8564 & .0214 & .8683 & .0575 & .6730 \\
D.C. Elite Appellant & -.0108 & .9325 & -.0429 & .6121 & -.0311 & .7320 \\
D.C. Elite Appellee & -.2030 & .1139 & -.1400 & .2088 & -.0578 & .5586 \\
Law Professor Appellant & .0754 & .5551 & .0623 & .6547 & .0704 & .6547 \\
Law Professor Appellee & -.0606 & .6354 & -.0801 & .5637 & -.0905 & .5637 \\
Clerk Appellant & .0385 & .7635 & .0051 & .9572 & -.0348 & .7591 \\
Clerk Appellee & -.1262 & .3242 & -.0529 & .5637 & -.0598 & .5637 \\
Elite Law School Appellant & .0275 & .8295 & -.0152 & .8878 & -.0971 & .3989 \\
Elite Law School Appellee & -.2615 & .0424 & -.1132 & .2609 & -.0329 & .7602 \\
Liberal Decision Below & .1646 & .1992 & .0629 & .5175 & -.1388 & .1422 \\
Relative Experience & .3606 & .0055 & .2186 & .0504 & -.0458 & .6431 \\
Case Complexity & -.1354 & .2905 & -.0666 & .5489 & -.0041 & .9726 \\
Court Median Ideology & .1704 & .1838 & .0442 & .6178 & .0123 & .8922 \\

\hline

\end{tabular}
\end{center}
\vspace{-6.38 mm}
\caption{Covariate balance for two matching specifications. The standardized difference of means (sdm) and a randomization inference-based $p$ value are presented for the unmatched sample and matching Specifications 1 and 2.  Petitioner-better, medium-difference cases are considered treated; the corresponding sensitivity analyses are presented in Table 2 and Supplemental Table \ref{sa.s1pbmd}.  See text for details.}
\label{ba.pbmd}
\end{table}

\begin{table}[th]\footnotesize
\label{bal.rbmd}
\begin{center}
\begin{tabular}{lrrrrrr}
\hline\\
\vspace{-6.38 mm}\\

& \multicolumn{2}{c}{Pre-Matching} & \multicolumn{2}{c}{Specification 1} & \multicolumn{2}{c}{Specification 2}\\
\hline\\
\vspace{-6.38 mm}\\
Covariate &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$}\\


\hline\\
\vspace{-6.38 mm}\\

U.S. Appellant & -.3371 & .0070 & -.2938 & .0098 & -.0669 & .5277 \\
U.S. Appellee & .1752 & .1570 & .1297 & .2212 & -.0223 & .8514 \\
S.G. Appellant & -.3247 & .0094 & -.2535 & .0153 & -.0150 & .8514 \\
S.G. Appellee & .2281 & .0662 & .2039 & .0504 & .0095 & .9281 \\
D.C. Elite Appellant & -.1105 & .3712 & -.1128 & .2438 & -.0938 & .3855 \\
D.C. Elite Appellee & -.0053 & .9660 & -.0088 & .9257 & -.0251 & .8029 \\
Law Professor Appellant & -.0019 & .9877 & .0662 & .3173 & .0759 & .3173 \\
Law Professor Appellee & .0535 & .6647 & .0594 & .3173 & -.0511 & .6858 \\
Clerk Appellant & -.1589 & .1991 & -.1213 & .1917 & -.0664 & .4817 \\
Clerk Appellee & .1604 & .1947 & .1169 & .1317 & -.0402 & .6115 \\
Elite Law School Appellant & -.1901 & .1249 & -.1278 & .1491 & -.0966 & .2761 \\
Elite Law School Appellee & .0046 & .9702 & .0537 & .4977 & .0143 & .8776 \\
Liberal Decision Below & -.0320 & .7956 & -.1223 & .2236 & -.1088 & .2832 \\
Relative Experience & -.4593 & .0003 & -.3467 & .0021 & -.0073 & .9228 \\
Case Complexity & -.1678 & .1753 & -.0277 & .6973 & .0376 & .6439 \\
Court Median Ideology & .0303 & .8060 & -.0329 & .7230 & -.0043 & .9673 \\
\hline

\end{tabular}
\end{center}
\vspace{-6.38 mm}
\caption{Covariate balance for two matching specifications. The standardized difference of means (sdm) and a randomization inference-based $p$ value are presented for the unmatched sample and matching Specifications 1 and 2.  Respondent-better, medium-difference cases are considered treated; the corresponding sensitivity analyses are presented in Supplemental Tables \ref{sa.s1rbmd} and \ref{sa.s2rbmd}.  See text for details.}
\label{ba.rbmd}
\end{table}



\begin{table}[th]\footnotesize
\label{bal.pbld}
\begin{center}
\begin{tabular}{lrrrrrr}
\hline\\
\vspace{-6.38 mm}\\

& \multicolumn{2}{c}{Pre-Matching} & \multicolumn{2}{c}{Specification 1} & \multicolumn{2}{c}{Specification 2}\\
\hline\\
\vspace{-6.38 mm}\\
Covariate &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$}\\


\hline\\
\vspace{-6.38 mm}\\

U.S. Appellant & .2543 & .0924 & .0362 & .7492 & -.1302 & .3564 \\
U.S. Appellee & -.2506 & .0972 & -.3124 & .0453 & -.0923 & .4857 \\
S.G. Appellant & .4094 & .0074 & .1426 & .1566 & -.0640 & .5949 \\
S.G. Appellee & -.0935 & .5333 & -.1129 & .4658 & -.0281 & .8557 \\
D.C. Elite Appellant & -.0598 & .6904 & -.1057 & .4294 & -.1099 & .3722 \\
D.C. Elite Appellee & -.1662 & .2693 & -.1527 & .2773 & -.0825 & .5338 \\
Law Professor Appellant & -.0037 & .9802 & -.0977 & .3173 & -.1105 & .3173 \\
Law Professor Appellee & -.0037 & .9802 & -.0488 & .7630 & -.1105 & .5637 \\
Clerk Appellant & .1174 & .4345 & -.0203 & .8886 & -.1339 & .3861 \\
Clerk Appellee & -.1366 & .3637 & -.1395 & .1573 & -.0789 & .3173 \\
Elite Law School Appellant & .0080 & .9572 & -.1156 & .3672 & -.1556 & .2105 \\
Elite Law School Appellee & -.1490 & .3216 & -.1445 & .2654 & -.0120 & .9346 \\
Liberal Decision Below & .1475 & .3267 & .1538 & .2545 & -.0055 & .9669 \\
Relative Experience & .4105 & .0072 & .2156 & .0596 & -.0608 & .6312 \\
Case Complexity & -.1307 & .3846 & -.0636 & .5940 & .0472 & .7177 \\
Court Median Ideology & .1070 & .4763 & .0584 & .6252 & -.0250 & .8541 \\

\hline

\end{tabular}
\end{center}
\vspace{-6.38 mm}
\caption{Covariate balance for two matching specifications. The standardized difference of means (sdm) and a randomization inference-based $p$ value are presented for the unmatched sample and matching Specifications 1 and 2.  Petitioner-better, large-difference cases are considered treated; the corresponding sensitivity analyses are presented in Table 3 and Supplemental Table \ref{sa.s1pbld}.  See text for details.}
\label{ba.pbld}
\end{table}


\begin{table}[th]\footnotesize
\label{bal.rbld}
\begin{center}
\begin{tabular}{lrrrrrr}
\hline\\
\vspace{-6.38 mm}\\

& \multicolumn{2}{c}{Pre-Matching} & \multicolumn{2}{c}{Specification 1} & \multicolumn{2}{c}{Specification 2}\\
\hline\\
\vspace{-6.38 mm}\\
Covariate &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$} &\multicolumn{1}{c}{sdm} &\multicolumn{1}{c}{$p$}\\


\hline\\
\vspace{-6.38 mm}\\

U.S. Appellant & -.4907 & .0005 & -.5338 & .0001 & -.0289 & .6467 \\
U.S. Appellee & .3027 & .0287 & .2075 & .0237 & .0091 & .9415 \\
S.G. Appellant & -.4180 & .0028 & -.3926 & .0019 & -.0027 & .9763 \\
S.G. Appellee & .3152 & .0228 & .2275 & .0112 & .0268 & .8150 \\
D.C. Elite Appellant & -.0624 & .6484 & -.0833 & .4922 & -.0814 & .4890 \\
D.C. Elite Appellee & .0456 & .7390 & .0157 & .8694 & -.0388 & .7663 \\
Law Professor Appellant & .0510 & .7096 & .0748 & .3173 & .0921 & .3173 \\
Law Professor Appellee & .0510 & .7096 & 0 & 1 & .0077 & .9334 \\
Clerk Appellant & -.0669 & .6248 & -.1310 & .2087 & -.1038 & .3972 \\
Clerk Appellee & .2141 & .1197 & .1278 & .1797 & -.0227 & .8457 \\
Elite Law School Appellant & -.2189 & .1117 & -.1713 & .0881 & .0106 & .9258 \\
Elite Law School Appellee & .0495 & .7174 & .1246 & .1573 & .0102 & .9198 \\
Liberal Decision Below & -.1520 & .2678 & -.0735 & .5008 & .0095 & .9305 \\
Relative Experience & -.6658 & .0000 & -.5307 & .0002 & -.0250 & .7976 \\
Case Complexity & -.2267 & .0996 & -.1141 & .3013 & .0517 & .6274 \\
Court Median Ideology & -.0391 & .7753 & -.0557 & .6572 & -.0072 & .9525 \\
\hline
\end{tabular}
\end{center}
\vspace{-6.38 mm}
\caption{Covariate balance for two matching specifications. The standardized difference of means (sdm) and a randomization inference-based $p$ value are presented for the unmatched sample and matching Specifications 1 and 2.  Respondent-better, large-difference cases are considered treated; the corresponding sensitivity analyses are presented in Supplemental Tables \ref{sa.s1rbld} and \ref{sa.s2rbld}.  See text for details.}
\label{ba.rbld}
\end{table}


%\section*{Additional Sensitivity Analyses}

\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                         
                                                                                          
        1    &  .0358   &  .0358   &  .0358   &  .0358   &  .0358   &  .0358   &  .0358   &  .0358 \\
      1.1    &  .0358   &  .0375   &  .0392   &  .0437   &  .0497   &  .0542   &  .0576   &  .0823 \\
      1.2    &  .0358   &  .0392   &  .0425   &  .0521   &  .0657   &  .0767   &  .0854    &  .1550 \\
      1.5    &  .0358   &  .0438   &  .0522    & .0790   &  .1239   &  .1638   &  .1973   &  .4751 \\
        2    &  .0358   &  .0501   &  .0668    & .1260   &  .2362   &  .3371   &  .4192   &  .8782 \\
      2.5    &  .0358   &  .0553   &  .0795   &  .1721   &  .3464   &  .4977   &  .6103   &  .9833 \\
        3    &  .0358   &  .0596   &  .0905   &  .2148   &  .4442   &  .6248   &  .7452   &  .9983 \\
 $\infty$    &  .0358   &  .1055    & .2260   &  .6984   &  .9853   &  .9998   &      1  &       1 \\

\hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 1.  Petitioner-better, positive-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.pbpd}.  See text for details.}
\label{sa.s1pbpd}
\end{table}

\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
        1    &  .0292   &  .0292   &  .0292   &  .0292   &  .0292   &  .0292   &  .0292   &  .0292 \\
      1.1    &  .0292   &  .0307   &  .0322   &  .0361   &  .0411   &  .0449   &  .0478   &  .0680 \\
      1.2    &  .0292   &  .0322   &  .0351   &  .0434   &  .0552   &  .0645   &  .0719   &  .1300 \\
      1.5    &  .0292   &  .0362   &  .0436   &  .0676   &  .1076   &  .1429   &  .1726   &  .4213 \\
        2    &  .0292   &  .0417   &  .0565   &  .1107   &  .2127   &  .3062   &  .3830   &  .8405 \\
      2.5    &  .0292   &  .0461   &  .0675   &  .1527   &  .3189   &  .4634   &  .5725   &  .9733 \\
        3    &  .0292   &  .0497   & .0770   	& .1914    & .4140    & .5923    & .7125    & .9966 \\
 $\infty$    &  .0292   &  .0877   &  .1924   &  .6434   &  .9760   &  .9995   &      1   &      1 \\
\hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 1.  Petitioner-better, medium-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.pbmd}.  See text for details.}
\label{sa.s1pbmd}
\end{table}
																																																						\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
        1    &  .0016   &  .0016   &  .0016   &  .0016   &  .0016   &  .0016   &  .0016   &  .0016  \\
      1.1    &  .0016   &  .0017   &  .0018   &  .0020   &  .0023   &  .0026   &  .0027   &  .0043 \\
      1.2    &  .0016   &  .0018   &  .0019   &  .0025   &  .0032   &  .0039   &  .0044   &  .0099 \\
      1.5    &  .0016   &  .0020   &  .0025   &  .0040   &  .0070   &  .0101   &  .0130   &  .0566 \\
        2    &  .0016   &  .0023   &  .0033   &  .0071   &  .0166   &  .0283   &  .0404   &  .2597 \\
      2.5    &  .0016   &  .0026   &  .0040   &  .0106   &  .0293   &  .0545   &  .0820   &  .5315 \\
        3    &  .0016   &  .0028   &  .0046   &  .0142   &  .0439   &  .0858   &  .1322   &  .7492 \\
 $\infty$    &  .0016   &  .0060   &  .0169   &  .1247   &  .5274   &  .8448   &  .9640   &       1 \\
           \hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 1.  Petitioner-better, large-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.pbld}.  See text for details.}
\label{sa.s1pbld}
\end{table}                                                                               


\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
                                                                                          
        1    &  .1804   &  .1804   &  .1804   &  .1804   &  .1804   &  .1804   &  .1804   &  .1804 \\
      1.1    &  .1804   &  .1872   &  .1935   &  .2098   &  .2299   &  .2440   &  .2543   &  .3213 \\
      1.2    &  .1804   &  .1935   &  .2059   &  .2388   &  .2809   &  .3110   &  .3332   &  .4785 \\
      1.5    &  .1804   &  .2102   &  .2395   &  .3210   &  .4292   &  .5059   &  .5605   &  .8414 \\
        2    &  .1804   &  .2321   &  .2852   &  .4361   &  .6258   &  .7422   &  .8123   &  .9913 \\
      2.5    &  .1804   &  .2488   &  .3209   &  .5251   &  .7542   &  .8689   &  .9244   &  .9997 \\
        3    &  .1804   &  .2620   &  .3493   &  .5928   &  .8348   &  .9317   &  .9691   &      1 \\
 $\infty$    &  .1804   &  .3791   &  .5964   &  .9525   &  .9998   &      1     &    1     &    1 \\
           \hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 1.  Respondent-better, positive-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.rbpd}.  See text for details.}
\label{sa.s1rbpd}
\end{table}                                                                               

\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
        1    &  .1424   &  .1424   &  .1424   &  .1424   &  .1424   &  .1424   &  .1424   &  .1424 \\
      1.1    &  .1424   &  .1478   &  .1529   &  .1662   &  .1829   &  .1948   &  .2034   &  .2588 \\
      1.2    &  .1424   &  .1529   &  .1630   &  .1900   &  .2254   &  .2513   &  .2705   &  .3966 \\
      1.5    &  .1424   &  .1665   &  .1905   &  .2590   &  .3541   &  .4247   &  .4766   &  .7651 \\
        2    &  .1424   &  .1845   &  .2287   &  .3599   &  .5389   &  .6591   &  .7371   &  .9773 \\
      2.5    &  .1424   &  .1984   &  .2590   &  .4416   &  .6720   &  .8040   &  .8757   &  .9987 \\
        3    &  .1424   &  .2094   &  .2836   &  .5065   &  .7631   &  .8859   &  .9410   &  .9999 \\
 $\infty$    &  .1424   &  .3081   &  .5063   &  .9109   &  .9991   &      1   &      1   &      1 \\
           \hline

\end{tabular}
\end{center}

\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 1.  Respondent-better, medium-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.rbmd}.  See text for details.}
\label{sa.s1rbmd}
\end{table}                                                                               

\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\        
				1    &  .0343   &  .0343   &  .0343   &  .0343   &  .0343   &  .0343   &  .0343   &  .0343 \\
      1.1    &  .0343   &  .0359   &  .0374   &  .0415   &  .0467   &  .0505   &  .0534   &  .0745 \\
      1.2    &  .0343   &  .0374   &  .0405   &  .0489   &  .0607   &  .0699   &  .0771   &  .1356 \\
      1.5    &  .0343   &  .0415   &  .0491   &  .0726   &  .1106   &  .1436   &  .1712   &  .4103 \\
        2    &  .0343   &  .0471   &  .0617   &  .1127   &  .2055   &  .2903   &  .3604   &  .8149 \\
      2.5    &  .0343   &  .0515   &  .0722   &  .1506   &  .2988   &  .4308   &  .5333   &  .9618 \\
        3    &  .0343   &  .0550   &  .0812   &  .1852   &  .3830   &  .5478   &  .6660   &  .9940 \\
 $\infty$    &  .0343   &  .0932   &  .1922   &  .6094   &  .9615   &  .9985   &      1   &     1 \\
           \hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 1.  Respondent-better, large-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.rbld}.  See text for details.}
\label{sa.s1rbld}
\end{table}
          
																																													
\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
        1    &  .3189   &  .3189   &  .3189   &  .3189   &  .3189   &  .3189   &  .3189   &  .3189 \\
      1.1    &  .3189   &  .3279   &  .3360   &  .3567   &  .3815   &  .3983   &  .4104   &  .4864 \\
      1.2    &  .3189   &  .3361   &  .3519   &  .3924   &  .4412   &  .4745   &  .4982   &  .6422 \\
      1.5    &  .3189   &  .3573   &  .3933   &  .4859   &  .5953   &  .6653   &  .7120   &  .9176 \\
        2    &  .3189   &  .3841   &  .4462   &  .6022   &  .7657   &  .8513   &  .8979   &  .9969 \\
      2.5    &  .3189   &  .4040   &  .4853   &  .6827   &  .8602   &  .9332   &  .9643   &  .9999 \\
        3    &  .3189   &  .4193   &  .5152   &  .7392   &  .9128   &  .9684   &  .9869    &     1 \\
 $\infty$    &  .3189   &  .5442   &  .7397   &  .9784   &  .9999  &       1  &       1   &      1  \\
\hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 2.  Respondent-better, positive-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.rbpd}.  See text for details.}
\label{sa.s2rbpd}
\end{table}
%6F50
\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
                                                                                          
        1     &  .1620    & .1620   &  .1620   &  .1620   &  .1620   &  .1620   &  .1620   &  .1620 \\
      1.1     &  .1620   &  .1673   &  .1721   &  .1847   &  .2004   &  .2116   &  .2198   &  .2753 \\
      1.2     &  .1620   &  .1721   &  .1817   &  .2071   &  .2397   &  .2635   &  .2813   &  .4041 \\
      1.5     &  .1620   &  .1850   &  .2075   &  .2701   &  .3553   &  .4184   &  .4652   &  .7470 \\
        2     &  .1620   &  .2018   &  .2425   &  .3600   &  .5186   &  .6280   &  .7018   &  .9677 \\
      2.5     &  .1620   &  .2146   &  .2699   &  .4318   &  .6384   &  .7648   &  .8391   &  .9974 \\
        3     &  .1620   &  .2247   &  .2918   &  .4887   &  .7237   &  .8485   &  .9120   &  .9998 \\
 $\infty$     &  .1620   &  .3185   &  .4984   &  .8848   &  .9974   &      1  &       1  &       1 \\
\hline

\end{tabular}
\end{center}

\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 2.  Respondent-better, medium-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.rbmd}.  See text for details.}
\label{sa.s2rbmd}

\end{table}                                                                                          
%6F100                                                                                          
\begin{table}[h]\footnotesize
\begin{center}
\begin{tabular}{lcccccccc}
\vspace{-6.38 mm}\\
\hline\\
\vspace{-6.38 mm}\\
    $\Gamma$&      $\Delta$=1 &    $\Delta$=1.1 &   $\Delta$=1.2 &    $\Delta$=1.5 &      $\Delta$=2 &    $\Delta$=2.5 &    $\Delta$=3 &    $\Delta$=$\infty$ \\
\hline\\
\vspace{-6.38 mm}\\                                                                                          
        1    &  .1283   &  .1283   &  .1283   &  .1283   &  .1283   &  .1283   &  .1283   &  .1283 \\
      1.1    &  .1283   &  .1317   &  .1348   &  .1428   &  .1530   &  .1604   &  .1660   &  .2087 \\ 
      1.2    &  .1283   &  .1348   &  .1408   &  .1570   &  .1780   &  .1937   &  .2057   &  .3027 \\
      1.5    &  .1283   &  .1430   &  .1572   &  .1967   &  .2519   &  .2949   &  .3286   &  .5932 \\
        2    &  .1283   &  .1537   &  .1794   &  .2545   &  .3632   &  .4478   &  .5123   &  .8870 \\
      2.5    &  .1283   &  .1621   &  .1972   &  .3029   &  .4550   &  .5691   &  .6506   &  .9753 \\
        3    &  .1283   &  .1687   &  .2117   &  .3432   &  .5295   &  .6602   &  .7479   &  .9954 \\
 $\infty$    &  .1283   &  .2373   &  .3689   &  .7386   &  .9722   &  .9984   &  .9999   &      1 \\
\hline

\end{tabular}
\end{center}
\vspace{-6.38mm}
\caption{Simultaneous sensitivity analysis for selected values of $\Delta$ and $\Gamma$, Matching Specification 2.  Respondent-better, large-difference cases are considered treated; balance is evaluated in Supplemental Table \ref{ba.rbld}.  See text for details.}
\label{sa.s2rbld}

\end{table}                                                                                          

%\section*{Regressions}
\begin{table}[ht]\scriptsize
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\begin{center}
\begin{tabular}{ldd}
\hline \\
\vspace{-6.38 mm}\\
       Covariate             &\multicolumn{1}{c}{Conference vote}         &\multicolumn{1}{c}{Report vote}         \\
\hline \\ 
\vspace{-6.38 mm}\\

Oral Argument Grade &         0.323\sym{***}&        0.339\sym{***}\\
                    &      (0.083)         &      (0.056)         \\
Ideological Affinity &        0.310\sym{***}&        0.354\sym{***}\\
                    &      (0.048)         &       (0.053)         \\

Case Complexity &       0.004         &    0.035         \\
                    &      (0.078)         &      (0.062)         \\
OAG $\times$ Case Complexity &       0.070         &        -0.041         \\
                    &      (0.157)         &      (0.121)         \\
OAG $\times$ Ideological Affinity   &        0.030\sym{*} &        0.037\sym{**}  \\
                    &   (0.014)         &      (0.012)         \\

US Appellant &         0.413\sym{***}&        0.411\sym{***}\\
                    &      (0.105)         &      (0.092)         \\

US Appellee & -0.839\sym{***}         &     -0.896\sym{***}     \\     
                &      (0.196)         &      (0.082)         \\

SG Appellant & 0.268\sym{*}     &     0.197  \sym{*}\\ 
                    &      (0.112)         &      (0.097)         \\
SG Appellee &  0.267     &     -0.070  \\
                    &      (0.073)         &      (0.154)         \\
Washington Elite Appellant & 0.227        &    0.209\sym{*}   \\
                    &      (0.128)         &      (0.089)         \\
Washington Elite Appellee & -0.048        &  0.075     \\
                    &      (0.177)         &      (0.144)         \\
Law Professor Appellant &   -0.385      &    -0.708   \\
                    &      (0.236)         &      (0.180)         \\
Law Professor Appellee &  -0.919\sym{*}       &  -1.085\sym{***}    \\
                    &      (0.416)         &      (0.204)         \\
Clerk Appellant &   0.382 \sym{**}     &     -0.116  \\
                    &      (0.128)         &      (0.102)         \\

Clerk Appellee &   -0.306      &    0.196   \\
                    &      (0.278)         &      (0.238)         \\

Elite Law School Appellant &  -0.135       &    -0.069   \\
                    &      (0.102)         &      (0.109)         \\
Elite Law School Appellee & -0.001        &     -0.066  \\
                    &      (0.115)         &      (0.074)         \\
Difference in Litigating Experience & -0.045         & -0.116      \\
                    &      (0.026)         &      (0.016)         \\
Constant &  0.254       &    0.436   \\
                    &      (0.073)         &      (0.054)         \\
\hline \\



\end{tabular}
\end{center}

\vspace{-6.38 mm}
\caption{Factors impacting conference merits vote and final, report vote. Dependent variable: Did justice vote to reverse? ($1$=yes.)  Logit coefficients; standard errors in parentheses, clustered on justice. $N=3471$ (conference vote); $N=3874$ (report vote).  \sym{*} \(p<0.05\), \sym{**} \(p<0.01\), \sym{***} \(p<0.001\). }
\label{regcr}
\end{table}

\clearpage
\bibliographystyle{apsrdl} %apsrdl.bst modifies apsr.bst to include space after ``issue(vol):'' and before page-page; but citationstyle (above--post-preamble) needs to be apsr.bst to get citations and biblography correct, for s/what unclear reasons 
\bibliography{C:/docs/master}


\end{document}