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\section{\texttt{summary()}: Numerical Summaries of Balance}
The \texttt{summary()} command returns numerical summaries of balance
diagnostics.
\subsubsection{Syntax}
\begin{verbatim}
summary(object, interactions = FALSE, addlvariables = NULL, standardize = FALSE, ...)
\end{verbatim}
\subsubsection{Arguments}
\begin{itemize}
\item \texttt{object}: the output from {\tt matchit()}.
\item \texttt{interactions}: an option to calculate summary statistics
in \texttt{sum.all} and \texttt{sum.matched} for all covariates,
their squares, and two-way interactions when \texttt{interactions =
TRUE} and only the covariates themselves when \texttt{interactions
= FALSE}, (DEFAULT = {\tt FALSE}).
\item \texttt{addlvariables}: additional variables on which to
calculate the diagnostic statistics (in addition to the variables
included in the matching procedure) (DEFAULT = {\tt NULL}).
\texttt{addlvariables}: a data frame containing additional variables
whose balance is examined. The data should come with the same number
of units and units in the same order as in the data set used for
{\tt matchit()}.
\item \texttt{standardize}: a logical variable indicating whether to
standardize balance measures, i.e., whether the difference in means should be divided
by the standard deviation in the original treated group. (DEFAULT = {\tt FALSE})
\end{itemize}
\subsubsection{Output Values}
The output from the \texttt{summary()} command includes the following
elements, when applicable:
\begin{itemize}
\item The original assignment model call.
\item \texttt{sum.all}: a data frame that contains variable names and
interactions down the row names, and summary statistics on \emph{all
observations} in each of the columns. The columns in
\texttt{sum.all} contain:
%\footnote{The output for full matching is
% slightly different from that described here; see Section
% \ref{subsubsec:full} for details.}
\begin{itemize}
\item means of all covariates $X$ for treated and control units,
where \texttt{Means Treated}$= \mu_{X|T=1} = \frac{1}{n_1}
\sum_{T=1} X_i$ and \texttt{Means Control}$= \mu_{X|T=0} =
\frac{1}{n_0} \sum_{T=0} X_i$,
\item standard deviation in the control group for all covariates $X$, where applicable,
$$\quad s_{x|T=0} = \sqrt{\frac{\sum_{i \in \{i: T_i=0\}}
(X_i - \mu_{X|T=0})^2}{n_0-1} }.$$
\item balance statistics of the original data (before matching),
which compare treated and control covariate distributions. If {\tt
standardize = FALSE}, balance measures will be presented on the
original scale. Specifically, mean differences (\texttt{Mean
Diff.}) as well as the median, mean, and maximum value of
differences in empirical quantile functions for each covariate
will be given (\texttt{eQQ Med}, \texttt{eQQ Mean}, and
\texttt{eQQ Max}, respectively). If {\tt standardize = TRUE}, the
balance measures will be standardized. Standardized mean
differences (\texttt{Std.\ Mean Diff.}), defined as
$\frac{\mu_{X|T=1} - \mu_{X|T=0}}{s_{x|T=1}}$, as well as the
median, mean, and maximum value of differences in empirical
cumulative distribution functions for each covariate will be given
(\texttt{eCDF Med}, \texttt{eCDF Mean}, and \texttt{eCDF Max},
respectively).
\end{itemize}
\item \texttt{sum.matched}: a data frame which contains variable names
down the row names, and summary statistics on only the \emph{matched
observations} in each of the columns. Specifically, the columns
in \texttt{sum.matched} contain the following elements:
%\footnote{The
% values output for full matching are slightly different from that
% described here; see Section \ref{subsubsec:full} for details}:
\begin{itemize}
\item weighted means for matched treatment units and matched control units of all covariates
$X$ and their interactions, where \texttt{Means Treated}$=
\mu_{wX|T=1} = \frac{1}{n_1} \sum_{T=1} w_iX_i$ and \texttt{Means
Control}$=\mu_{wX|T=0} = \frac{1}{n_0} \sum_{T=0} w_iX_i$,
\item weighted standard deviations in the matched control group for
all covariates $X$, where applicable, where \texttt{SD} $= s_{wX} =
\sqrt{\frac{1}{n} \sum_{i} (w_iX_i - \overline{X}^*)^2}$, where
$\overline{X}^*$ is the weighted mean of $X$ in the matched
control group, and
\item balance statistics of the matched data (after matching), which
compare treated and control covariate distributions. If {\tt
standardize = FALSE}, balance measures will be presented on the
original scale. Specifically, mean differences (\texttt{Mean
Diff.}) as well as the median, mean, and maximum value of
differences in empirical quantile functions for each covariate
will be given (\texttt{eQQ Med}, \texttt{eQQ Mean}, and
\texttt{eQQ Max}, respectively). If {\tt standardize = TRUE}, the
balance measures will be standardized. Standardized mean
differences (\texttt{Std.\ Mean Diff.}), defined as
$\frac{\mu_{wX|T=1} - \mu_{wX|T=0}}{s_{x|T=1}}$, as well as the
median, mean, and maximum value of differences in empirical
cumulative distribution functions for each covariate will be given
(\texttt{eCDF Med}, \texttt{eCDF Mean}, and \texttt{eCDF Max},
respectively).
\end{itemize}
where $w$ represents the vector of \texttt{weights}.
\item \texttt{reduction}: the percent reduction in the difference in
means achieved in each of the balance measures in \texttt{sum.all}
and \texttt{sum.matched}, defined as $100(|a|-|b|)/|a|$, where $a$
was the value of the balance measure before matching and $b$ is the
value of the balance measure after matching.
\item \texttt{nn}: the sample sizes in the full and matched samples
and the number of discarded units, by treatment and control.
\item \texttt{q.table}: an array that contains the same information
as \texttt{sum.matched} by subclass.
\item \texttt{qn}: the sample sizes in the full and matched
samples and the number of discarded units, by subclass and by
treatment and control.
\item \texttt{match.matrix}: the same object is contained in the
output of {\tt matchit()}.
\end{itemize}
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