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Statistical Overview

MATCHIT is designed for causal inference with a dichotomous treatment variable and a set of pretreatment control variables. Any number or type of dependent variables can be used. (If you are interested in the causal effect of more than one variable in your data set, run MATCHIT separately for each one; it is unlikely in any event that any one parametric model will produce valid causal inferences for more than one treatment variable at a time.) MATCHIT can be used for other types of causal variables by dichotomizing them, perhaps in multiple ways (see also Imai and van Dyk, 2004). MATCHIT works for experimental data, but is usually used for observational studies where the treatment variable is not randomly assigned by the investigator, or the random assignment goes awry.

We adopt the same notation as in Ho, Imai, King, and Stuart (2007). Unless otherwise noted, let $i$ index the $n$ units in the data set, $n_1$ denote the number of treated units, $n_0$ denote the number of control units (such that $n=n_0+n_1$ ), and $x_i$ indicate a vector of pretreatment (or control) variables for unit $i$ . Let $t_i=1$ when unit $i$ is assigned treatment, and $t_i=0$ when unit $i$ is assigned control. (The labels ``treatment'' and ``control'' and values 1 and 0 respectively are arbitrary and can be switched for convenience, except that some methods of matching are keyed to the definition of the treated group.) Denote $y_i(1)$ as the potential outcome of unit $i$ under treatment -- the value the outcome variable would take if $t_i$ were equal to 1, whether or not $t_i$ in fact is 0 or 1 - and $y_i(0)$ the potential outcome of unit $i$ under control -- the value the outcome variable would take if $t_i$ were equal to 0, regardless of its value in fact. The variables $y_i(1)$ and $y_i(0)$ are jointly unobservable, and for each $i$ , we observe one $y_i=t_iy_i(1)+(1-t_i)y_i(0)$ , and not the other.

Also denote a fixed vector of exogenous, pretreatment measured confounders as $X_i$ . These variables are defined in the hope or under the assumption that conditioning on them appropriately will make inferences ignorable. Measures of balance should be computed with respect to all of $X$ , even if some methods of matching only use some components.