match_on is generic. There are several supplied methods, all providing
the same basic output: a matrix (or similar) object with treated units on the
rows and control units on the columns. Each cell [i,j] then indicates the
distance from a treated unit i to control unit j. Entries that are Inf
are said to be unmatchable. Such units are guaranteed to never be in a
matched set. For problems with many Inf entries, so called sparse
matching problems, match_on uses a special data type that is more
space efficient than a standard R matrix. When problems are not
sparse (i.e. dense), match_on uses the standard matrix type.
match_on methods differ on the types of arguments they take, making
the function a one-stop location of many different ways of specifying
matches: using functions, formulas, models, and even simple scores. Many of
the methods require additional arguments, detailed below. All methods take a
within argument, a distance specification made using
exactMatch or caliper (or some additive
combination of these or other distance creating functions). All
match_on methods will use the finite entries in the within
argument as a guide for producing the new distance. Any entry that is
Inf in within will be Inf in the distance matrix
returned by match_on. This argument can reduce the processing time
needed to compute sparse distance matrices.
Details for each particular first type of argument follow:
First argument (x): glm. The model is assumed to be
a fitted propensity score model. From this it extracts distances on the
linear propensity score: fitted values of the linear predictor, the
link function applied to the estimated conditional probabilities, as opposed
to the estimated conditional probabilities themselves (Rosenbaum \& Rubin,
1985). For example, a logistic model (glm with
family=binomial()) has the logit function as its link, so from such
models match_on computes distances in terms of logits of the
estimated conditional probabilities, i.e. the estimated log odds.
Optionally these distances are also rescaled. The default is to rescale, by
the reciprocal of an outlier-resistant variant of the pooled s.d. of
propensity scores; see standardization_scale. (The
standardization.scale argument of this function can be used to
change how this dispersion is calculated, e.g. to calculate an ordinary not
an outlier-resistant s.d.; it will be passed down
to standardization_scale as its standardizer argument.)
To skip rescaling, set argument standardization.scale to 1.
The overall result records
absolute differences between treated and control units on linear, possibly
rescaled, propensity scores.
In addition, one can impose a caliper in terms of these distances by
providing a scalar as a caliper argument, forbidding matches between
treatment and control units differing in the calculated propensity score by
more than the specified caliper. For example, Rosenbaum and Rubin's (1985)
caliper of one-fifth of a pooled propensity score s.d. would be imposed by
specifying caliper=.2, in tandem either with the default rescaling
or, to follow their example even more closely, with the additional
specification standardization.scale=sd. Propensity calipers are
beneficial computationally as well as statistically, for reasons indicated
in the below discussion of the numeric method.
One can also specify exactMatching criteria by using strata(foo) inside
the formula to build the glm. For example, passing
glm(y ~ x + strata(s)) to match_on is equivalent to passing
within=exactMatch(y ~ strata(s)). Note that when combining with
the caliper argument, the standard deviation used for the caliper will be
computed across all strata, not within each strata.
If data used to fit the glm have missing values in the left-hand side
(dependent) variable, these observations are omitted from the output of
match_on. If there are observations with missing values in right hand
side (independent) variables, then a re-fit of the model after imputing
these variables using a simple scheme and adding indicator variables of
missingness will be attempted, via the scores function.
First argument (x): bigglm. This method works
analogously to the glm method, but with bigglm objects,
created by the bigglm function from package ‘biglm’, which can
handle bigger data sets than the ordinary glm function can.
First argument (x): formula. The formula must have
Z, the treatment indicator (Z=0 incidates control group,
Z=1 indicates treatment group), on the left hand side, and any
variables to compute a distance on on the right hand side. E.g. Z ~ X1
+ X2. The Mahalanobis distance is calculated as the square root of d'Cd,
where d is the vector of X-differences on a pair of observations and C is an
inverse (generalized inverse) of the pooled covariance of Xes. (The pooling
is of the covariance of X within the subset defined by Z==0 and
within the complement of that subset. This is similar to a Euclidean
distance calculated after reexpressing the Xes in standard units, such that
the reexpressed variables all have pooled SDs of 1; except that it addresses
redundancies among the variables by scaling down variables contributions in
proportion to their correlations with other included variables.)
Euclidean distance is also available, via method="euclidean", and
ranked, Mahalanobis distance, via method="rank_mahalanobis".
The treatment indicator Z as noted above must either be numeric
(1 representing treated units and 0 control units) or logical
(TRUE for treated, FALSE for controls). (Earlier versions of
the software accepted factor variables and other types of numeric variable; you
may have to update existing scripts to get them to run.)
As an alternative to specifying a within argument, when x is
a formula, the strata command can be used inside the formula to specify
exact matching. For example, rather than using within=exactMatch(y ~
z, data=data), you may update your formula as y ~ x + strata(z). Do
not use both methods (within and strata simultaneously. Note
that when combining with the caliper argument, the standard
deviation used for the caliper will be computed across all strata, not
separately by stratum.
A unit with NA treatment status (Z) is ignored and will not be included in the distance output.
Missing values in variables on the right hand side of the formula are handled as follows. By default
match_on will (1) create a matrix of distances between observations which
have only valid values for **all** covariates and then (2) append matrices of Inf values
for distances between observations either of which has a missing values on any of the right-hand-side variables.
(I.e., observations with missing values are retained in the output, but
matches involving them are forbidden.)
First argument (x): function. The passed function
must take arguments: index, data, and z. The
data and z arguments will be the same as those passed directly
to match_on. The index argument is a matrix of two columns,
representing the pairs of treated and control units that are valid
comparisons (given any within arguments). The first column is the row
name or id of the treated unit in the data object. The second column
is the id for the control unit, again in the data object. For each of
these pairs, the function should return the distance between the treated
unit and control unit. This may sound complicated, but is simple to
use. For example, a function that returned the absolute difference between
two units using a vector of data would be f <- function(index, data,
z) { abs(data[index[,1]] - data[index[,2]]) } . (Note: This simple case is
precisely handled by the numeric method.)
First argument (x): numeric. This returns
absolute differences between treated and control units' values of x.
If a caliper is specified, pairings with x-differences greater than
it are forbidden. Conceptually, those distances are set to Inf;
computationally, if either of caliper and within has been
specified then only information about permissible pairings will be stored,
so the forbidden pairings are simply omitted. Providing a caliper
argument here, as opposed to omitting it and afterward applying the
caliper function, reduces storage requirements and may
otherwise improve performance, particularly in larger problems.
For the numeric method, x must have names. If z is named
it must have the same names as x, though it allows for a different
ordering of names. x's name ordering is considered canonical.
First argument (x): matrix or InfinitySparseMatrix. These just return their
arguments as these objects are already valid distance specifications.