A function with which to produce matching distances, for instance Mahalanobis distances, propensity score discrepancies or calipers, or combinations thereof, for pairmatch or fullmatch to subsequently “match on”. Conceptually, the result of a call match_on is a treatment-by-control matrix of distances. Because these matrices can grow quite large, in practice match_on produces either an ordinary dense matrix or a special sparse matrix structure (that can make use of caliper and exact matching constraints to reduce storage requirements). Methods are supplied for these sparse structures, InfinitySparseMatrixes, so that they can be manipulated and modified in much the same way as dense matrices.

## Usage

match_on(x, within = NULL, caliper = NULL, exclude = NULL, data = NULL, ...)

# S3 method for glm
match_on(
x,
within = NULL,
caliper = NULL,
exclude = NULL,
data = NULL,
standardization.scale = NULL,
...
)

# S3 method for bigglm
match_on(
x,
within = NULL,
caliper = NULL,
exclude = NULL,
data = NULL,
standardization.scale = NULL,
...
)

# S3 method for formula
match_on(
x,
within = NULL,
caliper = NULL,
exclude = NULL,
data = NULL,
subset = NULL,
method = "mahalanobis",
...
)

# S3 method for function
match_on(
x,
within = NULL,
caliper = NULL,
exclude = NULL,
data = NULL,
z = NULL,
...
)

# S3 method for numeric
match_on(x, within = NULL, caliper = NULL, exclude = NULL, data = NULL, z, ...)

# S3 method for InfinitySparseMatrix
match_on(x, within = NULL, caliper = NULL, exclude = NULL, data = NULL, ...)

# S3 method for matrix
match_on(x, within = NULL, caliper = NULL, exclude = NULL, data = NULL, ...)

## Arguments

x

A model formula, fitted glm or other object implicitly specifying a distance; see blurbs on specific methods in Details.

within

A valid distance specification, such as the result of exactMatch or caliper. Finite entries indicate which distances to create. Including this argument can significantly speed up computation for sparse matching problems. Specify this filter either via within or via strata elements of a formula; mixing these methods will fail.

caliper

The width of a caliper to use to exclude treated-control pairs with values greater than the width. For some methods, there may be a speed advantage to passing a width rather than using the caliper function on an existing distance specification.

exclude

A list of units (treated or control) to exclude from the caliper argument (if supplied).

data

An optional data frame.

...

Other arguments for methods.

standardization.scale

Function for rescaling of scores(x), or NULL; defaults to mad. (See Details.)

subset

A subset of the data to use in creating the distance specification.

method

A string indicating which method to use in computing the distances from the data. The current possibilities are "mahalanobis", "euclidean" or "rank_mahalanobis".

z

A logical or binary vector indicating treatment and control for each unit in the study. TRUE or 1 represents a treatment unit, FALSE of 0 represents a control unit. Any unit with NA treatment status will be excluded from the distance matrix.

## Value

A distance specification (a matrix or similar object) which is suitable to be given as the distance argument to

fullmatch or pairmatch.

## Details

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 indicates 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.

## References

P.~R. Rosenbaum and D.~B. Rubin (1985), ‘Constructing a control group using multivariate matched sampling methods that incorporate the propensity score’, The American Statistician, 39 33--38.

fullmatch, pairmatch, exactMatch, caliper

scores

## Examples

data(nuclearplants)
match_on.examples <- list()
### Propensity score distances.
### Recommended approach:
(aGlm <- glm(pr~.-(pr+cost), family=binomial(), data=nuclearplants))
#>
#> Call:  glm(formula = pr ~ . - (pr + cost), family = binomial(), data = nuclearplants)
#>
#> Coefficients:
#> (Intercept)         date           t1           t2          cap           ne
#>   43.909419    -1.220088     0.938645     0.396002     0.001197    -0.995651
#>          ct           bw        cum.n           pt
#>   -2.615671    -0.088696     0.033575    -0.352112
#>
#> Degrees of Freedom: 31 Total (i.e. Null);  22 Residual
#> Null Deviance:	    39.75
#> Residual Deviance: 20.92 	AIC: 40.92
match_on.examples$ps1 <- match_on(aGlm) ### A second approach: first extract propensity scores, then separately ### create a distance from them. (Useful when importing propensity ### scores from an external program.) plantsPS <- predict(aGlm) match_on.examples$ps2 <- match_on(pr~plantsPS, data=nuclearplants)
### Full matching on the propensity score.
fm1 <- fullmatch(match_on.examples$ps1, data = nuclearplants) fm2 <- fullmatch(match_on.examples$ps2, data = nuclearplants)
### Because match_on.glm uses robust estimates of spread,
### the results differ in detail -- but they are close enough
### to yield similar optimal matches.
all(fm1 == fm2) # The same
#> [1] TRUE

### Mahalanobis distance:
match_on.examples$mh1 <- match_on(pr ~ t1 + t2, data = nuclearplants) ### Absolute differences on a scalar: tmp <- nuclearplants$t1
names(tmp) <- rownames(nuclearplants)

(absdist <- match_on(tmp, z = nuclearplants$pr, within = exactMatch(pr ~ pt, nuclearplants))) #>$0
#>        control
#> treated H  I  J K  L M  N  O P Q R S  T U V  W X Y  Z
#>       A 4  0  1 3  2 4  2  2 3 5 7 3  1 8 5  1 6 9 10
#>       B 3  1  0 2  1 3  1  1 2 4 6 2  0 7 4  0 5 8  9
#>       C 1  5  4 2  3 1  3  3 2 0 2 2  4 3 0  4 1 4  5
#>       D 1  5  4 2  3 1  3  3 2 0 2 2  4 3 0  4 1 4  5
#>       E 2  6  5 3  4 2  4  4 3 1 1 3  5 2 1  5 0 3  4
#>       F 8 12 11 9 10 8 10 10 9 7 5 9 11 4 7 11 6 3  2
#>       G 5  9  8 6  7 5  7  7 6 4 2 6  8 1 4  8 3 0  1
#>
#> $1 #> control #> treated d e f #> a 1 3 0 #> b 0 4 1 #> c 6 2 5 #> ### Pair matching on the variable t1: pairmatch(absdist, data = nuclearplants) #> H I A J B K L M C N O P Q R S T #> <NA> 0.1 0.1 <NA> 0.2 <NA> <NA> <NA> 0.3 <NA> <NA> <NA> 0.4 <NA> <NA> <NA> #> U D V E W F X G Y Z d e f a b c #> <NA> 0.4 0.3 0.5 0.2 0.6 0.5 0.7 0.7 0.6 1.2 1.3 1.1 1.1 1.2 1.3 ### Propensity score matching within subgroups: match_on.examples$ps3 <- match_on(aGlm, exactMatch(pr ~ pt, nuclearplants))
fullmatch(match_on.examples$ps3, data = nuclearplants) #> H I A J B K L M C N O P Q R S T U D V E #> 0.3 0.5 0.1 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.5 0.3 0.2 0.1 0.1 0.5 #> W F X G Y Z d e f a b c #> 0.3 0.1 0.3 0.2 0.3 0.3 1.2 1.1 1.3 1.1 1.2 1.3 ### Propensity score matching with a propensity score caliper: match_on.examples$pscal <- match_on.examples$ps1 + caliper(match_on.examples$ps1, 1)
fullmatch(match_on.examples$pscal, data = nuclearplants) # Note that the caliper excludes some units #> H I A J B K L M C N O P Q R S T #> <NA> 1.4 <NA> 1.2 1.1 <NA> <NA> 1.2 1.2 1.2 1.2 <NA> 1.2 1.2 1.4 1.2 #> U D V E W F X G Y Z d e f a b c #> 1.1 1.3 1.3 1.4 1.2 1.3 1.2 1.1 <NA> 1.2 1.4 <NA> 1.2 1.3 1.3 1.1 ### A Mahalanobis distance for matching within subgroups: match_on.examples$mh2 <- match_on(pr ~ t1 + t2 , data = nuclearplants,
within = exactMatch(pr ~ pt, nuclearplants))

### Mahalanobis matching within subgroups, with a propensity score
### caliper:
fullmatch(match_on.examples$mh2 + caliper(match_on.examples$ps3, 1), data = nuclearplants)
#>    H    I    A    J    B    K    L    M    C    N    O    P    Q    R    S    T
#> <NA>  0.1 <NA>  0.2  0.1 <NA> <NA>  0.2  0.2  0.2  0.2 <NA>  0.2  0.2  0.4  0.2
#>    U    D    V    E    W    F    X    G    Y    Z    d    e    f    a    b    c
#>  0.5  0.3  0.3  0.4  0.2  0.5  0.2  0.5 <NA>  0.4  1.1 <NA> <NA>  1.1 <NA>  1.1

### Alternative methods to matching without groups (exact matching)
m1 <- match_on(pr ~ t1 + t2, data=nuclearplants, within=exactMatch(pr ~ pt, nuclearplants))
m2 <- match_on(pr ~ t1 + t2 + strata(pt), data=nuclearplants)
# m1 and m2 are identical

m3 <- match_on(glm(pr ~ t1 + t2 + cost, data=nuclearplants,
family=binomial),
data=nuclearplants,
within=exactMatch(pr ~ pt, data=nuclearplants))
m4 <- match_on(glm(pr ~ t1 + t2 + cost + pt, data=nuclearplants,
family=binomial),
data=nuclearplants,
within=exactMatch(pr ~ pt, data=nuclearplants))
m5 <- match_on(glm(pr ~ t1 + t2 + cost + strata(pt), data=nuclearplants,
family=binomial), data=nuclearplants)
# Including strata(foo) inside a glm uses foo in the model as
# well, so here m4 and m5 are equivalent. m3 differs in that it does
# not include pt in the glm.