The functions compute a distance matrix, either for a single dataset (i.e., the distances between all pairs of units) or for two groups defined by a splitting variable (i.e., the distances between all units in one group and all units in the other). These distance matrices include the Mahalanobis distance, Euclidean distance, scaled Euclidean distance, and robust (rank-based) Mahalanobis distance. These functions can be used as inputs to the distance argument to matchit() and are used to compute the corresponding distance matrices within matchit() when named.

## Usage

mahalanobis_dist(
formula = NULL,
data = NULL,
s.weights = NULL,
var = NULL,
...
)

scaled_euclidean_dist(
formula = NULL,
data = NULL,
s.weights = NULL,
var = NULL,
...
)

robust_mahalanobis_dist(
formula = NULL,
data = NULL,
s.weights = NULL,
...
)

euclidean_dist(formula = NULL, data = NULL, ...)

## Arguments

formula

a formula with the treatment (i.e., splitting variable) on the left side and the covariates used to compute the distance matrix on the right side. If there is no left-hand-side variable, the distances will be computed between all pairs of units. If NULL, all the variables in data will be used as covariates.

data

a data frame containing the variables named in formula. If formula is NULL, all variables in data will be used as covariates.

s.weights

when var = NULL, an optional vector of sampling weights used to compute the variances used in the Mahalanobis, scaled Euclidean, and robust Mahalanobis distances.

var

for mahalanobis_dist(), a covariance matrix used to scale the covariates. For scaled_euclidean_dist(), either a covariance matrix (from which only the diagonal elements will be used) or a vector of variances used to scale the covariates. If NULL, these values will be calculated using formulas described in Details.

a logical vector denoting which units are to be discarded or not. This is used only when var = NULL. The scaling factors will be computed only using the non-discarded units, but the distance matrix will be computed for all units (discarded and non-discarded).

...

ignored. Included to make cycling through these functions easier without having to change the arguments supplied.

## Value

A numeric distance matrix. When formula has a left-hand-side (treatment) variable, the matrix will have one row for each treated unit and one column for each control unit. Otherwise, the matrix will have one row and one column for each unit.

## Details

The Euclidean distance (computed using euclidean_dist()) is the raw distance between units, computed as $$d_{ij} = \sqrt{(x_i - x_j)(x_i - x_j)'}$$ where $$x_i$$ and $$x_j$$ are vectors of covariates for units $$i$$ and $$j$$, respectively. The Euclidean distance is sensitive to the scales of the variables and their redundancy (i.e., correlation). It should probably not be used for matching unless all of the variables have been previously scaled appropriately or are already on the same scale. It forms the basis of the other distance measures.

The scaled Euclidean distance (computed using scaled_euclidean_dist()) is the Euclidean distance computed on the scaled covariates. Typically the covariates are scaled by dividing by their standard deviations, but any scaling factor can be supplied using the var argument. This leads to a distance measure computed as $$d_{ij} = \sqrt{(x_i - x_j)S_d^{-1}(x_i - x_j)'}$$ where $$S_d$$ is a diagonal matrix with the squared scaling factors on the diagonal. Although this measure is not sensitive to the scales of the variables (because they are all placed on the same scale), it is still sensitive to redundancy among the variables. For example, if 5 variables measure approximately the same construct (i.e., are highly correlated) and 1 variable measures another construct, the first construct will have 5 times as much influence on the distance between units as the second construct. The Mahalanobis distance attempts to address this issue.

The Mahalanobis distance (computed using mahalanobis_dist()) is computed as $$d_{ij} = \sqrt{(x_i - x_j)S^{-1}(x_i - x_j)'}$$ where $$S$$ is a scaling matrix, typically the covariance matrix of the covariates. It is essentially equivalent to the Euclidean distance computed on the scaled principal components of the covariates. This is the most popular distance matrix for matching because it is not sensitive to the scale of the covariates and accounts for redundancy between them. The scaling matrix can also be supplied using the var argument.

The Mahalanobis distance can be sensitive to outliers and long-tailed or otherwise non-normally distributed covariates and may not perform well with categorical variables due to prioritizing rare categories over common ones. One solution is the rank-based robust Mahalanobis distance (computed using robust_mahalanobis_dist()), which is computed by first replacing the covariates with their ranks (using average ranks for ties) and rescaling each ranked covariate by a constant scaling factor before computing the usual Mahalanobis distance on the rescaled ranks.

The Mahalanobis distance and its robust variant are computed internally by transforming the covariates in such a way that the Euclidean distance computed on the scaled covariates is equal to the requested distance. For the Mahalanobis distance, this involves replacing the covariates vector $$x_i$$ with $$x_iS^{-.5}$$, where $$S^{-.5}$$ is the Cholesky decomposition of the (generalized) inverse of the covariance matrix $$S$$.

When a left-hand-side splitting variable is present in formula and var = NULL (i.e., so that the scaling matrix is computed internally), the covariance matrix used is the "pooled" covariance matrix, which essentially is a weighted average of the covariance matrices computed separately within each level of the splitting variable to capture within-group variation and reduce sensitivity to covariate imbalance. This is also true of the scaling factors used in the scaled Euclidean distance.

## References

Rosenbaum, P. R. (2010). Design of observational studies. Springer.

Rosenbaum, P. R., & Rubin, D. B. (1985). Constructing a Control Group Using Multivariate Matched Sampling Methods That Incorporate the Propensity Score. The American Statistician, 39(1), 33–38. doi:10.2307/2683903

Rubin, D. B. (1980). Bias Reduction Using Mahalanobis-Metric Matching. Biometrics, 36(2), 293–298. doi:10.2307/2529981

distance, matchit(), dist() (which is used internally to compute Euclidean distances)

optmatch::match_on(), which provides similar functionality but with fewer options and a focus on efficient storage of the output.

Noah Greifer

## Examples


data("lalonde")

# Computing the scaled Euclidean distance between all units:
d <- scaled_euclidean_dist(~ age + educ + race + married,
data = lalonde)

# Another interface using the data argument:
dat <- subset(lalonde, select = c(age, educ, race, married))
d <- scaled_euclidean_dist(data = dat)

# Computing the Mahalanobis distance between treated and
# control units:
d <- mahalanobis_dist(treat ~ age + educ + race + married,
data = lalonde)

# Supplying a covariance matrix or vector of variances (note:
# a bit more complicated with factor variables)
dat <- subset(lalonde, select = c(age, educ, married, re74))
vars <- sapply(dat, var)

d <- scaled_euclidean_dist(data = dat, var = vars)

# Same result:
d <- scaled_euclidean_dist(data = dat, var = diag(vars))