  Multivariate matching algorithms "pair" similar study units in an
observational study to remove potential bias and confounding effects caused by
the absence of randomizations. In one-to-one multivariate matching algorithms,
a large number of "pairs" to be matched could mean both the information from a
large sample and a large number of tasks, and therefore, to best match the
pairs, such a matching algorithm with efficiency and comparatively limited
auxiliary matching knowledge provided through a "training" set of paired units
by domain experts, is practically intriguing.
  We proposed a novel one-to-one matching algorithm based on a quadratic score
function $S_{\beta}(x_i,x_j)= \beta^T (x_i-x_j)(x_i-x_j)^T \beta$. The weights
$\beta$, which can be interpreted as a variable importance measure, are
designed to minimize the score difference between paired training units while
maximizing the score difference between unpaired training units. Further, in
the typical but intricate case where the training set is much smaller than the
unpaired set, we propose a \underline{s}emisupervised \underline{c}ompanion
\underline{o}ne-\underline{t}o-\underline{o}ne \underline{m}atching
\underline{a}lgorithm (SCOTOMA) that makes the best use of the unpaired units.
The proposed weight estimator is proved to be consistent when the truth
matching criterion is indeed the quadratic score function. When the model
assumptions are violated, we demonstrate that the proposed algorithm still
outperforms some popular competing matching algorithms through a series of
simulations. We applied the proposed algorithm to a real-world study to
investigate the effect of in-person schooling on community Covid-19
transmission rate for policy making purpose.
