Abstract
The basic tools of multivariate matching are introduced, including the propensity score, distance matrices, calipers imposed using a penalty function, optimal matching, matching with multiple controls, and full matching. The tools are illustrated with a tiny example from genetic toxicology (nā=ā46), an example that is so small that one can keep track of individuals as they are matched using different techniques.
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Notes
- 1.
Cochran [12] discusses the relationship between the magnitude of a t-statistic for covariate imbalance and the coverage rate of a confidence interval for a treatment effect when no adjustment is made for the covariate. He concludes that problems begin to occur before the conventional 0.05 level of significance is reached, and that attention should be given to covariates exhibiting a t-statistic of 1.5 in absolute value.
- 2.
The distance need not be, and typically is not, a distance in the sense that the word is used in metric space topology: it need not satisfy the triangle inequality.
- 3.
Direct adjustment can be applied to pretty much anything, not just means and proportions. In particular, an empirical distribution function is little more than a sequence of proportions, and it is clear how to apply direct adjustment to proportions. From the weighted empirical distribution function, pretty much anything else can be computed; for instance, medians and quartiles. In [21], medians and quartiles from directly adjusted empirical distribution functions are used to construct directly adjusted boxplots. In the weighted empirical distribution function, the 44-year-old control in matched set #1 has mass 1ā21, whereas the 63-year-old control in matched set #18 has mass 1ā(4āĆā21).
- 4.
This is true because, when the number of controls matched is the same, optimal matching with variable controls solves a less constrained optimization problem than matching with a fixed ratio, yet the two problems have the same objective function, so the optimum is never worse.
- 5.
Although the proofs of these claims require some attention to detail, the underlying technique may be described briefly. Specifically, if a stratification is not a full match, then some stratum can be subdivided without increasing, and possibly decreasing, its average distance. Subdividing repeatedly terminates in a full match.
- 6.
There is a sense in which Pitmanās asymptotic relative efficiency is not well-defined (or is not meaningful) when there is a bias whose magnitude does not diminish with increasing sample size. See Chap. 16 where this subject is developed in a precise sense.
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R. Rosenbaum, P. (2020). Basic Tools of Multivariate Matching. In: Design of Observational Studies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-46405-9_9
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