Entire matching and its application in an observational study of treatment for melanoma
Keywords: Causal inference, design, matching, melanoma, observational study, propensity score
How many controls should be matched to a treated subject? Entire matching determines the optimal number, called the entire number, by using propensity scores in a new way. If a treated subject is matched to a number of controls bigger than the entire number, then bias may be induced in the estimate of the treatment effect; if fewer controls are used, then a smaller standard error is possible. The number is optimal because by it, all the bias can be removed on observed covariates while the match ratios are as uniform as possible. As such, entire matching produces an estimator of the treatment effect that is unbiased when treatment assignment is strongly ignorable and, subject to this, has minimum standard error, compared to traditional matched designs such as pair or fixed-ratio matching. In this paper entire matching is motivated by small examples and subsequent theoretical arguments, first in the univariate case, then extended using the theory of propensity scores. In these arguments a formal framework is established which allows proper comparison of matched designs, in which entire matching is shown to be the optimal design. In an application to an observational study of the sentinel lymph node (SLN) biopsy for melanoma, entire matching is shown to reduce more bias than pair matching, while maximizing the size of the control sample. The SLN biopsy has gained widespread use in the treatment of melanoma. According to the Surveillance, Epidemiology, and End Results (SEER) of the National Cancer Institute, almost 70% of patients with intermediate-stage melanomas undergo this procedure. Does it confer a survival benefit? A clinical trial (Morton et al., NEJM 2006) showed no significant improvement in survival, despite a widely held belief that it should. An observational study based on publicly available SEER data can contribute to this ongoing discussion.