If the treatment (T) and control (C) groups are different in observed variables x, then the difference in outcome y cannot be attributed to the difference in the treatment. The obvious solution is to compare only those subjects with the same (or similar) value of x across the two groups. Selecting subjects similar in x across the T and C groups is ‘matching’. Treatment-effect estimators with matching (or simply matching estimators) are introduced first, assuming that the matching has been done already; then how to do matching in practice is discussed. If x is high-dimensional, it is hard to find matched subjects, but there is a simple way to avoid the dimension problem, called ‘propensity score matching’. Matching also can be done to control for unobservables, e.g., using identical twins controls for genes. Combining before-after design with matching yields the popular ‘difference-in-differences’ design, which can be generalized to ‘difference in differences in differences’. These difference-based designs can deal with unobserved confounders to some extent; it is thus fitting that they are examined in the second half of this chapter before hidden bias is dealt with in the following chapters. This chapter employs terms used in medical science and epidemiology: a treated subject is called a ’case’, and a control subject a ‘control’.
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