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With many covariates it can be difficult to carefully examine numeric diagnostics for each; graphical diagnostics can be helpful for getting a quick assessment of the covariate balance. A first step is to examine the distribution of the propensity scores in the original and matched groups; this Three Matching Set is also useful for assessing common support. Figure 1 shows an example with adequate overlap of the propensity scores, with a good control match for each treated individual. For weighting or subclassification, plots such as this can show the dots with their size proportional to their weight.
As
the target string is scanned, REs separated by ‘|’ are tried from left to
right. This means
that once A matches, B will not be tested further, even if it would
produce a longer overall match. To match a literal ‘|’, use \|, or enclose it inside a
character class, as in [|]. Another nonstandard extension is that following the escape character with a letter or digit provides access to the escape sequences defined for POSIX regular expressions; see Table 9.20, Table 9.21, and Table 9.22 below. Weights need to be incorporated into the analysis for matching with replacement or variable ratio matching (Dehejia and Wahba, 1999; Hill et al., 2004). When matching with replacement, control group individuals receive a frequency weight that reflects the number of times they were selected as a match.
As for string literals, octal escapes are always at most
three digits in length. Matches characters considered whitespace in the ASCII character set;
this is equivalent to [ \t\n\r\f\v]. Character-entry escapes exist to make it easier to specify non-printing and other inconvenient characters in REs. The key word ILIKE can be used instead of LIKE to make the match case-insensitive according to the active locale. According to the SQL standard, omitting ESCAPE means there is no escape character (rather than defaulting to a backslash), and a zero-length ESCAPE value is disallowed.
One of the key benefits of randomized experiments for estimating causal effects is that the treated and control groups are guaranteed to be only randomly different from one another on all background covariates, both observed and unobserved. Work on matching methods has examined how to replicate this as much as possible for observed covariates with observational (non-randomized) data. Since early work in matching, which began in the 1940’s, the methods have increased in both complexity and use.
Rubin (2001) illustrates these diagnostics in an example with 146 covariates. For regression adjustment to be trustworthy, the absolute standardized differences of means should be less than 0.25 and the variance ratios should be between 0.5 and 2 (Rubin, 2001). It is also important to consider the implications of common support for the estimand of interest. Examining the common support may indicate that it is not possible to reliably estimate the ATE.
The best results are found when estimating marginal treatment effects, such as differences in means or differences in proportions. Greenland et al. (1999) and Austin (2007) discuss some of the challenges in estimating non-collapsible conditional treatment effects and which matching methods perform best for those situations. A third weighting technique, used primarily in economics, is kernel weighting, which averages over multiple individuals in the control group for each treated individual, with weights defined https://kelleysbookkeeping.com/ by their distance (Imbens, 2000). Heckman et al. (1997, 1998a,b) describe a local linear matching estimator that requires specifying a bandwidth parameter. Generally, larger bandwidths increase bias but reduce variance by putting weight on individuals that are further away from the treated individual of interest. A complication with these methods is this need to define a bandwidth or smoothing parameter, which does not generally have an intuitive meaning; Imbens (2004) provides some guidance on that choice.
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Make \w, \W, \b, \B, \d, \D, \s and \S
perform ASCII-only matching instead of full Unicode matching. This is only
meaningful for Unicode patterns, and is ignored for byte patterns. Matches Unicode whitespace characters (which includes
[ \t\n\r\f\v], and also many other characters, for example the
non-breaking spaces mandated by typography rules in many
languages). Matches if the current position in the string is not preceded by a match for
…. Similar to
positive lookbehind assertions, the contained pattern must only match strings of
some fixed length.
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