The CAUSALMED Procedure

Bootstrap Methods

If you specify the BOOTSTRAP statement, PROC CAUSALMED uses bootstrap resampling to compute standard errors and confidence intervals for causal mediation effects and decompositions. The procedure samples as many bootstrap sample data sets (replicates) as you specify in the NBOOT= option and then estimates the effects and decompositions for each replication.

Bootstrap confidence intervals are computed only for the effects and their corresponding percentages. These intervals are not computed for the parameters in the outcome or mediator models. You can specify one or more of the following types of bootstrap confidence intervals by using the BOOTCI option in the BOOTSTRAP statement:

  • The BOOTCI(NORMAL) option requests bootstrap confidence intervals that are based on the normal approximation method. The left-parenthesis 1 minus alpha right-parenthesis 100 percent-sign normal bootstrap confidence interval is given by

    ModifyingAbove mu With caret Subscript j Baseline plus-or-minus sigma Subscript mu Sub Subscript j Sub Superscript asterisk times normal z Subscript left-parenthesis 1 minus alpha slash 2 right-parenthesis

    where ModifyingAbove mu With caret Subscript jis the estimate of mu Subscript j from the original sample, sigma Subscript mu Sub Subscript j Sub Superscript asterisk is the standard deviation of the bootstrap parameter estimates, and normal z Subscript left-parenthesis 1 minus alpha slash 2 right-parenthesis is the 100 left-parenthesis 1 minus alpha slash 2 right-parenthesisth percentile of the standard normal distribution.

  • The BOOTCI(PERC) option requests bootstrap confidence intervals that are based on the percentile method. The confidence limits are the 100 left-parenthesis alpha slash 2 right-parenthesisth and 100 left-parenthesis 1 minus alpha slash 2 right-parenthesisth percentiles of the bootstrap parameter estimates, which are computed as follows. Let mu Subscript j comma 1 Superscript asterisk Baseline comma mu Subscript j comma 2 Superscript asterisk Baseline comma …comma mu Subscript j comma upper B Superscript asterisk represent the ordered values of the bootstrap estimates for the potential outcome mean mu Subscript j. Let the kth weighted average percentile be q, set p equals StartFraction k Over 100 EndFraction, and let

    n p equals l plus g

    where l is the integer part of n p and g is the fractional part of n p. Then the kth percentile, q, is computed as follows, which corresponds to the default percentile definition used by the UNIVARIATE procedure:

    q equals StartLayout Enlarged left-brace 1st Row 1st Column one-half left-parenthesis mu Subscript j comma l Superscript asterisk Baseline plus mu Subscript j comma l plus 1 Superscript asterisk Baseline right-parenthesis 2nd Column if g equals 0 2nd Row 1st Column mu Subscript j comma l plus 1 Superscript asterisk Baseline 2nd Column if g greater-than 0 EndLayout

  • The BOOTCI(BC) option requests bias-corrected bootstrap confidence intervals, which use the cumulative distribution function (CDF), upper G left-parenthesis mu Superscript asterisk Baseline right-parenthesis, of the bootstrap parameter estimates to determine the upper and lower endpoints of the confidence interval. The bias-corrected bootstrap confidence interval is given by

    upper G Superscript negative 1 Baseline left-parenthesis normal upper Phi left-parenthesis 2 z 0 plus-or-minus z Subscript alpha slash 2 Baseline right-parenthesis right-parenthesis

    where normal upper Phi is the standard normal CDF, z Subscript alpha slash 2 Baseline equals normal upper Phi Superscript negative 1 Baseline left-parenthesis alpha slash 2 right-parenthesis, and z 0 is a bias correction,

    z 0 equals normal upper Phi Superscript negative 1 Baseline left-parenthesis StartFraction normal upper N left-parenthesis mu Subscript j Superscript asterisk Baseline less-than-or-equal-to ModifyingAbove mu With caret Subscript j Baseline right-parenthesis Over normal upper B EndFraction right-parenthesis

    where ModifyingAbove mu With caret Subscript j is the original sample estimate of mu Subscript j from the input data set, normal upper N left-parenthesis mu Subscript j Superscript asterisk Baseline less-than-or-equal-to ModifyingAbove mu With caret Subscript j Baseline right-parenthesis is the number of bootstrap estimates (mu Subscript j Superscript asterisk) that are less than or equal to ModifyingAbove mu With caret Subscript j, and B is the number of bootstrap replicates for which an estimate for the treatment effect is obtained.

Bias-corrected bootstrap confidence intervals are the default.

PROC CAUSALMED requires at least 50 bootstrap samples for normal bootstrap confidence intervals and does not compute them if fewer than 40 of the samples produce usable estimates. The procedure requires at least 1,000 bootstrap samples for percentile and bias-corrected bootstrap confidence intervals and does not compute them if fewer than 900 of the samples produce usable estimates. If the number of samples n specified in the NBOOT=n option is less than 1,000 and percentile or bias-corrected bootstrap confidence intervals are requested, the value of n is ignored.

Last updated: December 09, 2022