Overcoming the Computing Barriers in Statistical Causal Inference

Abstract

The massive development in statistical causal inference to the era of big data commonly seen in public health applications can be always hindered due to the computational barriers. In this chapter we discuss a practical concern on computing barriers in statistical causal inference with example in optimal pair matching and consequently offer a novel solution by constructing a stratification tree based on exact matching and propensity scores. We demonstrate the implementation of this novel method with a large observational study from Philadelphia obstetric unit closure from 1995 to 2003 with 59 observed covariates in each of the 132,786 birth deliveries and 5,998,111 potential controls. Algorithms and R program code are also provided for interested readers.

Appendix: R Code for Propensity Score Stratification

The following R function opt_pstrat implements the PSS algorithm in Step 3 described above and forms the subclasses. The function takes three arguments as inputs:

• 1. indicator: This argument takes a binary vector which takes value 1 for treated units and 0 for control ones.

• 2. pscore: This argument takes a vector of propensity scores of each unit.

• 3. sizemax: This argument takes a preset tolerance level on the size of the distance matrix. The default value is 9,000,000.

The function opt_pstrat returns with the following values:

• 1. flag: This output returns 1 for successful stratification and 2 otherwise.

• 2. cutoffs: This output returns the cutoff points where the subclasses are split.

• 3. t.pstrata: This output returns a vector listing the number of treated units in each subclass formed.

• 4. c.pstrata: This output returns a vector listing the number of control units in each subclass formed.

• 5. prodsize.pstrata: This output returns a vector listing the size of distance matrix in each subclass formed.

opt_pstrat <−  function ( indicator, pscore, sizemax =9000000){

    cutoffs  <−  max( pscore )

    t. strata  <−  NULL

    c. strata  <−  NULL

    s i z e. strata  <−  NULL

    indicator_iter  <−  indicator

    pscore_iter  <−  pscore

    num_strata_formed <−  0

    while (sum( indicator_iter )>0 &  sum(1− indicator_iter )>0){

        n <−  length ( indicator_iter )

        t_ind <−  which ( indicator_iter==1)

        n_treated <−  sum( indicator_iter )

        treated_pscore_iter  <−  pscore_iter [ t_ind ]

        t_geq_t <−  n_treated+1−rank ( treated_pscore_iter,

        t i e s. method=”min”)

        c_geq_t <−  n +1−rank ( pscore_iter, t i e s. method=”min”

        ) [ t_ind]−t_geq_t

        matchable <−  c_geq_t >=  t_geq_t

        i f  (sum( matchable)>0){

            matchable_set <−  t_ind [ c_geq_t >=  t_geq_t ]

        } e l s e {

            print (”No way to s t r a t i f y:  c_geq_t <  t_geq_t.”

        ); stop}

        s i z e. dist  <−  t_geq_t * c_geq_t

        i f  (min( s i z e. dist [ matchable])> sizemax ){

            print (”No way to s t r a t i f y:  min( s i z e. dist)>sizemax. ” );

            return ( l i s t ( flag =2))

        }

        cutoff. s i z e. ind <−  which ( s i z e. dist== max( s i z e. dist [

        matchable ] [ s i z e. dist [ matchable]<sizemax ] ) ) [ 1 ]

        cutoff  <−  pscore_iter [ t_ind [ cutoff. s i z e. ind ] ]

        cutoffs  <−  c ( cutoffs, cutoff )

        t. strata  <−  c ( t. strata,sum( treated_pscore_iter >=cutoff ))

        c. strata  <−  c ( c. strata,sum( pscore_iter>=cutoff)−sum(

        treated_pscore_iter  >=  cutoff ))

        s i z e. strata  <−  c ( s i z e. strata,sum( treated_pscore_iter >=

        cutoff ) * ( sum( pscore_iter>=cutoff)−sum(

        treated_pscore_iter >=cutoff )))

        num_strata_formed <−  num_strata_formed+1

        print ( num_strata_formed )

        indicator_iter  <−  indicator_iter [ pscore_iter <cutoff ]

        pscore_iter  <−  pscore_iter [ pscore_iter <cutoff ]

    }

    i f  (sum( indicator_iter )==0){

        print (” S t r a t i f i c a t i o n  Finished:  Treated Units Used Up.”)

        return ( l i s t ( flag =1,num_pstrata = num_strata_formed,

        cutoffs=rev ( cutoffs ), t. pstrata=rev ( t. strata ), c. pstrata=

        rev ( c. strata ), prodsize. pstrata=rev ( s i z e. strata )))

    }

    i f  (sum ( ! indicator_iter )==0){

        print (” S t r a t i f i c a t i o n  Finished:  Control Units Used Up.

        Cannot Form New Strata.”)

        return ( l i s t ( flag =2,num_pstrata = num_strata_formed,

        cutoffs=rev ( cutoffs ), t. pstrata=rev ( t. strata ), c. pstrata=

        rev ( c. strata ), prodsize. pstrata=rev ( s i z e. strata )))

    }

}

As described in the main text, the function opt_pstrat is appliedwhen each stratum goes through Step 3. The outputs of this functionprovide useful information on whether to further split or match within thesubclasses.