Statistical Matching Workflow: Match, Check Balance, and (Possibly) Iterate
An Overview of the Family of Three Matching Functions match_2C, match_2C_mat, and match_2C_list
In the package match2C, three functions are primarily responsible for the main task statistical matching. These three functions are match_2C, match_2C_mat, and match_2C_list. We will examine more closely their differences and illustrate their usage with numerous examples in later sections. In this section we give a high-level outline of what each of them does. In short, the three functions have the same output format (details in the next section), but are different in their inputs.
Function match_2C_mat takes as input at least one distance matrix. A distance matrix is a n_t-by-b_c matrix whose ij-th entry encodes a measure of distance (or similarity) between the i-th treated and the j-th control subject. Hence, function match_2C_mat is most handy for users who are familiar with constructing and working with distance matrices. One commonly-used way to construct a distance matrix is to use the function match_on in the package optmatch (Hansen, 2007).
Function match_2C_list is similar to match_2C_mat except that it requires at least one distance list as input. A list representation of a treatment-by-control distance matrix consists of the following arguments:
- start_n: a vector containing the node numbers of the start nodes of each arc in the network.
- end_n: a vector containing the node numbers of the end nodes of each arc in the network.
- d: a vector containing the integer cost of each arc in the network.
Nodes 1,2,…,n_t correspond to n_t treatment nodes, and n_t + 1, n_t + 2, …, n_t + n_c correspond to n_c control nodes. Note that start_n, end_n, and d have the same lengths, all of which equal to the number of edges. Functions create_list_from_scratch and create_list_from_mat in the package allow users to construct a (possibly sparse) distance list with a possibly user-specified distance measure. We will discuss how to construct distance lists in later sections.
Function match_2C is a wrap-up of match_2C_list with pre-specified distance list structures. For the left network, a Mahalanobis distance between covariates X is adopted; For the right network, an L-1 distance between the propensity score is used. A large penalty is applied so that the algorithm prioritizes balancing the propensity score distributions in the treated and matched control groups, followed by minimizing the sum of within-matched-pair Mahalanobis distances. Function match_2C further allows fine-balancing the joint distribution of a few key covariates. The hierarchy goes in the order of fine-balance >> propensity score distribution >> within-pair Mahalanobis distance.
Object Returned by match_2C, match_2C_mat, and match_2C_list
Objects returned by the family of matching functions match_2C, match_2C_mat, and match_2C_list are the same in format: a list of the following three elements:
feasible: 0/1 depending on the feasibility of the matching problem;
data_with_matched_set_ind: a data frame that is the same as the original data frame, except that a column called matched_set and a column called distance are added to it. Variable matched_set assigns 1,2,…,n_t to each matched set, and NA to controls not matched to any treated. Variable distance records the control-to-treated distance in each matched pair, and assigns NA to all treated and controls that are left unmatched. If matching is not feasible, NULL will be returned;
matched_data_in_order: a data frame organized in the order of matched sets and otherwise the same as data_with_matched_set_ind. Null will be returned if the matching is unfeasible.
Let’s take a look at an example output returned by the function match_2C_list. The matching problem is indeed feasible:
Let’s take a look at the data frame data_with_matched_set_ind. Note that it is indeed the same as the original dataset except that a column matched_set and a column distance are appended. Observe that the first six instances belong to \(6\) different matched sets; therefore matched_set is from \(1\) to \(6\). The first six instances are all treated subjects so distance is NA.
# Check the original dataset with two new columns
head(matching_output_example$data_with_matched_set_ind, 6)
#> educ86 twoyr female black hispanic bytest dadsome dadcoll momsome momcoll
#> 1 12 1 1 1 0 41 0 0 0 0
#> 2 14 1 1 0 0 46 0 0 0 0
#> 3 12 1 1 0 0 60 0 0 0 0
#> 4 14 1 1 0 0 61 0 0 1 0
#> 5 16 1 1 0 0 46 0 0 0 0
#> 6 12 1 0 0 0 60 0 1 0 0
#> fincome fincmiss IV dadneither momneither dadeduc momeduc test_quartile
#> 1 9500 0 1 0 0 0 0 1
#> 2 18000 0 1 0 0 0 0 1
#> 3 22500 0 1 1 0 1 0 4
#> 4 22500 0 1 0 0 0 2 4
#> 5 0 1 1 1 0 1 0 1
#> 6 62000 0 1 0 0 3 0 4
#> income_quartile propensity matched_set distance
#> 1 1 0.42 1 NA
#> 2 2 0.41 2 NA
#> 3 3 0.35 3 NA
#> 4 3 0.35 4 NA
#> 5 0 0.43 5 NA
#> 6 4 0.31 6 NAFinally, matched_data_in_order is data_with_matched_set_ind organized in the order of matched sets. Note that the first \(2\) subjects belong to the same matched set; the next two subjects belong to the second matched set, and etc.
# Check dataframe organized in matched set indices
head(matching_output_example$matched_data_in_order, 6)
#> educ86 twoyr female black hispanic bytest dadsome dadcoll momsome momcoll
#> 1 12 1 1 1 0 41 0 0 0 0
#> 1779 15 0 1 1 0 42 0 0 0 0
#> 2 14 1 1 0 0 46 0 0 0 0
#> 2568 15 0 1 0 1 47 0 0 0 0
#> 3 12 1 1 0 0 60 0 0 0 0
#> 1828 16 0 1 0 0 59 0 0 0 0
#> fincome fincmiss IV dadneither momneither dadeduc momeduc test_quartile
#> 1 9500 0 1 0 0 0 0 1
#> 1779 3500 0 0 1 0 1 0 1
#> 2 18000 0 1 0 0 0 0 1
#> 2568 14000 0 0 0 0 0 0 2
#> 3 22500 0 1 1 0 1 0 4
#> 1828 31500 0 0 1 0 1 0 3
#> income_quartile propensity matched_set distance
#> 1 1 0.42 1 NA
#> 1779 1 0.42 1 1.99
#> 2 2 0.41 2 NA
#> 2568 1 0.41 2 0.29
#> 3 3 0.35 3 NA
#> 1828 3 0.35 3 0.44Checking Balance
Statistical matching belongs to the design stage of an observational study. The ultimate goal of statistical matching is to embed observational data into an approximate randomized controlled trial and the matching process should always be conducted without access to the outcome data. Not looking at the outcome at the design stage means researchers could in principle keep adjusting their matched design until some pre-specified design goal is achieved. A rule of thumb is that the standardized differences of each covariate, i.e., difference in means after matching divided by pooled standard error before matching, is less than 0.1.
Function check_balance in the package provides simple balance check and visualization. In the code chunk below, matching_output_example is an object returned by the family of matching functions match_2C_list/match_2C/match_2C_mat (we give details on how to use these functions later). Function check_balance then takes as input a vector of treatment status Z, an object returned by match_2C (or match_2C_mat or match_2C_list), a vector of covariate names for which we would like to check balance, and output a balance table.
There are six columns of the balance table:
Mean covariate values in the treated group (Z = 1) before matching.
Mean covariate values in the control group (Z = 0) before matching.
Standardized differences before matching.
Mean covariate values in the treated group (Z = 1) after matching.
Mean covariate values in the control group (Z = 0) after matching.
Standardized differences after matching.
tb_example = check_balance(Z, matching_output_example,
cov_list = c('female', 'black', 'bytest', 'fincome', 'dadeduc', 'momeduc', 'propensity'),
plot_propens = FALSE)
print(tb_example)
#> Z = 1 Z = 0 (Bef) Std. Diff (Bef) Z = 0 (Aft) Std. Diff (Aft)
#> female 0.58 0.56 0.032 0.58 0.0000
#> black 0.19 0.18 0.011 0.18 0.0177
#> bytest 51.88 53.04 -0.098 52.23 -0.0295
#> fincome 21630.12 23439.16 -0.072 21266.04 0.0145
#> dadeduc 1.10 1.17 -0.041 1.09 0.0047
#> momeduc 0.93 1.00 -0.038 0.91 0.0181
#> propensity 0.37 0.37 0.115 0.37 0.0119Function check_balance may also plot the distribution of the propensity score among the treated subjects, all conrol subjects, and the matched control subjects by setting option plot_propens = TRUE and supplying the option propens with estimated propensity scores as shown below. In the figure below, the blue curve corresponds to the propensity score distribution among 1,122 treated subjects, the red curve among 1,915 control subjects, and the green curve among 1,122 matched controls. It is evident that after matching, the propensity score distribution aligns better with that of the treated subjects.
tb_example = check_balance(Z, matching_output_example,
cov_list = c('female', 'black', 'bytest', 'fincome',
'dadeduc', 'momeduc', 'propensity'),
plot_propens = TRUE, propens = propensity)
