1 Setup

1.1 Software and Prerequisites

This is an intermediate R course:

  • Assumes working knowledge of R
  • Relatively fast-paced

Follow these R Installation instructions and ensure that you can successfully start RStudio.

Ask questions at any time during the workshop!

1.2 Goals

We will learn about the tools available in R to perform matching for the purpose of causal inference. We will also practice going through a recommended workflow for executing some of the matching algorithms. In particular, our goals are to learn about:

  • Estimating treatment effect in experimental data
  • Adjusting for covariates in observational studies
  • Balance checking between exposure group and control group
  • Identifying regions of common support
  • R tools for matching
  • Matching workflow

1.3 Launch an R session

Start RStudio and create a new project:

  • On Windows click the start button and search for RStudio. On Mac RStudio will be in your applications folder.
  • In Rstudio go to File -> New Project.
  • Choose Existing Directory and browse to the workshop materials directory on your desktop.
  • Choose File -> New File and select an R Script.

1.4 Packages

Let’s install (if needed) and load R packages that will aid with data loading and wrangling:

# install.packages("tidyverse")
library(tidyverse)

# install.packages("haven")
library(haven)

Next, lets add some packages that will help with matching:

# install.packages("MatchIt")
library(MatchIt) # for propensity score matching

# install.packages("cem")
library(cem) # for coarsened exact matching

# install.packages("RItools")
library(RItools) # for displaying balance statistics

# install.packages("cobalt")
library(cobalt) # for displaying balance statistics

2 NSW Example

We will use a classic example data set from a National Supported Work (NSW) Study conducted in 1970’s and designed to test whether and to what extent a temporary employment in a supportive but performance-oriented environment would help hard-to-employ people to get and hold normal jobs.

2.1 Background

Here are some facts about the study and subsequent use of the data it generated:

  • Multisite randomized experiment; found that training raised yearly earnings by about $800 for male participants.
  • Lalonde (1986) tried to use (then) existing non-experimental methods to recover this effect:
    • Used randomized treatment group, but replaced the control group with comparison groups from large publicly available datasets, including the Current Population Survey (CPS) and the Population Survey of Income Dynamics (PSID).
    • Caveat - used almost everyone from the selected surveys!
    • Found that none of the tried methods did very well; results all over the place (-$16,000 to $7,000)
  • Dehejia and Wahba (DW, 1999) used the matching approach to select people from the CPS who looked the most similar to the treated individuals and were able to recover the effect from the experiment.

Let’s load the data and examine it.

## Data is hosted on one of the authors' web-site
## Note that this data set is a subset of the original data, 
## excluding subject with missing information on prior earnings
nsw_dw <- haven::read_dta("http://www.nber.org/~rdehejia/data/nsw_dw.dta") 

PSID <- haven::read_dta("http://www.nber.org/~rdehejia/data/psid_controls.dta")

## If the server seems overloaded, please load locally:
# nsw_dw <- haven::read_dta("data/nsw_dw.dta")
# PSID <- haven::read_dta("data/psid_controls.dta")
# Examine the NSW data
dim(nsw_dw)
## [1] 445  11
str(nsw_dw)
## tibble [445 × 11] (S3: tbl_df/tbl/data.frame)
##  $ data_id  : chr [1:445] "Dehejia-Wahba Sample" "Dehejia-Wahba Sample" "Dehejia-Wahba Sample" "Dehejia-Wahba Sample" ...
##   ..- attr(*, "format.stata")= chr "%20s"
##  $ treat    : num [1:445] 1 1 1 1 1 1 1 1 1 1 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ age      : num [1:445] 37 22 30 27 33 22 23 32 22 33 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ education: num [1:445] 11 9 12 11 8 9 12 11 16 12 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ black    : num [1:445] 1 0 1 1 1 1 1 1 1 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ hispanic : num [1:445] 0 1 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ married  : num [1:445] 1 0 0 0 0 0 0 0 0 1 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ nodegree : num [1:445] 1 1 0 1 1 1 0 1 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ re74     : num [1:445] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ re75     : num [1:445] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ re78     : num [1:445] 9930 3596 24909 7506 290 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
head(nsw_dw)
## # A tibble: 6 x 11
##   data_id treat   age education black hispanic married nodegree  re74  re75
##   <chr>   <dbl> <dbl>     <dbl> <dbl>    <dbl>   <dbl>    <dbl> <dbl> <dbl>
## 1 Deheji…     1    37        11     1        0       1        1     0     0
## 2 Deheji…     1    22         9     0        1       0        1     0     0
## 3 Deheji…     1    30        12     1        0       0        0     0     0
## 4 Deheji…     1    27        11     1        0       0        1     0     0
## 5 Deheji…     1    33         8     1        0       0        1     0     0
## 6 Deheji…     1    22         9     1        0       0        1     0     0
## # … with 1 more variable: re78 <dbl>
summary(nsw_dw)
##    data_id              treat             age          education   
##  Length:445         Min.   :0.0000   Min.   :17.00   Min.   : 3.0  
##  Class :character   1st Qu.:0.0000   1st Qu.:20.00   1st Qu.: 9.0  
##  Mode  :character   Median :0.0000   Median :24.00   Median :10.0  
##                     Mean   :0.4157   Mean   :25.37   Mean   :10.2  
##                     3rd Qu.:1.0000   3rd Qu.:28.00   3rd Qu.:11.0  
##                     Max.   :1.0000   Max.   :55.00   Max.   :16.0  
##      black           hispanic          married          nodegree    
##  Min.   :0.0000   Min.   :0.00000   Min.   :0.0000   Min.   :0.000  
##  1st Qu.:1.0000   1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:1.000  
##  Median :1.0000   Median :0.00000   Median :0.0000   Median :1.000  
##  Mean   :0.8337   Mean   :0.08764   Mean   :0.1685   Mean   :0.782  
##  3rd Qu.:1.0000   3rd Qu.:0.00000   3rd Qu.:0.0000   3rd Qu.:1.000  
##  Max.   :1.0000   Max.   :1.00000   Max.   :1.0000   Max.   :1.000  
##       re74              re75            re78      
##  Min.   :    0.0   Min.   :    0   Min.   :    0  
##  1st Qu.:    0.0   1st Qu.:    0   1st Qu.:    0  
##  Median :    0.0   Median :    0   Median : 3702  
##  Mean   : 2102.3   Mean   : 1377   Mean   : 5301  
##  3rd Qu.:  824.4   3rd Qu.: 1221   3rd Qu.: 8125  
##  Max.   :39570.7   Max.   :25142   Max.   :60308
table(nsw_dw$treat)
## 
##   0   1 
## 260 185
# Examine the PSID data
dim(PSID)
## [1] 2490   11
str(PSID)
## tibble [2,490 × 11] (S3: tbl_df/tbl/data.frame)
##  $ data_id  : chr [1:2490] "PSID" "PSID" "PSID" "PSID" ...
##   ..- attr(*, "format.stata")= chr "%9s"
##  $ treat    : num [1:2490] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ age      : num [1:2490] 47 50 44 28 54 55 47 25 44 50 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ education: num [1:2490] 12 12 12 12 12 12 12 12 12 12 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ black    : num [1:2490] 0 1 0 1 0 0 0 1 0 1 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ hispanic : num [1:2490] 0 0 0 0 0 1 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ married  : num [1:2490] 0 1 0 1 1 1 1 0 1 1 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ nodegree : num [1:2490] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ re74     : num [1:2490] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ re75     : num [1:2490] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
##  $ re78     : num [1:2490] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "format.stata")= chr "%9.0g"
head(PSID)
## # A tibble: 6 x 11
##   data_id treat   age education black hispanic married nodegree  re74  re75
##   <chr>   <dbl> <dbl>     <dbl> <dbl>    <dbl>   <dbl>    <dbl> <dbl> <dbl>
## 1 PSID        0    47        12     0        0       0        0     0     0
## 2 PSID        0    50        12     1        0       1        0     0     0
## 3 PSID        0    44        12     0        0       0        0     0     0
## 4 PSID        0    28        12     1        0       1        0     0     0
## 5 PSID        0    54        12     0        0       1        0     0     0
## 6 PSID        0    55        12     0        1       1        0     0     0
## # … with 1 more variable: re78 <dbl>
summary(PSID)
##    data_id              treat        age          education    
##  Length:2490        Min.   :0   Min.   :18.00   Min.   : 0.00  
##  Class :character   1st Qu.:0   1st Qu.:26.00   1st Qu.:11.00  
##  Mode  :character   Median :0   Median :33.00   Median :12.00  
##                     Mean   :0   Mean   :34.85   Mean   :12.12  
##                     3rd Qu.:0   3rd Qu.:44.00   3rd Qu.:14.00  
##                     Max.   :0   Max.   :55.00   Max.   :17.00  
##      black           hispanic          married          nodegree     
##  Min.   :0.0000   Min.   :0.00000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:1.0000   1st Qu.:0.0000  
##  Median :0.0000   Median :0.00000   Median :1.0000   Median :0.0000  
##  Mean   :0.2506   Mean   :0.03253   Mean   :0.8663   Mean   :0.3052  
##  3rd Qu.:1.0000   3rd Qu.:0.00000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :1.00000   Max.   :1.0000   Max.   :1.0000  
##       re74             re75             re78       
##  Min.   :     0   Min.   :     0   Min.   :     0  
##  1st Qu.: 10776   1st Qu.:  9847   1st Qu.: 11526  
##  Median : 18417   Median : 17903   Median : 20688  
##  Mean   : 19429   Mean   : 19063   Mean   : 21554  
##  3rd Qu.: 26450   3rd Qu.: 26497   3rd Qu.: 29555  
##  Max.   :137149   Max.   :156653   Max.   :121174

2.2 Data Set

Dehejia-Wahha’s sample represents a subset of the original experimental data, where subjects have information on earnings in 1974 (important for ignorability assumption):

  • 445 subjects, with 185 treated
  • Treatment: participation in job training (treat)
  • Outcome: income in 1978 dollars (re78)
  • Controls/confounders: age, years of education, high school degree (nodegree), black, hispanic, married, real earnings in 1974 (re74) and 1975 (re75)

PSID: Non-experimental comparison group constructed by Lalonde from the PSID

  • ~2500 subjects
  • Treatment (treat, all 0)
  • Same controls and outcome as in NSW

3 Matching workflow

As we discussed during the lecture portion, there is a suggested sequence of
steps when applying matching techniques:

  1. Pre-analysis of non-matched data
  2. Matching
  3. Checking balance post-matching; repeat steps 2-3 as needed

Throughout this workshop we will repeat some of the steps and try out several matching methods.

3.1 Exercise 0

Treatment effect from experimental data

Since the original data set came from a randomized experiment, a gold standard of causal inference, we can estimate the true treatment effect.

  1. Use OLS regression to estimate the difference between average outcomes.
##
  1. Repeat the calculation controlling for pre-intervention covariates.
##
  1. Discuss the following:
  1. What do you notice about the treatment effect estimate from the two models?
  2. Pros/cons of controlling for pre-intervention variables in a context of a randomized experiment.
Click for Exercise 0 Solution
  1. Use OLS regression to estimate the difference between average outcomes.
# Marginal difference between means
marg_mod <- lm(re78 ~ treat, data=nsw_dw)
summary(marg_mod)
## 
## Call:
## lm(formula = re78 ~ treat, data = nsw_dw)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##  -6349  -4555  -1829   2917  53959 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4554.8      408.0  11.162  < 2e-16 ***
## treat         1794.3      632.9   2.835  0.00479 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6580 on 443 degrees of freedom
## Multiple R-squared:  0.01782,    Adjusted R-squared:  0.01561 
## F-statistic: 8.039 on 1 and 443 DF,  p-value: 0.004788
confint(marg_mod)
##                 2.5 %   97.5 %
## (Intercept) 3752.8550 5356.747
## treat        550.5745 3038.110
  1. Repeat the calculation controlling for pre-intervention covariates.
# Treatment effect using regression adjustment
cond_mod <- lm(re78 ~ treat + age + education + black + hispanic + 
                 nodegree + married + re74 + re75, 
                data = nsw_dw)
summary(cond_mod)
## 
## Call:
## lm(formula = re78 ~ treat + age + education + black + hispanic + 
##     nodegree + married + re74 + re75, data = nsw_dw)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##  -9776  -4425  -1645   2942  54107 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  7.851e+02  3.375e+03   0.233  0.81617   
## treat        1.676e+03  6.387e+02   2.625  0.00898 **
## age          5.532e+01  4.528e+01   1.222  0.22254   
## education    3.957e+02  2.274e+02   1.740  0.08254 . 
## black       -2.160e+03  1.169e+03  -1.847  0.06539 . 
## hispanic     1.640e+02  1.549e+03   0.106  0.91574   
## nodegree    -7.068e+01  1.004e+03  -0.070  0.94393   
## married     -1.387e+02  8.797e+02  -0.158  0.87477   
## re74         8.214e-02  7.742e-02   1.061  0.28930   
## re75         5.276e-02  1.355e-01   0.389  0.69712   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6513 on 435 degrees of freedom
## Multiple R-squared:  0.05483,    Adjusted R-squared:  0.03527 
## F-statistic: 2.804 on 9 and 435 DF,  p-value: 0.00329
confint(cond_mod)
##                     2.5 %       97.5 %
## (Intercept) -5.848211e+03 7418.3343412
## treat        4.210563e+02 2931.6289692
## age         -3.368585e+01  144.3192055
## education   -5.123410e+01  842.7027152
## black       -4.457183e+03  138.1387526
## hispanic    -2.881320e+03 3209.3857810
## nodegree    -2.044736e+03 1903.3743248
## married     -1.867771e+03 1590.3205044
## re74        -7.002578e-02    0.2343082
## re75        -2.135045e-01    0.3190327
  1. Discuss the following:
  1. What do you notice about treatment effect estimates from the two models?
  2. Pros/cons of controlling for pre-intervention variables in a context of a randomized experiment.

Answer:

  1. The estimated treatment effects are very similar ($1,794.3 vs. $1,676). There is also considerable overlap between their 95% intervals: (551, 3038) vs. (421, 2932). The estimates are statistically (and practically) indistinguishable.
  2. Randomization does not guarantee a perfect balance, and experiments usually do not go as planned. Regression adjustment in experiments may improve precision over a simple difference in means between the treated and control outcomes if the covariates are sufficiently predictive of the outcome. Nevertheless, regression adjustment is not uniformly accepted due to the introduced reliance on the “correctness” of the model.

Note that, regardless of the trade-offs discussed above, it is only appropriate to control for predictors that are not affected by the treatment (i.e., no “intermediate outcomes” on the RHS!).

4 Preliminary analysis

4.1 Analytic summaries

Covariate balance is typically assessed and reported using statistical measures, including standardized mean differences, variance ratios, t- and Kolmogorov-Smirnov tests. The balance can be reported graphically and analytically.

R packages that aid with matching usually have functions to assess balance between treated and control groups. However, most of them are designed to contrast the balance before and after matching rather than help the researcher figure out whether matching is necessary in the first place.

Conveniently, there are a couple of packages, including RItools and cobalt, that allow examining covariate balance tables and plots prior to matching.

xBal_res <- RItools::xBalance(treat ~ ., 
                              data = nsw_dw %>% select(-data_id, -re78), 
                              report="all")

xBal_res
##           strata   unstrat                                                                   
##           stat     treat=0   treat=1  adj.diff adj.diff.null.sd  std.diff      z             
## vars                                                                                         
## age              2.51e+01  2.58e+01  7.62e-01         6.83e-01  1.07e-01  1.12e+00           
## education        1.01e+01  1.03e+01  2.57e-01         1.72e-01  1.44e-01  1.49e+00           
## black            8.27e-01  8.43e-01  1.63e-02         3.59e-02  4.37e-02  4.55e-01           
## hispanic         1.08e-01  5.95e-02  -4.82e-02        2.72e-02  -1.71e-01 -1.77e+00 .        
## married          1.54e-01  1.89e-01  3.53e-02         3.60e-02  9.43e-02  9.80e-01           
## nodegree         8.35e-01  7.08e-01  -1.27e-01        3.98e-02  -3.09e-01 -3.18e+00 **       
## re74             2.11e+03  2.10e+03  -1.15e+01        5.16e+02  -2.13e-03 -2.22e-02          
## re75             1.27e+03  1.53e+03  2.65e+02         3.03e+02  8.41e-02  8.75e-01           
## ---Overall Test---
##         chisquare df p.value
## unstrat      16.8  8  0.0325
## ---
## Signif. codes:  0 '***' 0.001 '** ' 0.01 '*  ' 0.05 '.  ' 0.1 '   ' 1
# round(xBal_res$result,4)

bal_res <- cobalt::bal.tab(nsw_dw %>% select(-data_id, -re78), 
                           treat = nsw_dw$treat,
                           disp.means = T, 
                           disp.sds = T,
                           stats = c("mean.diffs", "variance.ratios", "ks.statistics"))
## Note: 's.d.denom' not specified; assuming pooled.
bal_res
## Balance Measures
##              Type    M.0.Un   SD.0.Un    M.1.Un   SD.1.Un Diff.Un V.Ratio.Un
## age       Contin.   25.0538    7.0577   25.8162    7.1550  0.1073     1.0278
## education Contin.   10.0885    1.6143   10.3459    2.0107  0.1412     1.5513
## black      Binary    0.8269              0.8432            0.0163           
## hispanic   Binary    0.1077              0.0595           -0.0482           
## married    Binary    0.1538              0.1892            0.0353           
## nodegree   Binary    0.8346              0.7081           -0.1265           
## re74      Contin. 2107.0267 5687.9056 2095.5737 4886.6204 -0.0022     0.7381
## re75      Contin. 1266.9090 3102.9821 1532.0553 3219.2509  0.0839     1.0763
##            KS.Un
## age       0.0652
## education 0.1265
## black     0.0163
## hispanic  0.0482
## married   0.0353
## nodegree  0.1265
## re74      0.0471
## re75      0.1075
## 
## Sample sizes
##     Control Treated
## All     260     185

While both xBalance() and bal.tab() allow to display various statistics for multiple covariates simultaneously, the latter deliberately omits performing significance tests as they can be misleading and their use is discouraged by leading methodologists (e.g., Ali et al. (2015)). bal.tab() also identifies types of each variable and applies appropriate techniques for calculating (or not) standardized difference and other summaries.

Examine available arguments in ?xBalance and ?bal.tab to see what other functionality is available.

Clearly, the above output indicates that the balance in the experimental data is very good.

4.2 Visual comparisons

Ideally, we are hoping to achieve balance in the joint distributions of covariates. As we saw above, most packages assess numerical summaries of balance by comparing moments (first or second) between the treated and control groups. However, those may not reflect a full picture and it is useful to complement such analysis with visual comparison of covariate distributions. Note that, Kolmogorov-Smirnov statistics and the overlapping coefficient available in cobalt (stats = "ovl.coefficients") also address some aspects of distributional balance, beyond the first few moments.

# cobalt package has the most intuitive visualization functions
cobalt::bal.plot(nsw_dw %>% select(-data_id), 
                treat = nsw_dw$treat,
                var.name = "age")

cobalt::bal.plot(nsw_dw %>% select(-data_id), 
                treat = nsw_dw$treat,
                var.name = "education",
                type = "histogram")

cobalt::bal.plot(nsw_dw %>% select(-data_id), 
                treat = nsw_dw$treat,
                var.name = "nodegree")

Once again, it is fairy evident that the experiment achieved good balance between the two groups.You can read about other functionalities available in cobalt in the following tutorial: Guide to the cobalt Package.

4.3 Merging PSID controls

Next, we remove experimental controls and combine treated subjects from nsw_dw with the PSID sample. This follows the steps taken in Lalonde (1986) to emulate a typical data collection process in an observational study.

nsw_PSID <- nsw_dw %>%
  filter(treat==1) %>%
  bind_rows(PSID)

summary(nsw_PSID)
##    data_id              treat              age          education    
##  Length:2675        Min.   :0.00000   Min.   :17.00   Min.   : 0.00  
##  Class :character   1st Qu.:0.00000   1st Qu.:25.00   1st Qu.:10.00  
##  Mode  :character   Median :0.00000   Median :32.00   Median :12.00  
##                     Mean   :0.06916   Mean   :34.23   Mean   :11.99  
##                     3rd Qu.:0.00000   3rd Qu.:43.50   3rd Qu.:14.00  
##                     Max.   :1.00000   Max.   :55.00   Max.   :17.00  
##      black           hispanic          married          nodegree     
##  Min.   :0.0000   Min.   :0.00000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:1.0000   1st Qu.:0.0000  
##  Median :0.0000   Median :0.00000   Median :1.0000   Median :0.0000  
##  Mean   :0.2916   Mean   :0.03439   Mean   :0.8194   Mean   :0.3331  
##  3rd Qu.:1.0000   3rd Qu.:0.00000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :1.00000   Max.   :1.0000   Max.   :1.0000  
##       re74             re75             re78       
##  Min.   :     0   Min.   :     0   Min.   :     0  
##  1st Qu.:  8817   1st Qu.:  7605   1st Qu.:  9243  
##  Median : 17437   Median : 17008   Median : 19432  
##  Mean   : 18230   Mean   : 17851   Mean   : 20502  
##  3rd Qu.: 25470   3rd Qu.: 25584   3rd Qu.: 28816  
##  Max.   :137149   Max.   :156653   Max.   :121174
sum(nsw_PSID$treat)
## [1] 185

The new combined data set contains 2,675 subjects, including 185 treated from the NSW data.

4.4 Exercise 1

Balance in the combined data set

  1. Assess the balance in the combined data set using any routines outlined above, both analytic and graphical. Discuss your observations.
##
  1. Estimate the treatment effect using OLS, controlling for available background covariates.
##
  1. Discuss the following questions:
  1. How does the estimated effect compare to the one from the randomized experiment?
  2. Why do you think there is such discrepancy, even though we “controlled” for all pre-treatment variables?
Click for Exercise 1 Solution
  1. Assess the balance in the combined data set using any routines outlined above, both analytic and graphical. Discuss your observations.
## Let's use bal.tab
bal.tab(nsw_PSID %>% select(-data_id, -re78), 
        treat = nsw_PSID$treat,
        disp.means = T, 
        disp.sds = T,
        stats = c("mean.diffs", "variance.ratios", "ks.statistics"))
## Note: 's.d.denom' not specified; assuming pooled.
## Balance Measures
##              Type     M.0.Un    SD.0.Un    M.1.Un   SD.1.Un Diff.Un V.Ratio.Un
## age       Contin.    34.8506    10.4408   25.8162    7.1550 -1.0094     0.4696
## education Contin.    12.1169     3.0824   10.3459    2.0107 -0.6805     0.4255
## black      Binary     0.2506               0.8432            0.5926           
## hispanic   Binary     0.0325               0.0595            0.0269           
## married    Binary     0.8663               0.1892           -0.6771           
## nodegree   Binary     0.3052               0.7081            0.4029           
## re74      Contin. 19428.7458 13406.8772 2095.5737 4886.6204 -1.7178     0.1329
## re75      Contin. 19063.3377 13596.9549 1532.0553 3219.2509 -1.7744     0.0561
##            KS.Un
## age       0.3771
## education 0.4029
## black     0.5926
## hispanic  0.0269
## married   0.6771
## nodegree  0.4029
## re74      0.7292
## re75      0.7736
## 
## Sample sizes
##     Control Treated
## All    2490     185
nsw_PSID_posRE74 <- nsw_PSID %>% filter(re74>0)
bal.plot(nsw_PSID_posRE74, 
         treat = nsw_PSID_posRE74$treat,
         var.name = "re74")

bal.plot(nsw_PSID, 
         treat = nsw_PSID$treat,
         var.name = "married")

The treated subjects tend to be younger, have fewer years of education, are less likely to be married, and earn a lot less in 1974 and 1975.

  1. Estimate the treatment effect using OLS, controlling for available background covariates.
# Treatment effect using regression adjustment
raw_mod <- lm(re78 ~ treat + age + education + black + 
                hispanic + nodegree + married + re74 + re75, 
                data = nsw_PSID)
summary(raw_mod)
## 
## Call:
## lm(formula = re78 ~ treat + age + education + black + hispanic + 
##     nodegree + married + re74 + re75, data = nsw_PSID)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -64870  -4302   -435   3786 110412 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -129.74267 1688.51705  -0.077   0.9388    
## treat        751.94636  915.25722   0.822   0.4114    
## age          -83.56559   20.81380  -4.015 6.11e-05 ***
## education    592.61019  103.30278   5.737 1.07e-08 ***
## black       -570.92798  495.17772  -1.153   0.2490    
## hispanic    2163.28116 1092.29036   1.981   0.0478 *  
## nodegree     590.46694  646.78416   0.913   0.3614    
## married     1240.51951  586.25390   2.116   0.0344 *  
## re74           0.27812    0.02792   9.960  < 2e-16 ***
## re75           0.56809    0.02756  20.613  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10070 on 2665 degrees of freedom
## Multiple R-squared:  0.5864, Adjusted R-squared:  0.585 
## F-statistic: 419.8 on 9 and 2665 DF,  p-value: < 2.2e-16
confint(raw_mod)
##                     2.5 %       97.5 %
## (Intercept) -3440.6789935 3181.1936528
## treat       -1042.7399231 2546.6326360
## age          -124.3784144  -42.7527564
## education     390.0484754  795.1719123
## black       -1541.8994550  400.0434933
## hispanic       21.4586495 4305.1036682
## nodegree     -677.7827254 1858.7166030
## married        90.9608842 2390.0781379
## re74            0.2233653    0.3328768
## re75            0.5140503    0.6221322
  1. Discuss the following questions:
  1. How does the estimated effect compare to the one from the randomized experiment?
  2. Why do you think there is such discrepancy, even though we “controlled” for all pre-treatment variables?

Answer:

  1. The estimated effect ($752) is about half of the effect obtained from the experiment ($1,676). It is also not significant.
  2. Drawing causal inference from OLS estimates is only appropriate when we can assume ignorability (i.e., we believe that treatment assignments only depend on information included in the model), and when the chosen outcome model is correct. Lack of overlap and balance forces stronger reliance on our modeling than if covariate distributions were the same across treatment groups. Considerable nonoverlap also makes the assumption about a constant treatment effect less plausible as well as results in heavy extrapolations.

5 Tools for Matching in R

There are a few R packages that perform data preprocessing and causal effect estimation:

  • MatchIt Nonparametric Preprocessing for Parametric Causal Inference (Ho, Imai, King, & Stuart, 2011)
  • twang Toolkit for Weighting and Analysis of Nonequivalent Groups (Ridgeway, McCaffrey, Morral, Burgette, & Griffin, 2020)
  • Matching Multivariate and Propensity Score Matching with Balance Optimization (Sekhon, 2011)
  • cem Coarsened Exact Matching (Iacus, King, & Porro, 2018)
  • optmatch Functions for Optimal Matching (Hansen & Klopfer, 2006)
  • CBPS Covariate Balancing Propensity Score (Fong, Ratkovic, Hazlett, Yang, & Imai, 2019),
  • ebal Entropy reweighting to create balanced samples (Hainmueller, 2014)
  • sbw Stable Balancing Weights for Causal Inference and Estimation with Incomplete Outcome Data (Zubizarreta & Li, 2019)
  • designmatch Matched Samples that are Balanced and Representative by Design (Zubizarreta, Kilcioglu, & Vielma, 2018)
  • WeightIt Weighting for Covariate Balance in Observational Studies (Greifer, 2020)
  • MatchThem Matching and Weighting Multiply Imputed Datasets (Pishgar & Greifer, 2020)
  • PanelMatch Matching Methods for Causal Inference with Time-Series Cross-Sectional Data (Imai, Kim, and Wang, 2018)
  • multilevelPSA Multilevel Propensity Score Analysis (Bryer & Pruzek, 2011)
  • party A Laboratory for Recursive Partytioning (Hothorn, Hornik, & Zeileis, 2006)
  • rpart Recursive Partitioning (Therneau, Atkinson, & Ripley, 2012)
  • rbounds An Overview of rebounds: An R Package for Rosenbaum bounds sensitivity analysis with matched data (Keele, 2010)
  • TriMatchPropensity Score Matching for Non-Binary Treatments (Bryer, 2013)

The packages vary in ranges of available methods and functionalities but, together, provide a rich set of preprocessing tools. For a thorough review of some of them see Keller & Tipton (2016). Next we will explore functions from packages MatchIt and cem.

5.1 PS Matching with MatchIt

The default method in the MatchIt package is the Nearest Neighbor Matching, which (sequentially) selects the best control for each treated subject using distance between their estimated propensity scores. The end result is two groups with (hopefully) similar distributions of covariates.

m_NNM_out <- MatchIt::matchit(treat ~ age + education + black + hispanic + 
                                married + nodegree + re74 + re75,
                              data = nsw_PSID) # Try discard = "both"
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
summary(m_NNM_out) # Can add standardize = TRUE
## 
## Call:
## MatchIt::matchit(formula = treat ~ age + education + black + 
##     hispanic + married + nodegree + re74 + re75, data = nsw_PSID)
## 
## Summary of balance for all data:
##           Means Treated Means Control SD Control   Mean Diff    eQQ Med
## distance         0.6364        0.0270     0.0992      0.6094     0.7267
## age             25.8162       34.8506    10.4408     -9.0344     8.0000
## education       10.3459       12.1169     3.0824     -1.7709     2.0000
## black            0.8432        0.2506     0.4334      0.5926     1.0000
## hispanic         0.0595        0.0325     0.1774      0.0269     0.0000
## married          0.1892        0.8663     0.3404     -0.6771     1.0000
## nodegree         0.7081        0.3052     0.4606      0.4029     0.0000
## re74          2095.5737    19428.7458 13406.8772 -17333.1721 18417.1074
## re75          1532.0553    19063.3377 13596.9549 -17531.2824 17903.2266
##             eQQ Mean     eQQ Max
## distance      0.6075      0.8817
## age           9.0432     17.0000
## education     1.8595      5.0000
## black         0.5892      1.0000
## hispanic      0.0270      1.0000
## married       0.6757      1.0000
## nodegree      0.4000      1.0000
## re74      17662.9118 102108.6172
## re75      17977.5835 131510.9941
## 
## 
## Summary of balance for matched data:
##           Means Treated Means Control SD Control  Mean Diff  eQQ Med  eQQ Mean
## distance         0.6364        0.2934     0.2319     0.3430    0.377    0.3430
## age             25.8162       30.4811    11.1077    -4.6649    2.000    4.6649
## education       10.3459       10.3784     2.9187    -0.0324    0.000    0.6919
## black            0.8432        0.7568     0.4302     0.0865    0.000    0.0865
## hispanic         0.0595        0.0649     0.2470    -0.0054    0.000    0.0054
## married          0.1892        0.4595     0.4997    -0.2703    0.000    0.2703
## nodegree         0.7081        0.6216     0.4863     0.0865    0.000    0.0865
## re74          2095.5737     4499.8428  4654.9584 -2404.2691 2006.289 2645.6165
## re75          1532.0553     3204.3968  3745.1931 -1672.3415 1994.419 1737.5551
##              eQQ Max
## distance      0.5791
## age          16.0000
## education     4.0000
## black         1.0000
## hispanic      1.0000
## married       1.0000
## nodegree      1.0000
## re74      14655.8574
## re75       4660.9512
## 
## Percent Balance Improvement:
##           Mean Diff.  eQQ Med eQQ Mean eQQ Max
## distance     43.7119  48.1282  43.5326 34.3220
## age          48.3654  75.0000  48.4160  5.8824
## education    98.1686 100.0000  62.7907 20.0000
## black        85.4066 100.0000  85.3211  0.0000
## hispanic     79.9274   0.0000  80.0000  0.0000
## married      60.0827 100.0000  60.0000  0.0000
## nodegree     78.5333   0.0000  78.3784  0.0000
## re74         86.1291  89.1064  85.0216 85.6468
## re75         90.4608  88.8600  90.3349 96.4558
## 
## Sample sizes:
##           Control Treated
## All          2490     185
## Matched       185     185
## Unmatched    2305       0
## Discarded       0       0
## `cobalt` package offers additional summaries of distributional differences
## Note that it can process objects returned by matchit()!
cobalt::bal.tab(m_NNM_out,
                disp.means = T,
                disp.sds = T,
                stats = c("mean.diffs", "variance.ratios", "ks.statistics"))
## Call
##  MatchIt::matchit(formula = treat ~ age + education + black + 
##     hispanic + married + nodegree + re74 + re75, data = nsw_PSID)
## 
## Balance Measures
##               Type   M.0.Adj  SD.0.Adj   M.1.Adj  SD.1.Adj Diff.Adj V.Ratio.Adj
## distance  Distance    0.2934    0.2319    0.6364    0.2812   1.2200      1.4702
## age        Contin.   30.4811   11.1077   25.8162    7.1550  -0.6520      0.4149
## education  Contin.   10.3784    2.9187   10.3459    2.0107  -0.0161      0.4745
## black       Binary    0.7568              0.8432             0.0865            
## hispanic    Binary    0.0649              0.0595            -0.0054            
## married     Binary    0.4595              0.1892            -0.2703            
## nodegree    Binary    0.6216              0.7081             0.0865            
## re74       Contin. 4499.8428 4654.9584 2095.5737 4886.6204  -0.4920      1.1020
## re75       Contin. 3204.3968 3745.1931 1532.0553 3219.2509  -0.5195      0.7389
##           KS.Adj
## distance  0.5568
## age       0.1784
## education 0.0919
## black     0.0865
## hispanic  0.0054
## married   0.2703
## nodegree  0.0865
## re74      0.4162
## re75      0.2973
## 
## Sample sizes
##           Control Treated
## All          2490     185
## Matched       185     185
## Unmatched    2305       0

This should run fairly quickly! The next step is to investigate the results of the matching procedure. Did we improve balance? Are the selected comparison units similar to treated units across age, race, education, marital status, and prior earnings? The quickest way to check this is to plot a summary of the results, like so:

# The default plotting function in `MatchIt`
# Compare the distribution of propensity scores before and after matching
plot(m_NNM_out, type = "hist")

# Compare standardized mean differences before/after
s.out <- summary(m_NNM_out, standardize = TRUE)
plot(s.out, interactive = F)

The default plotting function in MatchIt is somewhat clunky. Again, we can use functions from cobalt to make pretty plots using objects generated by matchit().

The Love plot is a summary plot of covariate balance before and after conditioning popularized by Dr. Thomas Love. In a visually appealing and clear way, it can help demonstrate that balance has been met within a threshold, and that balance has improved after conditioning.

cobalt::love.plot(m_NNM_out, 
                  stats = c("mean.diffs", "ks.statistics"), 
                  threshold = c(m = .5, ks = .25), 
                  binary = "std", #display standardized mean difference for all of the covariates
                  #abs = TRUE,
                  var.order = "unadjusted", #how to order the variables in the plot
                  grid = T, 
                  sample.names = c("Matched", "Unmatched"), #new names to be given to the samples
                  position = "top", 
                  colors = c("blue", "red"))
## Warning: Large mean differences detected; you may not be using standardized mean
## differences for continuous variables.

# Propensity Scores before and after weighting (which = "both")
cobalt::bal.plot(m_NNM_out, 
                 var.name = "distance", 
                 which = "both", #what distributional balance to display
                 type = "histogram", 
                 mirror = TRUE)

Even though we haven’t achieved a perfect balance, we will hope that the remaining discrepancy is adequately controlled by an OLS regression. For that, we first extract the matched data set using MatchIt::match.data.

# Extracting the matched data set
m.data.NNM <- MatchIt::match.data(m_NNM_out)
summary(m.data.NNM)
##    data_id              treat          age          education         black    
##  Length:370         Min.   :0.0   Min.   :17.00   Min.   : 3.00   Min.   :0.0  
##  Class :character   1st Qu.:0.0   1st Qu.:21.00   1st Qu.: 9.00   1st Qu.:1.0  
##  Mode  :character   Median :0.5   Median :25.00   Median :11.00   Median :1.0  
##                     Mean   :0.5   Mean   :28.15   Mean   :10.36   Mean   :0.8  
##                     3rd Qu.:1.0   3rd Qu.:31.00   3rd Qu.:12.00   3rd Qu.:1.0  
##                     Max.   :1.0   Max.   :55.00   Max.   :17.00   Max.   :1.0  
##     hispanic          married          nodegree           re74         
##  Min.   :0.00000   Min.   :0.0000   Min.   :0.0000   Min.   :    0.00  
##  1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:    0.00  
##  Median :0.00000   Median :0.0000   Median :1.0000   Median :    8.82  
##  Mean   :0.06216   Mean   :0.3243   Mean   :0.6649   Mean   : 3297.71  
##  3rd Qu.:0.00000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.: 5681.87  
##  Max.   :1.00000   Max.   :1.0000   Max.   :1.0000   Max.   :35040.07  
##       re75              re78          distance            weights 
##  Min.   :    0.0   Min.   :    0   Min.   :0.0002345   Min.   :1  
##  1st Qu.:    0.0   1st Qu.:    0   1st Qu.:0.1783501   1st Qu.:1  
##  Median :  164.1   Median : 4697   Median :0.4178355   Median :1  
##  Mean   : 2368.2   Mean   : 6406   Mean   :0.4648903   Mean   :1  
##  3rd Qu.: 3795.5   3rd Qu.: 9961   3rd Qu.:0.7924988   3rd Qu.:1  
##  Max.   :25142.2   Max.   :60308   Max.   :0.9350264   Max.   :1
m.data.NNM$treat %>% table
## .
##   0   1 
## 185 185

Now we are ready to rerun the OLS and estimate the causal treatment effect on all subjects:

# Re-estimating the treatment effect using regression adjustment
NNM_mod <- lm(re78 ~ treat + age + education + black + hispanic + 
                nodegree + married + re74 + re75, 
              data = m.data.NNM)
summary(NNM_mod)
## 
## Call:
## lm(formula = re78 ~ treat + age + education + black + hispanic + 
##     nodegree + married + re74 + re75, data = m.data.NNM)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -12395  -4563  -1013   2915  54877 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  2.058e+03  3.698e+03   0.556   0.5782  
## treat        1.084e+03  8.235e+02   1.316   0.1890  
## age         -6.179e+01  4.605e+01  -1.342   0.1805  
## education    4.528e+02  2.197e+02   2.061   0.0401 *
## black       -2.726e+02  1.141e+03  -0.239   0.8113  
## hispanic    -3.588e+02  1.811e+03  -0.198   0.8431  
## nodegree    -1.080e+03  1.091e+03  -0.990   0.3228  
## married      1.497e+03  8.741e+02   1.712   0.0877 .
## re74         1.835e-01  9.599e-02   1.911   0.0567 .
## re75         3.047e-01  1.299e-01   2.346   0.0195 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7020 on 360 degrees of freedom
## Multiple R-squared:  0.1322, Adjusted R-squared:  0.1105 
## F-statistic: 6.093 on 9 and 360 DF,  p-value: 5.55e-08
confint(NNM_mod)
##                     2.5 %       97.5 %
## (Intercept) -5.214064e+03 9329.3285743
## treat       -5.356871e+02 2703.3411299
## age         -1.523510e+02   28.7726453
## education    2.065763e+01  884.8567013
## black       -2.517094e+03 1971.8477442
## hispanic    -3.920666e+03 3203.0848552
## nodegree    -3.224450e+03 1064.9646654
## married     -2.221203e+02 3215.7927406
## re74        -5.289760e-03    0.3722581
## re75         4.927121e-02    0.5600429

Note that the estimated treatment effect ($1,084) is closer to the one obtained from the experiment ($1,676) and its 95% CI contains this value.

MatchIt offers other matching techniques, including exact matching, full matching, and k:1 matching with replacement, some of which are enabled by loading other add-on packages in the background. Read more about all the available options here: Ho, Imai, King, Stuart 2011, “MatchIt: Nonparametric Preprocessing for Parametric Causal Inference.” Journal of Statistical Software.

Next, we will practice matching using another package called cem.

5.2 Matching with cem

Package cem performs an automatic Coarsened Exact Matching (CEM) using the following steps:

  1. Temporarily coarsen each covariate in a user-defined (or automated) way.

  2. Sort all units into strata, each of which has the same values of the coarsened X.

  3. Discard all units in those strata that do not include at least one treated and one control unit.

cem also offers its own functions for checking balance between groups:

## While not as consequential for propensity-score matching, it is important 
## to preprocess the data set before running CEM and convert to factors all 
## non-numeric covariates to ensure that "binning" is performed correctly:

nsw_PSID_cem <- 
  nsw_PSID %>%
    mutate_at(vars(black, hispanic, married, nodegree), ~as.factor(.)) %>% 
    as.data.frame

# Balance checking via cem package
cem::imbalance(group = nsw_PSID_cem$treat, 
               data  = nsw_PSID_cem,
               drop  = c("treat", "re78", "data_id"))
## 
## Multivariate Imbalance Measure: L1=0.986
## Percentage of local common support: LCS=1.4%
## 
## Univariate Imbalance Measures:
## 
##               statistic   type         L1 min        25%       50%       75%
## age           -9.034386 (diff) 0.18616086  -1     -6.000     -8.00    -15.00
## education     -1.770922 (diff) 0.44048627   4     -2.000     -1.00     -2.00
## black        289.940097 (Chi2) 0.59264083  NA         NA        NA        NA
## hispanic       2.993226 (Chi2) 0.02692934  NA         NA        NA        NA
## married      528.991963 (Chi2) 0.67707587  NA         NA        NA        NA
## nodegree     124.024343 (Chi2) 0.40288722  NA         NA        NA        NA
## re74      -17333.172112 (diff) 0.00000000   0 -10775.967 -18417.11 -25158.63
## re75      -17531.282355 (diff) 0.00000000   0  -9846.774 -17903.23 -24679.49
##                 max
## age            -7.0
## education      -1.0
## black            NA
## hispanic         NA
## married          NA
## nodegree         NA
## re74      -102108.6
## re75      -131511.0

L1 statistic from the output above measures differences between the uni- (and multi-) dimensional histograms of all covariates in the two groups. Perfect (up to discretization) global balance result in L1 = 0. Whereas, L1 = 1 indicates complete separation of the multidimensional histograms. Any value in the interval (0, 1) indicates the amount of difference between frequencies of units in strata for each coarsened covariate of the two groups.

In our case, the multivariate imbalanced measure (L1) is close to 1, suggesting an almost complete separation. The differences in the empirical quantiles of the two distributions (same as eQQ statistics reported by MatchIt) also indicate a large amount of imbalance for many covariates.

Let’s now perform CEM on our observational data:

## It is advised to prespecify relevant cutpoints for continuous covariates 
edu_cut <- c(0, 8.5, 12.5, 17)
re_cut <- c(0, 2500, 5000, 7500, 10000, 15000, 20000, 50000, 100000, 200000)

m_cem_out <-
  cem::cem(treatment = "treat",
           data = nsw_PSID_cem, 
           verbose = TRUE, 
           keep.all = TRUE, # coarsened dataset is returned
           drop = c("re78", "data_id"),
           cutpoints = list(education = edu_cut,
                            re75 = re_cut))  
## 
## Using 'treat'='1' as baseline group
m_cem_out
##             G0  G1
## All       2490 185
## Matched     71 111
## Unmatched 2419  74

Note that, unlike NNM example in MatchIt, CEM method dropped a few (74) treated units and a vast majority of control units in strata that did not have any comparables from the other groups. (Note that we could’ve specified discard = "both" in matchit() to remove units that fall outside of some measure of support before matching).

cem() used prespecified breaks for the corresponding variables and calculated the breaks automatically for the rest:

m_cem_out$breaks
## $age
##  [1] 17.00000 20.16667 23.33333 26.50000 29.66667 32.83333 36.00000 39.16667
##  [9] 42.33333 45.50000 48.66667 51.83333 55.00000
## 
## $education
## [1]  0.0  8.5 12.5 17.0
## 
## $re74
##  [1]      0.00  11429.06  22858.11  34287.17  45716.23  57145.29  68574.34
##  [8]  80003.40  91432.46 102861.52 114290.57 125719.63 137148.69
## 
## $re75
##  [1]      0   2500   5000   7500  10000  15000  20000  50000 100000 200000

We can also explore how the subjects were matched:

# Strata that had at least one treatment and one control subject
m_cem_out$mstrataID
##  [1]  1  6  7  8 10 11 12 13 14 16 19 22 26 27 31 33 38 41 44 49 51 59 60 61 62
## [26] 64 68 71 75 76 77 84 86 88 89 90
nsw_PSID_cem %>%
  mutate(strata_num = m_cem_out$mstrata) %>%
  arrange(strata_num) %>%
  select(-data_id) %>%
  kable %>%
  kable_styling() %>%
  scroll_box(width = "800", height = "400px")
treat age education black hispanic married nodegree re74 re75 re78 strata_num
1 37 11 1 0 1 1 0.0000 0.00000 9930.04590 1
0 38 10 1 0 1 1 5877.8003 0.00000 0.00000 1
1 22 9 1 0 0 1 0.0000 0.00000 4056.49390 6
1 23 10 1 0 0 1 0.0000 0.00000 7693.39990 6
1 23 11 1 0 0 1 0.0000 0.00000 0.00000 6
1 21 9 1 0 0 1 0.0000 0.00000 0.00000 6
1 22 11 1 0 0 1 0.0000 0.00000 6456.69678 6
0 23 9 1 0 0 1 0.0000 0.00000 218.70354 6
0 23 11 1 0 0 1 0.0000 0.00000 0.00000 6
0 23 11 1 0 0 1 0.0000 0.00000 0.00000 6
0 22 11 1 0 0 1 2351.1201 0.00000 879.24731 6
1 23 12 1 0 0 0 0.0000 0.00000 0.00000 7
1 21 12 1 0 0 0 0.0000 0.00000 1254.58203 7
1 23 12 1 0 0 0 0.0000 0.00000 4843.17578 7
1 22 12 1 0 0 0 0.0000 0.00000 18678.08008 7
1 21 12 1 0 0 0 0.0000 0.00000 9983.78418 7
1 23 12 1 0 0 0 5506.3081 501.07410 671.33179 7
1 22 12 1 0 0 0 5605.8521 936.17731 0.00000 7
0 23 12 1 0 0 0 0.0000 0.00000 4728.72510 7
1 32 11 1 0 0 1 0.0000 0.00000 8472.15820 8
1 30 11 1 0 0 1 0.0000 0.00000 0.00000 8
1 31 9 1 0 0 1 0.0000 1698.60706 10363.26953 8
0 30 9 1 0 0 1 1959.2668 162.91936 16809.14062 8
1 33 12 0 0 1 0 0.0000 0.00000 12418.07031 10
0 34 12 0 0 1 0 0.0000 0.00000 0.00000 10
0 33 12 0 0 1 0 0.0000 0.00000 0.00000 10
0 34 12 0 0 1 0 0.0000 0.00000 0.00000 10
0 33 12 0 0 1 0 2351.1201 0.00000 2770.73730 10
1 19 9 1 0 0 1 0.0000 0.00000 8173.90820 11
1 19 10 1 0 0 1 0.0000 0.00000 3228.50293 11
1 18 10 1 0 0 1 0.0000 0.00000 11163.16992 11
1 17 10 1 0 0 1 0.0000 0.00000 16218.04004 11
1 17 10 1 0 0 1 0.0000 0.00000 0.00000 11
1 20 11 1 0 0 1 0.0000 0.00000 3972.54004 11
1 17 9 1 0 0 1 0.0000 0.00000 0.00000 11
1 18 11 1 0 0 1 0.0000 0.00000 0.00000 11
1 17 10 1 0 0 1 0.0000 0.00000 0.00000 11
1 17 10 1 0 0 1 0.0000 0.00000 0.00000 11
1 18 9 1 0 0 1 0.0000 0.00000 4482.84521 11
1 18 11 1 0 0 1 0.0000 0.00000 4814.62695 11
1 19 11 1 0 0 1 0.0000 0.00000 7458.10498 11
1 17 9 1 0 0 1 0.0000 0.00000 1953.26794 11
1 20 9 1 0 0 1 6083.9941 0.00000 8881.66504 11
1 18 11 1 0 0 1 858.2543 214.56360 929.88391 11
1 19 10 1 0 0 1 0.0000 385.27411 8124.71484 11
1 18 10 1 0 0 1 0.0000 798.90790 9737.15430 11
1 19 9 1 0 0 1 0.0000 798.90790 17685.17969 11
1 20 11 1 0 0 1 3637.4980 1220.83606 1085.43994 11
1 17 9 1 0 0 1 1716.5090 1253.43896 5445.20020 11
1 18 10 1 0 0 1 2143.4109 1784.27405 11141.38965 11
0 19 11 1 0 0 1 1567.4135 0.00000 0.00000 11
1 18 8 1 0 0 1 0.0000 0.00000 0.00000 12
1 17 7 1 0 0 1 0.0000 0.00000 3023.87891 12
1 17 8 1 0 0 1 0.0000 0.00000 8061.48486 12
1 17 8 1 0 0 1 0.0000 0.00000 0.00000 12
1 17 8 1 0 0 1 0.0000 0.00000 0.00000 12
1 18 6 1 0 0 1 0.0000 0.00000 0.00000 12
0 20 8 1 0 0 1 2938.9001 1838.66125 1226.51306 12
0 19 6 1 0 0 1 3526.6802 0.00000 0.00000 12
1 27 10 1 0 1 1 0.0000 0.00000 18739.92969 13
1 29 11 1 0 1 1 0.0000 0.00000 9642.99902 13
1 27 9 1 0 1 1 0.0000 934.44537 1773.42297 13
0 29 11 1 0 1 1 783.7067 0.00000 23643.62500 13
1 27 13 1 0 0 0 0.0000 0.00000 14581.86035 14
1 29 13 1 0 0 0 0.0000 0.00000 7479.65576 14
1 28 15 1 0 0 0 0.0000 0.00000 9598.54102 14
1 27 13 1 0 0 0 0.0000 0.00000 0.00000 14
1 27 13 1 0 0 0 0.0000 0.00000 34099.28125 14
1 29 14 1 0 0 0 0.0000 679.67340 17814.98047 14
0 28 15 1 0 0 0 7445.2139 0.00000 12708.44922 14
1 26 12 1 0 0 0 0.0000 0.00000 10747.34961 16
1 25 12 1 0 0 0 0.0000 0.00000 3191.75293 16
1 25 12 1 0 0 0 0.0000 0.00000 2348.97290 16
1 25 12 1 0 0 0 0.0000 0.00000 11965.80957 16
1 25 12 1 0 0 0 0.0000 0.00000 0.00000 16
0 25 12 1 0 0 0 0.0000 0.00000 0.00000 16
0 24 12 1 0 0 0 783.7067 0.00000 10344.08594 16
1 24 11 1 0 0 1 0.0000 0.00000 1991.40002 19
1 25 11 1 0 0 1 0.0000 0.00000 9897.04883 19
1 24 11 1 0 0 1 0.0000 0.00000 995.70020 19
1 24 10 1 0 0 1 0.0000 0.00000 0.00000 19
1 26 11 1 0 0 1 0.0000 0.00000 17230.96094 19
1 25 11 1 0 0 1 0.0000 0.00000 0.00000 19
1 25 10 1 0 0 1 0.0000 0.00000 0.00000 19
1 24 10 1 0 0 1 0.0000 0.00000 0.00000 19
1 26 10 1 0 0 1 0.0000 0.00000 9265.78809 19
1 25 11 1 0 0 1 0.0000 0.00000 485.22980 19
1 24 9 1 0 0 1 9154.7002 2288.67505 4849.55908 19
1 24 10 1 0 0 1 4250.4019 2421.94702 1660.50806 19
0 25 11 1 0 0 1 0.0000 0.00000 3694.31641 19
1 25 11 1 0 1 1 0.0000 0.00000 1574.42395 22
1 25 11 1 0 1 1 0.0000 0.00000 0.00000 22
1 26 10 1 0 1 1 2027.9990 0.00000 0.00000 22
1 26 11 1 0 1 1 0.0000 1392.85303 1460.35999 22
1 24 11 1 0 1 1 824.3886 1666.11304 4032.70801 22
0 24 11 1 0 1 1 5427.1689 0.00000 1329.95386 22
0 26 10 1 0 1 1 8816.7002 0.00000 7388.63281 22
1 23 12 1 0 1 0 0.0000 0.00000 5911.55078 26
0 22 12 1 0 1 0 1269.6049 0.00000 0.00000 26
0 22 12 1 0 1 0 2006.2892 2470.64526 0.00000 26
1 46 8 1 0 1 1 0.0000 0.00000 3094.15601 27
0 46 6 1 0 1 1 0.0000 0.00000 0.00000 27
1 31 11 1 0 1 1 0.0000 0.00000 8087.48682 31
1 30 11 1 0 1 1 0.0000 0.00000 590.78180 31
1 31 11 1 0 1 1 0.0000 0.00000 14509.92969 31
0 32 11 1 0 1 1 6524.3584 0.00000 930.96771 31
1 41 4 1 0 1 1 0.0000 0.00000 7284.98584 33
0 41 6 1 0 1 1 0.0000 0.00000 0.00000 33
1 45 5 1 0 1 1 0.0000 0.00000 8546.71484 38
0 45 7 1 0 1 1 0.0000 0.00000 11821.81348 38
0 43 3 1 0 1 1 17.6334 0.00000 0.00000 38
1 40 11 1 0 0 1 0.0000 0.00000 23005.59961 41
0 42 10 1 0 0 1 7837.0674 1718.70972 0.00000 41
1 21 12 0 0 0 0 0.0000 0.00000 8048.60303 44
1 21 12 0 0 0 0 3670.8721 334.04941 12558.01953 44
0 21 12 0 0 0 0 0.0000 0.00000 0.00000 44
1 27 12 1 0 1 0 3670.8721 334.04929 0.00000 49
0 28 12 1 0 1 0 0.0000 0.00000 0.00000 49
0 27 12 1 0 1 0 3918.5337 1994.41943 17732.71875 49
0 27 12 1 0 1 0 6855.4746 0.00000 8275.26855 49
0 27 12 1 0 1 0 9796.3340 0.00000 11821.81348 49
1 27 13 0 0 1 0 9381.5664 853.72247 0.00000 51
0 27 13 0 0 1 0 0.0000 0.00000 28076.80469 51
0 28 17 0 0 1 0 0.0000 0.00000 0.00000 51
0 27 16 0 0 1 0 0.0000 0.00000 0.00000 51
0 27 17 0 0 1 0 0.0000 0.00000 0.00000 51
0 29 16 0 0 1 0 0.0000 0.00000 8396.44238 51
0 29 14 0 0 1 0 0.0000 0.00000 0.00000 51
0 27 17 0 0 1 0 0.0000 0.00000 0.00000 51
0 27 16 0 0 1 0 1469.4501 1253.22583 4363.72656 51
0 27 15 0 0 1 0 4702.2402 0.00000 0.00000 51
0 27 14 0 0 1 0 6269.6538 0.00000 26007.98828 51
1 19 11 1 0 0 1 2305.0259 2615.27588 4146.60303 59
1 20 10 1 0 0 1 5005.7310 2777.35498 5615.18896 59
1 18 9 1 0 0 1 0.0000 3287.37500 5010.34180 59
0 19 10 1 0 0 1 4702.2402 3437.41943 2364.36255 59
1 19 8 1 0 0 1 0.0000 2657.05688 9970.68066 60
1 19 8 1 0 0 1 2636.3530 2937.26392 7535.94189 60
0 20 8 1 0 0 1 1880.8961 3652.25806 1018.15363 60
1 27 11 1 0 0 1 2206.9399 2666.27393 0.00000 61
1 28 10 1 0 0 1 0.0000 2836.50610 3196.57104 61
1 27 10 1 0 0 1 1001.1460 3550.07495 0.00000 61
1 29 10 1 0 0 1 0.0000 4398.95020 0.00000 61
0 29 10 1 0 0 1 2037.6375 2685.48389 0.00000 61
1 26 11 1 0 1 1 0.0000 2754.64600 26372.27930 62
0 25 10 1 0 1 1 4702.2402 4282.45166 4947.42871 62
0 26 10 1 0 1 1 9404.4805 4923.38721 17732.71875 62
1 23 12 1 0 0 0 6269.3408 3039.95996 8484.23926 64
1 21 12 1 0 0 0 6473.6831 3332.40894 9371.03711 64
0 23 12 1 0 0 0 1077.5967 3222.58057 5910.90625 64
0 22 12 1 0 0 0 6583.1367 4475.80664 0.00000 64
0 23 12 1 0 0 0 11328.4814 2685.48389 29216.13281 64
1 22 9 1 0 0 1 2192.8770 3836.98608 3462.56396 68
0 22 9 1 0 0 1 3134.8269 4345.11279 13299.53906 68
1 29 12 1 0 0 0 9748.3867 4878.93701 10976.50977 71
0 28 12 1 0 0 0 2938.9001 2864.51611 0.00000 71
0 27 12 1 0 0 0 10623.1445 3222.58057 8866.35938 71
1 22 10 1 0 1 1 1468.3480 5588.66406 13228.28027 75
0 22 11 1 0 1 1 4449.4951 6187.35498 14570.38379 75
0 22 11 1 0 1 1 7837.0674 5012.90332 3694.31641 75
0 23 11 1 0 1 1 9980.5049 6266.12891 10344.08594 75
1 21 9 1 0 0 1 6416.4702 5749.33105 743.66663 76
0 23 10 1 0 0 1 1469.4501 5370.96777 0.00000 76
0 23 10 1 0 0 1 3918.5337 7161.29053 6649.76953 76
0 22 11 1 0 0 1 10971.8936 7161.29053 7675.31201 76
0 22 11 1 0 0 1 11069.8574 5191.93555 9605.22266 76
1 17 10 1 0 0 1 1291.4680 5793.85205 5522.78809 77
1 19 10 1 0 0 1 4121.9492 6056.75391 0.00000 77
0 18 11 1 0 0 1 5877.8003 6932.12891 9457.45020 77
1 22 12 0 0 0 0 6759.9941 8455.50391 12590.70996 84
0 21 12 0 0 0 0 11363.7471 9667.74219 13299.53906 84
1 33 12 1 0 1 0 20279.9492 10941.34961 15952.59961 86
0 35 12 1 0 1 0 21551.9355 14322.58105 15638.78027 86
0 35 12 1 0 1 0 21551.9355 14322.58105 15638.78027 86
1 35 9 1 0 1 1 13602.4297 13830.63965 12803.96973 88
0 34 10 1 0 1 1 12343.3809 11637.09668 9900.76758 88
0 36 11 1 0 1 1 13140.8018 12532.25781 12811.88867 88
0 34 10 1 0 1 1 21551.9355 10083.09668 18471.58203 88
1 35 8 1 0 1 1 13732.0703 17976.15039 3786.62793 89
0 36 7 1 0 1 1 14694.5010 15754.83887 19210.44531 89
1 33 11 1 0 1 1 14660.7100 25142.24023 4181.94189 90
0 36 11 1 0 1 1 18025.2559 21483.87109 26599.07812 90
1 22 9 0 1 0 1 0.0000 0.00000 3595.89404 NA
1 30 12 1 0 0 0 0.0000 0.00000 24909.44922 NA
1 27 11 1 0 0 1 0.0000 0.00000 7506.14600 NA
1 33 8 1 0 0 1 0.0000 0.00000 289.78989 NA
1 22 16 1 0 0 0 0.0000 0.00000 2164.02197 NA
1 21 13 1 0 0 0 0.0000 0.00000 17094.64062 NA
1 40 12 1 0 0 0 0.0000 0.00000 10804.32031 NA
1 41 14 0 0 0 0 0.0000 0.00000 5149.50098 NA
1 38 9 0 0 0 1 0.0000 0.00000 6408.95020 NA
1 27 10 0 1 0 1 0.0000 0.00000 11142.87012 NA
1 48 4 1 0 0 1 0.0000 0.00000 6551.59180 NA
1 20 12 1 0 0 0 0.0000 0.00000 0.00000 NA
1 42 14 1 0 0 0 0.0000 0.00000 20505.92969 NA
1 25 5 1 0 0 1 0.0000 0.00000 6181.87988 NA
1 19 9 0 0 0 1 0.0000 0.00000 13188.83008 NA
1 18 8 0 1 1 1 0.0000 0.00000 2787.95996 NA
1 25 5 1 0 0 1 0.0000 0.00000 12187.41016 NA
1 28 8 1 0 0 1 0.0000 0.00000 0.00000 NA
1 37 9 1 0 0 1 0.0000 0.00000 1067.50598 NA
1 42 14 1 0 1 0 0.0000 0.00000 13167.51953 NA
1 22 11 0 0 0 1 0.0000 0.00000 1048.43201 NA
1 29 8 1 0 0 1 0.0000 0.00000 1923.93799 NA
1 35 10 1 0 0 1 0.0000 0.00000 4666.23584 NA
1 27 11 1 0 0 1 0.0000 0.00000 549.29840 NA
1 29 4 1 0 0 1 0.0000 0.00000 762.91461 NA
1 28 9 1 0 0 1 0.0000 0.00000 10694.29004 NA
1 27 11 1 0 0 1 0.0000 0.00000 0.00000 NA
1 23 7 0 0 0 1 0.0000 0.00000 0.00000 NA
1 27 9 1 0 0 1 0.0000 0.00000 0.00000 NA
1 46 13 1 0 0 0 0.0000 0.00000 647.20459 NA
1 25 11 0 0 0 1 0.0000 0.00000 18783.34961 NA
1 18 12 1 0 0 0 0.0000 0.00000 2321.10693 NA
1 38 12 0 0 0 0 0.0000 0.00000 4941.84912 NA
1 27 8 1 0 0 1 0.0000 0.00000 0.00000 NA
1 38 11 1 0 0 1 0.0000 0.00000 0.00000 NA
1 23 8 0 1 0 1 0.0000 0.00000 3881.28394 NA
1 25 8 1 0 0 1 0.0000 0.00000 0.00000 NA
1 44 11 1 0 0 1 0.0000 0.00000 0.00000 NA
1 25 12 0 0 0 0 0.0000 0.00000 5587.50293 NA
1 42 12 1 0 0 0 0.0000 0.00000 2456.15308 NA
1 31 9 0 1 0 1 0.0000 0.00000 26817.59961 NA
1 43 9 1 0 0 1 0.0000 0.00000 0.00000 NA
1 17 9 0 1 0 1 445.1704 74.34345 6210.66992 NA
1 20 12 1 0 0 0 989.2678 165.20770 0.00000 NA
1 27 12 1 0 0 0 2143.4131 357.94989 22163.25000 NA
1 20 12 1 0 0 0 0.0000 377.56860 1652.63696 NA
1 18 11 0 0 0 1 3678.2310 919.55792 4321.70508 NA
1 23 10 1 0 1 1 0.0000 936.43860 11233.25977 NA
1 23 12 0 1 0 0 9385.7402 1117.43896 559.44318 NA
1 28 11 1 0 0 1 0.0000 1284.07898 60307.92969 NA
1 20 11 1 0 0 1 16318.6201 1484.99402 6943.34180 NA
1 23 8 0 0 1 1 0.0000 1713.15002 4232.30908 NA
1 29 12 1 0 0 0 10881.9404 1817.28406 0.00000 NA
1 26 11 0 0 0 1 0.0000 2226.26611 13385.86035 NA
1 25 12 1 0 0 0 14426.7900 2409.27393 0.00000 NA
1 46 8 1 0 0 1 3165.6580 2594.72290 0.00000 NA
1 31 12 0 0 0 0 0.0000 2611.21802 2484.54907 NA
1 24 12 1 0 0 0 13765.7500 2842.76392 6167.68115 NA
1 42 9 1 0 1 1 0.0000 3058.53101 1294.40906 NA
1 25 13 1 0 0 0 12362.9297 3090.73193 0.00000 NA
1 21 8 1 0 0 1 989.2678 3695.89697 4279.61279 NA
1 31 4 1 0 0 1 8517.5889 4023.21094 7382.54883 NA
1 24 10 1 0 1 1 11703.2002 4078.15210 0.00000 NA
1 19 10 0 0 0 1 0.0000 5324.10889 13829.62012 NA
1 19 11 0 1 1 1 5424.4849 5463.80322 6788.46289 NA
1 31 9 1 0 0 1 10717.0303 5517.84082 9558.50098 NA
1 26 12 1 0 1 0 8408.7617 5794.83105 1424.94397 NA
1 20 9 0 1 0 1 12260.7803 5875.04883 1358.64294 NA
1 26 10 1 0 0 1 25929.6797 6788.95801 672.87732 NA
1 28 11 1 0 0 1 1929.0291 6871.85596 0.00000 NA
1 22 12 0 1 1 0 492.2305 7055.70215 10092.83008 NA
1 33 11 1 0 0 1 0.0000 7867.91602 6281.43311 NA
1 29 10 0 1 0 1 0.0000 8853.67383 5112.01416 NA
1 25 14 1 0 1 0 35040.0703 11536.57031 36646.94922 NA
0 47 12 0 0 0 0 0.0000 0.00000 0.00000 NA
0 50 12 1 0 1 0 0.0000 0.00000 0.00000 NA
0 44 12 0 0 0 0 0.0000 0.00000 0.00000 NA
0 54 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 55 12 0 1 1 0 0.0000 0.00000 0.00000 NA
0 47 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 44 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 50 12 1 0 1 0 0.0000 0.00000 0.00000 NA
0 49 14 0 0 1 0 0.0000 0.00000 0.00000 NA
0 55 14 0 0 1 0 0.0000 0.00000 0.00000 NA
0 34 14 1 0 1 0 0.0000 0.00000 0.00000 NA
0 34 16 0 0 1 0 0.0000 0.00000 11821.81348 NA
0 55 13 0 1 1 0 0.0000 0.00000 0.00000 NA
0 52 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 52 15 0 0 1 0 0.0000 0.00000 0.00000 NA
0 30 16 0 0 1 0 0.0000 0.00000 0.00000 NA
0 30 17 0 0 0 0 0.0000 0.00000 0.00000 NA
0 44 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 46 12 0 0 0 0 0.0000 0.00000 0.00000 NA
0 53 17 0 0 1 0 0.0000 0.00000 0.00000 NA
0 26 16 0 0 0 0 0.0000 0.00000 9014.13184 NA
0 49 12 0 0 1 0 0.0000 0.00000 13299.53906 NA
0 39 15 0 0 1 0 0.0000 0.00000 1921.04456 NA
0 26 17 0 0 1 0 0.0000 0.00000 11821.81348 NA
0 32 16 0 0 1 0 0.0000 0.00000 0.00000 NA
0 31 14 0 0 1 0 0.0000 0.00000 0.00000 NA
0 39 17 0 1 1 0 0.0000 0.00000 8866.35938 NA
0 40 17 0 0 1 0 0.0000 0.00000 0.00000 NA
0 31 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 48 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 41 13 0 0 1 0 0.0000 0.00000 0.00000 NA
0 26 14 0 0 0 0 0.0000 0.00000 0.00000 NA
0 50 17 0 0 0 0 0.0000 0.00000 0.00000 NA
0 48 12 0 0 1 0 0.0000 0.00000 12930.10840 NA
0 45 16 0 0 1 0 0.0000 0.00000 0.00000 NA
0 37 13 0 0 1 0 0.0000 0.00000 0.00000 NA
0 31 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 36 12 1 0 1 0 0.0000 0.00000 17732.71875 NA
0 44 13 0 0 1 0 0.0000 0.00000 0.00000 NA
0 30 17 0 0 1 0 0.0000 0.00000 0.00000 NA
0 42 16 0 0 1 0 0.0000 0.00000 0.00000 NA
0 30 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 37 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 37 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 45 12 0 0 1 0 0.0000 0.00000 22165.89844 NA
0 35 16 0 0 1 0 0.0000 0.00000 0.00000 NA
0 37 17 0 0 1 0 0.0000 0.00000 66497.69531 NA
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0 50 14 0 0 1 0 0.0000 0.00000 0.00000 NA
0 50 12 0 0 1 0 0.0000 0.00000 0.00000 NA
0 21 14 0 1 0 0 0.0000 71.61290 3657.37329 NA
0 26 12 0 0 1 0 0.0000 71.61290 0.00000 NA
0 21 13 0 0 0 0 0.0000 692.85486 2659.90771 NA
0 23 16 0 0 1 0 0.0000 1521.77417 0.00000 NA
0 21 15 0 0 0 0 0.0000 3222.58057 14777.26562 NA
0 23 14 0 0 0 0 0.0000 3401.61279 5319.81592 NA
0 27 16 0 0 1 0 0.0000 4833.87109 0.00000 NA
0 25 12 0 0 1 0 0.0000 5729.03223 26599.07812 NA
0 27 17 0 0 0 0 0.0000 6087.09668 7388.63281 NA
0 55 12 0 0 1 0 0.0000 6266.12891 10344.08594 NA
0 20 12 0 0 1 0 0.0000 7161.29053 0.00000 NA
0 18 12 0 0 1 0 0.0000 8056.45166 19210.44531 NA
0 27 12 0 0 1 0 0.0000 8235.48438 20688.17188 NA
0 28 16 0 0 1 0 0.0000 8951.61328 36943.16406 NA
0 36 12 1 0 1 0 0.0000 8951.61328 11821.81348 NA
0 33 13 1 0 1 0 0.0000 10741.93457 20688.17188 NA
0 36 12 0 0 1 0 0.0000 10741.93457 0.00000 NA
0 27 13 0 0 1 0 0.0000 16112.90332 29554.53125 NA
0 45 17 0 0 1 0 0.0000 16112.90332 17732.71875 NA
0 45 17 0 0 1 0 0.0000 16112.90332 17732.71875 NA
0 45 17 0 0 1 0 0.0000 16112.90332 17732.71875 NA
0 31 12 0 0 1 0 0.0000 17903.22656 0.00000 NA
0 34 12 0 0 1 0 0.0000 17903.22656 63542.24219 NA
0 34 12 0 0 1 0 0.0000 17903.22656 63542.24219 NA
0 47 14 0 0 1 0 0.0000 17903.22656 22165.89844 NA
0 35 16 0 0 1 0 0.0000 19693.54883 17732.71875 NA
0 43 17 0 0 1 0 0.0000 21161.61328 29554.53125 NA
0 25 12 1 0 1 0 0.0000 21483.87109 1220.60217 NA
0 29 15 0 0 0 0 0.0000 21483.87109 22313.67188 NA
0 31 12 0 0 1 0 0.0000 21483.87109 20688.17188 NA
0 52 12 0 0 1 0 0.0000 21483.87109 22165.89844 NA
0 22 16 0 0 0 0 0.0000 22558.06445 32509.98438 NA
0 53 15 0 0 1 0 0.0000 24169.35547 28076.80469 NA
0 29 17 0 0 1 0 0.0000 25959.67773 36898.83203 NA
0 31 13 0 0 1 0 0.0000 25959.67773 39898.61719 NA
0 35 12 0 0 1 0 0.0000 26138.71094 29554.53125 NA
0 29 16 0 0 1 0 0.0000 26854.83984 23643.62500 NA
0 39 12 0 0 1 0 0.0000 28645.16016 31032.25781 NA
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0 54 7 0 0 1 1 15674.1338 16112.90332 18841.01367 NA
0 37 9 1 0 1 1 15674.1338 16112.90332 19210.44531 NA
0 24 9 0 0 1 1 15674.1338 16112.90332 36943.16406 NA
0 53 6 1 0 1 1 15674.1338 16112.90332 9974.65430 NA
0 21 10 0 0 1 1 15674.1338 16112.90332 20141.41211 NA
0 35 10 1 0 1 1 15674.1338 16470.96875 23643.62500 NA
0 54 2 1 0 1 1 15674.1338 16990.16016 18096.24023 NA
0 43 10 0 0 1 1 15674.1338 17903.22656 19709.91797 NA
0 46 8 0 0 1 1 15674.1338 17903.22656 17732.71875 NA
0 35 11 1 0 1 1 15674.1338 17903.22656 0.00000 NA
0 50 8 0 0 1 1 15674.1338 18109.11328 17437.17383 NA
0 47 10 0 0 1 1 15674.1338 19693.54883 22165.89844 NA
0 47 10 0 0 1 1 15674.1338 19693.54883 22165.89844 NA
0 21 9 0 0 1 1 15674.1338 19693.54883 1773.27185 NA
0 18 10 0 0 1 1 15674.1338 21483.87109 24973.57812 NA
0 37 8 0 0 1 1 15674.1338 26317.74219 36943.16406 NA
0 41 7 1 0 1 1 15674.1338 32225.80664 11082.94922 NA
0 31 11 0 0 1 1 15870.0605 15602.66113 17732.71875 NA
0 35 10 0 0 1 1 15921.0020 16148.70996 16606.69141 NA
0 51 8 1 0 1 1 15987.6172 5800.64502 0.00000 NA
0 30 7 1 0 1 1 15995.4541 15387.82324 13151.76660 NA
0 30 11 0 0 1 1 16065.9883 17989.16016 23052.53516 NA
0 30 11 0 0 1 1 16065.9883 17989.16016 23052.53516 NA
0 53 4 1 0 1 1 16142.3994 984.67743 0.00000 NA
0 24 10 0 0 1 1 16144.3584 20767.74219 22904.76172 NA
0 48 8 1 0 1 1 16261.9141 11637.09668 5172.04297 NA
0 48 8 1 0 1 1 16261.9141 11637.09668 5172.04297 NA
0 47 6 1 0 1 1 16261.9141 14412.09668 14260.06055 NA
0 45 4 0 1 1 1 16261.9141 28108.06445 28076.80469 NA
0 47 8 1 0 1 1 16301.0996 17129.80664 17732.71875 NA
0 47 8 1 0 1 1 16301.0996 17129.80664 17732.71875 NA
0 31 11 1 0 1 1 16453.9238 19693.54883 14777.26562 NA
0 46 8 0 0 1 1 16457.8418 16112.90332 16254.99219 NA
0 22 11 0 0 0 1 16653.7676 3759.67749 0.00000 NA
0 21 11 0 0 1 1 16653.7676 13427.41895 10344.08594 NA
0 33 10 0 0 1 1 16653.7676 16112.90332 12560.67578 NA
0 28 9 0 0 1 1 16653.7676 17158.45117 17957.33203 NA
0 20 10 0 0 1 1 16849.6953 11995.16113 17880.49219 NA
0 47 11 0 0 1 1 16928.0645 15409.30566 17732.71875 NA
0 23 11 0 0 1 1 17045.6211 11986.20996 5910.90625 NA
0 35 10 1 0 0 1 17045.6211 14322.58105 16846.08203 NA
0 25 7 1 0 0 1 17045.6211 16137.96777 11821.81348 NA
0 23 11 1 0 1 1 17116.1543 10741.93457 8866.35938 NA
0 31 8 0 0 1 1 17241.5488 15754.83887 16210.66113 NA
0 55 8 1 0 1 1 17241.5488 17903.22656 16254.99219 NA
0 33 11 1 0 1 1 17370.8594 15844.35547 14753.62207 NA
0 31 9 1 0 0 1 17437.4746 13427.41895 11326.77441 NA
0 34 7 0 0 0 1 17619.6855 20481.28906 19432.10352 NA
0 22 11 0 0 1 1 17633.4004 0.00000 13299.53906 NA
0 21 9 0 0 1 1 17633.4004 3202.88721 3250.99854 NA
0 42 8 1 0 0 1 17633.4004 10562.90332 0.00000 NA
0 46 10 0 1 1 1 17633.4004 10741.93457 5910.90625 NA
0 23 9 1 0 1 1 17633.4004 12532.25781 14777.26562 NA
0 33 8 0 0 1 1 17633.4004 12890.32324 22165.89844 NA
0 43 11 1 0 1 1 17633.4004 14322.58105 17732.71875 NA
0 43 11 1 0 1 1 17633.4004 14322.58105 17732.71875 NA
0 29 8 0 0 1 1 17633.4004 16112.90332 11821.81348 NA
0 35 9 0 0 1 1 17633.4004 16112.90332 17732.71875 NA
0 50 9 1 0 1 1 17633.4004 16112.90332 16254.99219 NA
0 18 9 0 0 1 1 17633.4004 16112.90332 17584.94531 NA
0 44 11 0 0 1 1 17633.4004 16470.96875 14555.60742 NA
0 34 6 0 0 1 1 17633.4004 17903.22656 4876.49756 NA
0 26 8 0 0 1 1 17633.4004 18440.32227 886.63593 NA
0 47 10 0 0 1 1 17633.4004 19693.54883 30884.48438 NA
0 25 11 1 0 1 1 17633.4004 21483.87109 26599.07812 NA
0 44 8 1 0 1 1 17633.4004 21483.87109 19210.44531 NA
0 36 10 1 0 0 1 17633.4004 21483.87109 17732.71875 NA
0 40 2 1 0 1 1 17645.1562 13427.41895 8866.35938 NA
0 46 11 0 0 1 1 17684.3418 16112.90332 19210.44531 NA
0 44 5 1 0 1 1 17688.2617 17903.22656 19210.44531 NA
0 38 9 0 0 1 1 17829.3281 7877.41943 20688.17188 NA
0 54 11 0 0 1 1 17829.3281 15754.83887 18102.15039 NA
0 32 8 0 0 1 1 17829.3281 21483.87109 20688.17188 NA
0 23 10 0 0 1 1 17829.3281 21483.87109 23643.62500 NA
0 46 11 1 0 1 1 17868.5137 17903.22656 13326.13770 NA
0 44 11 1 0 1 1 17929.2520 18588.91992 19210.44531 NA
0 41 10 1 0 1 1 18025.2559 17366.12891 21427.03516 NA
0 44 8 0 0 1 1 18025.2559 18798.38672 28076.80469 NA
0 23 8 0 0 1 1 18221.1816 25073.46875 12117.35840 NA
0 20 9 0 0 1 1 18285.8379 20767.74219 23643.62500 NA
0 46 3 1 0 1 1 18319.1445 14322.58105 19949.30859 NA
0 36 10 1 0 1 1 18417.1074 16112.90332 22165.89844 NA
0 40 10 0 0 1 1 18417.1074 17903.22656 20688.17188 NA
0 40 10 0 0 1 1 18417.1074 17903.22656 20688.17188 NA
0 27 8 1 0 1 1 18417.1074 22916.12891 31032.25781 NA
0 24 11 1 0 0 1 18613.0352 20230.64453 31032.25781 NA
0 34 11 1 0 1 1 18808.9609 17903.22656 14777.26562 NA
0 47 10 0 0 1 1 19004.8887 17903.22656 21722.58008 NA
0 47 10 0 0 1 1 19004.8887 17903.22656 21722.58008 NA
0 49 4 0 0 1 1 19200.8145 7161.29053 2955.45312 NA
0 35 9 0 0 1 1 19200.8145 17706.28906 8866.35938 NA
0 33 9 1 0 1 1 19396.7402 16112.90332 19210.44531 NA
0 51 11 0 0 1 1 19592.6680 6982.25830 26599.07812 NA
0 33 10 0 0 1 1 19592.6680 10741.93457 26599.07812 NA
0 20 7 1 0 1 1 19592.6680 11637.09668 14777.26562 NA
0 36 8 0 1 1 1 19592.6680 11995.16113 16993.85547 NA
0 36 8 0 1 1 1 19592.6680 11995.16113 16993.85547 NA
0 28 11 1 0 1 1 19592.6680 12890.32324 7130.03076 NA
0 44 6 1 0 1 1 19592.6680 14322.58105 5615.36084 NA
0 22 10 0 0 1 1 19592.6680 15217.74219 0.00000 NA
0 50 10 0 0 1 1 19592.6680 15933.87109 29554.53125 NA
0 52 9 0 0 1 1 19592.6680 16112.90332 24045.56641 NA
0 25 5 1 0 1 1 19592.6680 16650.00000 28076.80469 NA
0 52 8 0 0 1 1 19592.6680 17903.22656 17289.40039 NA
0 38 8 0 0 1 1 19592.6680 17903.22656 30207.68750 NA
0 53 7 0 0 1 1 19592.6680 19693.54883 22165.89844 NA
0 38 8 0 0 1 1 19592.6680 19693.54883 23643.62500 NA
0 38 8 0 0 1 1 19592.6680 19693.54883 23643.62500 NA
0 22 11 0 0 1 1 19592.6680 20588.71094 27929.03125 NA
0 41 9 0 0 1 1 19592.6680 21483.87109 20303.96289 NA
0 30 11 1 0 1 1 19592.6680 24547.11328 22165.89844 NA
0 24 10 0 0 1 1 19592.6680 25064.51562 29554.53125 NA
0 43 6 1 0 1 1 19912.0293 16112.90332 20688.17188 NA
0 55 6 1 0 1 1 19962.9688 26854.83984 14777.26562 NA
0 27 7 0 0 1 1 19984.5215 20409.67773 31032.25781 NA
0 23 10 0 0 1 1 19984.5215 21483.87109 17732.71875 NA
0 53 10 0 0 1 1 19984.5215 21483.87109 4433.17969 NA
0 48 8 0 0 1 1 19984.5215 21483.87109 19210.44531 NA
0 53 10 1 0 1 1 20062.8926 15502.40332 13299.53906 NA
0 50 7 0 0 1 1 20180.4473 14322.58105 0.00000 NA
0 40 11 1 0 1 1 20278.4102 16112.90332 19210.44531 NA
0 50 7 1 0 1 1 20384.2129 17008.06445 17732.71875 NA
0 38 9 1 0 1 1 20388.1309 22407.67773 16254.99219 NA
0 52 5 0 0 1 1 20572.3008 16112.90332 19210.44531 NA
0 36 10 0 0 1 1 20572.3008 17366.12891 14481.71973 NA
0 40 11 0 0 1 1 20572.3008 21483.87109 27485.71484 NA
0 38 11 0 0 1 1 20572.3008 21483.87109 19210.44531 NA
0 41 8 0 0 1 1 20572.3008 22379.03125 19633.07422 NA
0 43 11 0 0 1 1 20629.1191 31330.64453 26599.07812 NA
0 46 10 0 0 1 1 20944.5625 21338.85547 21131.49023 NA
0 53 8 0 0 1 1 20964.1543 22033.50000 23195.87305 NA
0 25 10 0 0 1 1 20964.1543 23632.25781 23495.85352 NA
0 44 10 1 0 1 1 20993.5449 19190.46875 25068.15430 NA
0 44 10 1 0 1 1 20993.5449 19190.46875 25068.15430 NA
0 37 8 0 0 1 1 21481.4004 14322.58105 23643.62500 NA
0 46 11 0 0 1 1 21551.9355 16112.90332 13299.53906 NA
0 48 6 1 0 1 1 21551.9355 16112.90332 17732.71875 NA
0 42 10 0 0 1 1 21551.9355 18798.38672 22165.89844 NA
0 55 10 0 0 1 1 21551.9355 19693.54883 14777.26562 NA
0 39 9 0 0 1 1 21551.9355 19693.54883 20688.17188 NA
0 41 10 1 0 1 1 21551.9355 20767.74219 25121.35156 NA
0 33 8 1 0 1 1 21551.9355 22379.03125 27485.71484 NA
0 22 10 1 0 1 1 21551.9355 25064.51562 31032.25781 NA
0 25 8 0 0 1 1 21551.9355 25959.67773 25121.35156 NA
0 54 9 1 0 0 1 21551.9355 30435.48438 33987.71094 NA
0 49 8 0 0 0 1 21551.9355 36316.69531 44331.79688 NA
0 35 10 0 0 1 1 21583.2832 23847.09766 22756.98828 NA
0 43 9 0 0 1 1 21630.3047 11565.48438 11821.81348 NA
0 39 7 0 0 1 1 21712.5957 24065.51562 23643.62500 NA
0 47 10 0 1 1 1 21943.7871 18977.41992 19210.44531 NA
0 51 10 0 0 1 1 21943.7871 23274.19336 25860.21484 NA
0 51 10 0 0 1 1 21943.7871 23274.19336 25860.21484 NA
0 49 5 0 1 1 1 22041.7520 23274.19336 20392.62695 NA
0 48 11 0 0 1 1 22178.9004 19693.54883 16254.99219 NA
0 48 11 0 0 1 1 22178.9004 19693.54883 16254.99219 NA
0 30 10 1 0 1 1 22335.6426 0.00000 121173.57812 NA
0 31 9 0 0 1 1 22531.5684 21483.87109 0.00000 NA
0 28 9 0 0 1 1 22727.4941 20767.74219 22904.76172 NA
0 49 10 0 0 1 1 22727.4941 23095.16016 24086.94336 NA
0 48 10 0 0 1 1 22727.4941 26854.83984 27042.39648 NA
0 49 9 0 0 1 1 22727.4941 30435.48438 29554.53125 NA
0 49 4 1 0 1 1 22968.4844 28645.16016 22165.89844 NA
0 32 10 0 0 1 1 23025.3027 21483.87109 33987.71094 NA
0 30 11 0 0 1 1 23119.3477 20051.61328 22018.12695 NA
0 47 10 0 0 1 1 23236.9043 17903.22656 35364.95312 NA
0 31 10 0 0 1 1 23315.2754 2467.06445 15220.58398 NA
0 48 10 0 0 1 1 23511.2012 2864.51611 0.00000 NA
0 28 11 0 0 1 1 23511.2012 5370.96777 21796.46680 NA
0 33 10 1 0 1 1 23511.2012 12532.25781 18471.58203 NA
0 52 8 1 0 0 1 23511.2012 15217.74219 7093.08740 NA
0 53 5 0 0 1 1 23511.2012 17903.22656 20215.30078 NA
0 42 6 0 0 1 1 23511.2012 17903.22656 23643.62500 NA
0 42 6 0 0 1 1 23511.2012 17903.22656 23643.62500 NA
0 42 6 0 0 1 1 23511.2012 17903.22656 23643.62500 NA
0 42 8 0 0 1 1 23511.2012 18798.38672 26569.52344 NA
0 30 9 0 0 1 1 23511.2012 19441.11328 26599.07812 NA
0 36 7 1 0 1 1 23511.2012 21483.87109 13299.53906 NA
0 54 8 0 0 1 1 23511.2012 21483.87109 19210.44531 NA
0 42 9 0 0 1 1 23511.2012 21841.93555 26303.53320 NA
0 51 8 1 0 1 1 23511.2012 23274.19336 23643.62500 NA
0 24 10 0 0 1 1 23511.2012 25062.72656 29678.66016 NA
0 23 10 0 0 1 1 23511.2012 25064.51562 25121.35156 NA
0 50 8 0 0 1 1 23511.2012 25243.54883 26155.75977 NA
0 51 11 0 0 1 1 23511.2012 25780.64453 27190.16992 NA
0 51 8 0 0 1 1 23511.2012 26407.25781 22165.89844 NA
0 51 8 0 0 1 1 23511.2012 27019.54883 27337.94141 NA
0 30 10 0 0 1 1 23511.2012 28645.16016 28047.25000 NA
0 52 11 0 0 1 1 23568.0195 0.00000 0.00000 NA
0 50 8 1 0 1 1 23707.1289 28913.71094 31032.25781 NA
0 29 10 0 0 1 1 23711.0469 26854.83984 13299.53906 NA
0 48 9 0 0 1 1 23859.9512 25064.51562 26968.50977 NA
0 30 9 0 0 1 1 23903.0547 23900.80664 23791.39844 NA
0 37 8 0 0 1 1 24098.9824 22823.03125 23145.63086 NA
0 28 11 0 0 1 1 24098.9824 24169.35547 35465.43750 NA
0 44 11 1 0 1 1 24294.9082 35806.45312 29554.53125 NA
0 44 11 1 0 1 1 24294.9082 35806.45312 29554.53125 NA
0 54 3 0 0 1 1 24530.0195 19514.51562 30549.04102 NA
0 52 9 0 0 1 1 24627.9844 25064.51562 19210.44531 NA
0 35 9 1 0 1 1 24686.7617 16829.03125 20392.62695 NA
0 32 10 0 0 1 1 24686.7617 27033.87109 22904.76172 NA
0 47 7 0 0 1 1 24784.7246 23811.28906 25860.21484 NA
0 30 11 0 0 1 1 24882.6875 26854.83984 33248.84766 NA
0 49 6 0 0 1 1 25274.5430 21841.93555 29111.21484 NA
0 48 8 0 0 1 1 25470.4688 0.00000 13299.53906 NA
0 32 9 0 0 1 1 25470.4688 1432.25806 24803.64062 NA
0 44 6 0 0 1 1 25470.4688 7619.61279 11821.81348 NA
0 48 8 0 1 1 1 25470.4688 16112.90332 23643.62500 NA
0 52 7 1 0 0 1 25470.4688 17903.22656 25121.35156 NA
0 49 11 0 0 1 1 25470.4688 19693.54883 481.73886 NA
0 38 9 0 0 1 1 25470.4688 21483.87109 47287.25000 NA
0 46 8 1 0 1 1 25470.4688 21483.87109 11390.31641 NA
0 40 8 0 0 1 1 25470.4688 21483.87109 16254.99219 NA
0 44 3 1 0 1 1 25470.4688 21483.87109 26599.07812 NA
0 40 10 1 0 0 1 25470.4688 23274.19336 20688.17188 NA
0 26 8 1 0 1 1 25470.4688 23274.19336 23643.62500 NA
0 49 11 0 0 1 1 25470.4688 25064.51562 25121.35156 NA
0 51 11 0 0 1 1 25470.4688 25064.51562 26599.07812 NA
0 44 8 0 0 1 1 25470.4688 26854.83984 25860.21484 NA
0 47 8 0 0 1 1 25707.5391 25064.51562 19062.67383 NA
0 49 8 0 0 1 1 25862.3223 22916.12891 16254.99219 NA
0 40 8 0 0 1 1 25862.3223 26138.71094 26155.75977 NA
0 48 10 0 0 0 1 26058.2480 26854.83984 33043.44531 NA
0 35 9 1 0 1 1 26134.6602 24169.35547 26599.07812 NA
0 44 6 0 1 1 1 26450.1016 3580.64526 0.00000 NA
0 42 11 0 0 1 1 26450.1016 22379.03125 23327.39258 NA
0 31 8 0 0 1 1 26450.1016 32225.80664 26953.73242 NA
0 46 11 0 0 1 1 26841.9551 29361.28906 36943.16406 NA
0 51 10 1 0 1 1 27037.8828 21483.87109 26894.62305 NA
0 50 8 0 0 0 1 27084.9043 21125.80664 26599.07812 NA
0 46 11 0 0 1 1 27429.7344 9846.77441 16993.85547 NA
0 51 10 1 0 1 1 27429.7344 25064.51562 24382.48828 NA
0 41 7 1 0 1 1 27429.7344 26708.03125 14777.26562 NA
0 39 8 1 0 1 1 27429.7344 26854.83984 28076.80469 NA
0 48 10 1 0 1 1 27429.7344 30435.48438 29554.53125 NA
0 48 10 1 0 1 1 27429.7344 30435.48438 29554.53125 NA
0 50 10 0 0 1 1 27429.7344 32225.80664 33987.71094 NA
0 49 6 1 0 1 1 27429.7344 34016.12891 32509.98438 NA
0 49 8 0 0 1 1 27482.6348 25064.51562 27337.94141 NA
0 47 7 0 0 1 1 27821.5898 27929.03125 28925.01953 NA
0 42 9 0 0 1 1 28017.5156 23453.22656 26377.41992 NA
0 42 9 0 0 1 1 28017.5156 23453.22656 26377.41992 NA
0 45 10 0 0 1 1 28017.5156 32225.80664 36943.16406 NA
0 55 10 0 0 1 1 28330.9980 28602.19336 28815.66797 NA
0 40 11 0 0 1 1 28409.3691 12532.25781 13299.53906 NA
0 48 8 0 0 1 1 28409.3691 26317.74219 28076.80469 NA
0 36 10 0 1 1 1 28409.3691 28645.16016 39159.75391 NA
0 46 9 1 0 1 1 28605.2949 32225.80664 41376.34375 NA
0 46 10 0 0 1 1 28765.9551 19514.51562 25676.97656 NA
0 45 9 1 0 1 1 28863.9199 26854.83984 36943.16406 NA
0 49 7 0 0 1 1 29247.9355 20767.74219 21341.32617 NA
0 37 10 0 0 1 1 29369.4082 27750.00000 39898.61719 NA
0 48 9 0 0 1 1 29389.0020 19693.54883 23052.53516 NA
0 37 11 1 0 1 1 29389.0020 21483.87109 13063.10254 NA
0 28 10 0 0 1 1 29389.0020 23274.19336 22904.76172 NA
0 36 11 1 0 0 1 29389.0020 25982.95117 45809.52344 NA
0 40 11 0 0 1 1 29389.0020 26407.25781 31032.25781 NA
0 44 10 0 0 1 1 29389.0020 26854.83984 23643.62500 NA
0 47 10 1 0 1 1 29389.0020 26854.83984 0.00000 NA
0 47 10 1 0 1 1 29389.0020 26854.83984 0.00000 NA
0 30 11 0 1 1 1 29389.0020 26854.83984 36943.16406 NA
0 33 10 0 0 1 1 29389.0020 28645.16016 36499.84766 NA
0 48 9 0 0 1 1 29389.0020 32225.80664 31032.25781 NA
0 19 11 1 0 0 1 29725.9961 8806.59668 9457.45020 NA
0 38 9 0 0 1 1 29976.7812 25064.51562 47287.25000 NA
0 42 9 0 0 0 1 30368.6348 19693.54883 35465.43750 NA
0 32 10 1 0 1 1 30930.9453 35806.45312 38125.34766 NA
0 55 11 0 0 1 1 31048.5020 31053.14453 28076.80469 NA
0 37 11 0 0 1 1 31250.3047 26854.83984 36943.16406 NA
0 21 10 0 0 1 1 31348.2695 10741.93457 44331.79688 NA
0 51 11 1 0 1 1 31348.2695 21483.87109 26007.98828 NA
0 42 6 1 0 1 1 31348.2695 28645.16016 20688.17188 NA
0 42 5 1 0 1 1 31348.2695 32225.80664 23643.62500 NA
0 52 11 0 0 1 1 31348.2695 33120.96875 28076.80469 NA
0 45 5 0 0 1 1 31348.2695 37596.77344 31477.05469 NA
0 49 8 0 0 1 1 31348.2695 37596.77344 38420.89062 NA
0 48 8 0 0 1 1 31510.8887 35795.71094 28076.80469 NA
0 48 8 0 0 1 1 31510.8887 35795.71094 28076.80469 NA
0 55 7 1 0 1 1 32504.2363 26854.83984 31918.89453 NA
0 55 7 1 0 1 1 32504.2363 26854.83984 31918.89453 NA
0 47 11 0 0 1 1 32719.7559 30435.48438 36943.16406 NA
0 51 8 0 0 1 1 33111.6094 27750.00000 32362.21289 NA
0 54 11 0 0 1 1 33158.6328 32225.80664 36943.16406 NA
0 42 9 0 0 1 1 33221.3281 32225.80664 33987.71094 NA
0 42 9 0 0 1 1 33221.3281 32225.80664 33987.71094 NA
0 51 8 1 0 1 1 33299.6992 25064.51562 25121.35156 NA
0 55 2 1 0 1 1 33307.5352 20946.77344 28076.80469 NA
0 32 10 0 0 1 1 33307.5352 25064.51562 29554.53125 NA
0 38 8 0 0 1 1 33307.5352 35448.38672 33987.71094 NA
0 42 11 0 0 1 1 33307.5352 44758.06641 54675.88281 NA
0 49 8 0 0 1 1 33699.3906 28645.16016 2955.45312 NA
0 35 7 0 0 1 1 34091.2422 28287.09766 25121.35156 NA
0 46 11 0 0 1 1 34287.1680 30077.41992 39898.61719 NA
0 46 11 0 0 1 1 34287.1680 30077.41992 39898.61719 NA
0 31 11 0 0 1 1 34483.0977 35806.45312 36943.16406 NA
0 52 10 0 0 1 1 35266.8008 17187.09766 39159.75391 NA
0 51 8 1 0 1 1 35266.8008 25064.51562 1182.18127 NA
0 51 9 0 0 0 1 35266.8008 32225.80664 39898.61719 NA
0 52 8 1 0 1 1 35266.8008 33120.96875 44147.08203 NA
0 49 6 0 0 1 1 35266.8008 33120.96875 26894.62305 NA
0 48 10 0 0 1 1 35266.8008 34911.28906 38568.66406 NA
0 31 10 0 0 1 1 35266.8008 35806.45312 44331.79688 NA
0 42 11 0 0 1 1 35266.8008 44758.06641 38420.89062 NA
0 50 11 0 0 1 1 36205.2891 38563.54688 31032.25781 NA
0 50 11 0 0 1 1 36205.2891 38563.54688 31032.25781 NA
0 37 10 0 0 1 1 36246.4375 17545.16016 17732.71875 NA
0 37 10 0 0 1 1 36246.4375 17545.16016 17732.71875 NA
0 55 11 0 0 1 1 36638.2891 37417.74219 32177.49609 NA
0 50 11 0 0 1 1 36677.4766 46548.38672 47287.25000 NA
0 41 8 1 0 1 1 36691.1875 32888.22656 42262.98047 NA
0 49 8 0 0 1 1 37226.0703 14322.58105 23643.62500 NA
0 48 8 1 0 1 1 37226.0703 26854.83984 30569.73047 NA
0 48 8 1 0 1 1 37226.0703 26854.83984 30569.73047 NA
0 48 6 0 1 1 1 38670.0508 42967.74219 36943.16406 NA
0 48 6 0 1 1 1 38670.0508 42967.74219 36943.16406 NA
0 46 11 0 0 1 1 39185.3359 39387.09766 35465.43750 NA
0 29 10 0 0 1 1 41144.6016 8951.61328 31032.25781 NA
0 55 8 0 0 1 1 43025.5000 37972.74219 51580.04688 NA
0 32 10 0 0 1 1 43103.8711 39387.09766 36943.16406 NA
0 47 8 0 0 1 1 44667.3633 33837.09766 38568.66406 NA
0 32 8 0 0 1 1 47022.4023 67137.09375 59109.06250 NA
0 47 10 0 0 1 1 48197.9648 47968.11328 55710.29297 NA
0 54 0 0 1 1 1 49228.5391 44220.96875 20540.39844 NA
0 40 8 0 0 1 1 50940.9375 55500.00000 53198.15625 NA

Let’s examine the resulting balance:

cobalt::bal.tab(m_cem_out,
                data =  nsw_PSID_cem, # Data must be specified for `cem` objects
                disp.means = T,
                disp.sds = T,
                stats = c("mean.diffs", "variance.ratios", "ks.statistics"))
## Balance Measures
##              Type   M.0.Adj SD.0.Adj   M.1.Adj  SD.1.Adj Diff.Adj V.Ratio.Adj
## age       Contin.   24.6697    6.065   24.1261    6.0394  -0.0760      0.9916
## education Contin.   10.7868    1.956   10.4324    1.6982  -0.1762      0.7538
## black      Binary    0.9550             0.9550             0.0000            
## hispanic   Binary    0.0000             0.0000             0.0000            
## married    Binary    0.2252             0.2252             0.0000            
## nodegree   Binary    0.7387             0.7387             0.0000            
## re74      Contin. 3172.9223 4193.724 1609.9595 3549.5576  -0.3198      0.7164
## re75      Contin. 1400.0887 3510.904 1491.9022 3579.8715   0.0285      1.0397
##           KS.Adj
## age       0.1802
## education 0.1899
## black     0.0000
## hispanic  0.0000
## married   0.0000
## nodegree  0.0000
## re74      0.4284
## re75      0.1275
## 
## Sample sizes
##                      Control Treated
## All                  2490.       185
## Matched (ESS)          14.67     111
## Matched (Unweighted)   71.       111
## Unmatched            2419.        74
cobalt::love.plot(m_cem_out, 
                  stats = c("mean.diffs", "variance.ratios", "ks.statistics"), 
                  data = nsw_PSID_cem,
                 # threshold = c(m = .5, ks = .25), 
                  binary = "std", 
                  abs = TRUE,
                  var.order = "unadjusted", 
                  grid = T,
                  sample.names = c("Matched", "Unmatched"),
                  colors = c("blue", "red"))
## Warning: Large mean differences detected; you may not be using standardized mean
## differences for continuous variables.

We’ve achieved a much better balance. However, lost many degrees of freedom by considerably pruning the treated group. Also, note that since we dropped quite a few units from the treatment group, our target population of treated units have changed and we need to acknowledge that and disclose when reporting the results of the analysis.

5.3 Estimating ATT with cem

Because cem results in a stratified data set, we have to use a special function cem::att() (which stands for “Average Treatment effect on the Treated”) to estimate the treatment effect. It takes CEM output and applies a model of user’s choice, using a specified formula:

## Need to exclude hispanic from the model as none were matched
est <- att(m_cem_out, 
           re78 ~ treat + age + education + black + #hispanic
             nodegree + married + re74 + re75, 
           data = nsw_PSID_cem)
summary(est)
## 
## Treatment effect estimation for data:
## 
##             G0  G1
## All       2490 185
## Matched     71 111
## Unmatched 2419  74
## 
## Linear regression model estimated on matched data only
## 
## Coefficients:
##                Estimate  Std. Error t value p-value  
## (Intercept) -7276.10189  5524.08616 -1.3172 0.18953  
## treat        1767.34678  1000.09179  1.7672 0.07896 .
## age           222.38810    97.22261  2.2874 0.02338 *
## education     547.90582   343.28229  1.5961 0.11230  
## black1       1153.87426  2409.26565  0.4789 0.63259  
## nodegree1   -1614.53273  1491.54480 -1.0825 0.28056  
## married1      596.96804  1425.24741  0.4189 0.67584  
## re74            0.11987     0.19858  0.6036 0.54688  
## re75            0.21496     0.20719  1.0375 0.30096  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The estimate of the treatment effect ($1767) is very close to the one obtained from the randomized experiment.

For more on cem package check out Iacus, King, Porro, 2009, “CEM: Software for Coarsened Exact Matching.” Journal of Statistical Software.

5.4 Exercise 2*

Progressive coarsening with cem

Although the maximum imbalance is fixed by the user’s coarsening choices, the number of observations matched is determined as a consequence of the matching procedure. If you are dissatisfied with the number of observations available after matching, and you feel that it is substantively appropriate to coarsen further, then just increase the coarsening (by using fewer cutpoints). The result will be additional matches, though the maximum possible imbalance between the treated and control groups would also go up.

This is easy with CEM because CEM is a monotonic imbalance bounding (MIB) method, which means that increasing the imbalance on one variable (by widening the coarsened bin sizes) will not change the maximum imbalance on any other variable. MIB thus enables you to tinker with the solution one variable at a time to quickly produce a satisfactory result, if one is feasible.

cem offers an automated way to relax the initial coarsening with a function relax.cem(). This function starts with the output of cem() and relaxes variables one (depth = 1), two (depth = 2), or three (depth = 3) at a time, while optionally keeping unchanged a chosen subset of the variables which we know well or have important effects on the outcome (fixed). The function also allows to specify the minimal number of breaks for each variable (minimal, set to 1 by default).

Review the documentation on relax.cem() and perform coarsened matching with depth = 1. Examine the resulting tables and a summary graph. Which variables are the hardest to match on?

##
Click for Exercise 2* Solution 
## Let's use bal.tab
prog_cem  <- relax.cem(m_cem_out, nsw_PSID_cem, depth = 1)
## Warning in pretty.default(range(data[[i]], na.rm = TRUE), n =
## nclass.scott(na.omit(data[[i]])), : NAs introduced by coercion
## Executing 33 different relaxations
## 
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prog_cem
## $G0
##    G0  G1   PercG0   PercG1      Relaxed        L1       var
## 1  71 111 2.851406 60.00000      <start> 0.7792158   <start>
## 2  78 117 3.132530 63.24324      age(11) 0.7735043       age
## 3  80 115 3.212851 62.16216  hispanic(1) 0.7967391  hispanic
## 4  83 129 3.333333 69.72973       age(8) 0.7907911       age
## 5  84 118 3.373494 63.78378      re74(6) 0.7962066      re74
## 6  84 118 3.373494 63.78378      re74(7) 0.7962066      re74
## 7  84 118 3.373494 63.78378      re74(8) 0.7962066      re74
## 8  84 118 3.373494 63.78378      re74(9) 0.7962066      re74
## 9  86 117 3.453815 63.24324       age(9) 0.7866229       age
## 10 89 125 3.574297 67.56757  nodegree(1) 0.8044045  nodegree
## 11 91 130 3.654618 70.27027      re75(7) 0.7681319      re75
## 12 91 130 3.654618 70.27027      re75(8) 0.7681319      re75
## 13 93 118 3.734940 63.78378 education(2) 0.8077274 education
## 14 93 120 3.734940 64.86486      re74(4) 0.8008065      re74
##  [ reached 'max' / getOption("max.print") -- omitted 20 rows ]
## 
## $G1
##    G0  G1   PercG0   PercG1      Relaxed        L1       var
## 1  71 111 2.851406 60.00000      <start> 0.7792158   <start>
## 3  80 115 3.212851 62.16216  hispanic(1) 0.7967391  hispanic
## 2  78 117 3.132530 63.24324      age(11) 0.7735043       age
## 9  86 117 3.453815 63.24324       age(9) 0.7866229       age
## 5  84 118 3.373494 63.78378      re74(6) 0.7962066      re74
## 6  84 118 3.373494 63.78378      re74(7) 0.7962066      re74
## 7  84 118 3.373494 63.78378      re74(8) 0.7962066      re74
## 8  84 118 3.373494 63.78378      re74(9) 0.7962066      re74
## 13 93 118 3.734940 63.78378 education(2) 0.8077274 education
## 14 93 120 3.734940 64.86486      re74(4) 0.8008065      re74
## 15 93 120 3.734940 64.86486      re74(5) 0.8008065      re74
## 18 97 121 3.895582 65.40541      re74(1) 0.8059981      re74
## 19 97 121 3.895582 65.40541      re74(2) 0.8059981      re74
## 20 97 121 3.895582 65.40541      re74(3) 0.8059981      re74
##  [ reached 'max' / getOption("max.print") -- omitted 20 rows ]
## 
## $L1breaks
## $L1breaks$data_id
## character(0)
## 
## $L1breaks$treat
##  [1] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
## [16] 0.75 0.80 0.85 0.90 0.95 1.00
## 
## $L1breaks$age
##  [1] 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56
## 
## $L1breaks$education
##  [1]  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## 
## $L1breaks$black
## NULL
## 
## $L1breaks$hispanic
## NULL
## 
## $L1breaks$married
## NULL
## 
## $L1breaks$nodegree
## NULL
## 
## $L1breaks$re74
##  [1]      0   5000  10000  15000  20000  25000  30000  35000  40000  45000
## [11]  50000  55000  60000  65000  70000  75000  80000  85000  90000  95000
## [21] 100000 105000 110000 115000 120000 125000 130000 135000 140000
## 
## $L1breaks$re75
##  [1]      0   5000  10000  15000  20000  25000  30000  35000  40000  45000
## [11]  50000  55000  60000  65000  70000  75000  80000  85000  90000  95000
## [21] 100000 105000 110000 115000 120000 125000 130000 135000 140000 145000
## [31] 150000 155000 160000
## 
## $L1breaks$re78
##  [1]      0   5000  10000  15000  20000  25000  30000  35000  40000  45000
## [11]  50000  55000  60000  65000  70000  75000  80000  85000  90000  95000
## [21] 100000 105000 110000 115000 120000 125000
## 
## 
## attr(,"class")
## [1] "relax.cem"

From the output we can see where the difficulties in matching this data are found and what choices would produce which outcomes in terms of the numbers of observations matched. Here, as expected, re74, married, and age variables were some of the hardest to match on. Note that, for categorical variables, the algorithm considers the numerical values associated to each level of the variable.

5.5 Conclusion

We’ve only scratches the surface on available R packages, methods that they offer, and options available for these methods. We found that R documentation for many of them is sparse. However, it helps to review the accompanying publications as well as search around for examples developed by other users.

5.6 Wrap-up

5.6.1 Feedback

This workshop is a work in progress. Please provide any feedback to: .

5.6.2 Resources