Turn up the Heat! A Tutorial for the MAGMA R-package.
Summary
Balancing quasi-experimental field research for effects of covariates is fundamental for drawing causal inference. Propensity Score Matching addresses this issue but current techniques are restricted to binary treatment variables. Moreover, they offer a variety of solutions without providing a comprehensive framework on choosing the best model. The MAGMA R-package addresses these restrictions by providing nearest neighbor matching for up to four groups using the Many-group-Matching (MAGMA) algorithm. It also includes the option to match data of a 2x2 design. In addition, MAGMA provides a framework for evaluating the post-matching balance. This vignette is a tutorial on MAGMA. It explains and illustrates the main MAGMA functions with an included simulated dataset. In total, this vignette includes a two-group, a three-group, and a 2x2-design, which is equivalent to a four-group example. These three grouping scenarios are illustrated in a standard matching process, as well as with exact matching, where only cases sharing the same value on a variable can be matched.
Introduction
Lack of experimental control is an issue in field research, observational, and non-randomized studies. A popular method to control for the effects of covariates post-hoc is propensity score matching (PSM; Rosenbaum & Rubin, 1983, 1985). By balancing out the effects of covariates, PSM approxiamtes causal inference. Since causal inference is fundamental in science, PSM is used in many scientific fields such as in organizational (e.g., Li, 2013), educational (e.g., Powell et al., 2020), and social science (e.g., Thoemmes & Kim, 2011) as well as in medical research (e.g., Austin, 2014).
PSM builds statistical pairs of individuals, for example from a treatment group and a control group, based on the distance on a specific variable. Propensity scores are often used in representation of said distance (for a review, see Stuart, 2010). They express the conditional probability of belonging to a group (e.g., the treatment group) given a set of covariates (Rosenbaum & Rubin, 1985). Although a plethora of matching techniques exist, a commonly applied method is nearest neighbor matching (Austin, 2014; Jacovidis, 2017). By matching a treated individual to its non-treated nearest neighbor, PSM controls for the effects of covariates.
PSM’s potential is nonetheless limited due to three reasons. First, current techniques are restricted to binary treatment variables, and hence, to two-group designs (for first approaches, see Imai & van Dyk, 2004; McCaffrey et al., 2013). Second, researchers must successively extract and compare several arbitrary matching solutions by hand, instead of having access to a comprehensive, unambiguous overview of possible matching solutions. Third, there is no comprehensive framework for evaluating the quality of matching solutions, especially when multiple groups are involved. We addressed these issues with our MAGMA package. For an extensive overview of MAGMA and its balance framework, see Feuchter at al. (2022).
The aims of this vignette are to provide a tutorial for the MAGMA R-package. This includes examples for all currently available group scenarios, namely two, three, and four groups, as well as 2x2 designs. We conducted these examples for standard and exact matching. Moreover, this tutorial contains examples on how to report matching relevant statistics. Lastly, we demonstrate how to compute and report balance of individual PSM solutions, using our newly developed framework.
The MAGMA Simulated Dataset
We used a simulated data set for all examples. This simulated dataset
contains 800 cases and 14 variables. It is available after installing
the MAGMA package. Associations between the variables were modeled to
increase the usability and comprehensibility of this tutorial, and are,
therefore, not representative of specific real-life phenomena. For
information about this simulated dataset, use
?MAGMA_sim_data. Below is a brief look at the data and its
structure. Note that the last three variables are propensity scores that
were estimated using the twang package (Ridgeway et al.,
2015). For a detailed overview and tutorial on twang, see
(Ridgeway et al., 2015). Propensity scores estimated via twang serve as
distance indicators for all subsequent examples.
For the two group example, we use the propensity score
ps_gifted. This score expresses the conditional probability
of receiving gifted_support. It was computed using the
ps function of twang. The propensity scores
ps_tar (three group example) and ps_2x2 (four
group / 2x2 design example) are the result of the mnps
function of twang. They express the conditional probability
for the groups teacher_rated_ability (below average,
average, above average) and the combination of receiving
gifted_support and participating in afternoon
enrichment respectively.
str(MAGMA_sim_data)
#> 'data.frame': 800 obs. of 17 variables:
#> $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ gender : int 1 0 1 1 1 0 0 0 1 0 ...
#> $ gifted_support : int 1 1 1 1 1 1 0 1 0 1 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 2 2 1 1 ...
#> $ enrichment : int 0 1 0 1 0 0 0 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 1 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 1 3 1 4 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ step_gifted : num 56 140 249 117 52 231 266 304 NA 29 ...
#> $ weight_gifted : num 1 1 1 1 1 1 1 1 NA 1 ...
#> $ distance_gifted : num 4.55e-08 3.88e-06 5.06e-01 1.43e-06 3.27e-08 ...Two-Group Example
This is a fictitious two-group example. The independent variable
(i.e., the treatment) is gifted_support. We are interested
in how receiving giftedness support affects college GPA. However, other
variables, such as intelligence, previous achievement, and motivation
are associated with both giftedness support and college GPA. This is
because gifted students were chosen to participate in this support
program based on their previous achievement or intelligence test
results. Therefore, we aim to reduce differences on these covariates
between the two groups by PSM. For this example, the covariates are
high school GPA, intelligence, motivation,
gender, and whether at least one parent has an academic
background. Note that MAGMA can only consider metric and binary
variables for balance estimation.
Defining all covariates beforehand (see above)nfacilitates
estimations of descriptive statistics, standardized mean differences,
and hence, the initial unbalance across all cases of our dataset. To do
this we can use the functions MAGMA_desc and
initial_unbalance. The first function –
MAGMA_desc – computes sample size, mean, and standard
deviation of all covariates for the overall sample and the two groups.
Moreover, it estimates the standardized mean differences using the
pooled standard deviation (i.e., Cohen’s d) between the two
groups. This function needs the dataset (MAGMA_sim_data),
the grouping variable (gifted_support), and all binary and
metric covariates of interest (covariates_gifted). The
function results in a table that contains these statistics for each
covariate.
The second function – initial_unbalance - estimates the
four balance criteria of our balance framework. These criteria are (1)
Pillai’s Trace, which estimates group differences across all
variables, (2) d-ratio, which expresses the percentage of how
many pairwise effects are negligible, (3) mean g, which
specifies the average pariwise effect using meta-analytical techniques,
and (4) adjusted d-ratio, an adjusted version of (2) that
respects that effects are point estimates. For more information, see
Feuchter et al. (2022). The function initial_unbalance
indicates the unbalance in the sample before matching. To this end, it
is necessary to define the dataset (MAGMA_sim_data), the
grouping variable (gifted_support), and all binary and
metric covariates of interest (covariates_gifted). This
results in a vector of length four - the four balance criteria.
# Estimate overall and group specific descriptive statistics and Cohen’s d
descs_gifted_pre <- MAGMA_desc(Data = MAGMA_sim_data,
group = "gifted_support",
covariates = covariates_gifted,
filename = "stats_gifted_pre.docx")
descs_gifted_pre %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"No Support N", "No Support Mean", "No Support SD",
"Support N", "Support Mean", "Support SD",
"d"))
#> Overall N Overall Mean Overall SD No Support N No Support Mean
#> gifted_support 800 0.38 0.49 495 0.00
#> GPA_school 800 3.29 0.93 495 3.13
#> IQ_score 800 100.51 15.00 495 96.32
#> Motivation 800 3.82 0.89 495 3.81
#> parents_academic 800 0.40 0.49 495 0.35
#> gender 800 0.50 0.50 495 0.50
#> No Support SD Support N Support Mean Support SD d
#> gifted_support 0.00 305 1.00 0.00 -Inf
#> GPA_school 0.91 305 3.56 0.89 -0.48
#> IQ_score 14.20 305 107.30 13.75 -0.78
#> Motivation 0.90 305 3.83 0.88 -0.02
#> parents_academic 0.48 305 0.48 0.50 -0.27
#> gender 0.50 305 0.49 0.50 0.02
# Estimating the four balance criteria
unbalance_gifted <- initial_unbalance(Data = MAGMA_sim_data,
group = "gifted_support",
covariates = covariates_gifted)
#> Mean g was computed using random effects meta-analysis with metafor.
unbalance_gifted
#> Pillai's Trace d-ratio Mean g adj. d-ratio
#> Unbalance 0.13 0.4 0.31 0.44As may be expected from the descriptive statistics, the cases that
received giftedness support differed substantially from the cases that
did not receive giftedness support on at least some covariates. Before
we start matching, we check the possibility of eligible matches.
Therefore, we estimate and plot the common support in propensity score
via the kernel-density (Pastore et al., 2022) and the function
density_overlap. This function requires the dataset
(MAGMA_sim_data), the variable for which the overlap should
be computed (ps_gifted), and the grouping variable
(gifted_support). The remaining arguments of the functions
define aesthetics of the plot, namely the name of the variable
(variable_name), the name of the group
(group_name), and the labels of the respective groups
(group_labels). The function returns a density plot and a
value expressing the overlapping area.
# Density overlap in propensity scores for gifted before matching
Density_overlap(Data = MAGMA_sim_data,
variable = "ps_gifted",
group = "gifted_support",
variable_name = "Propensity Score",
group_labels = c("No Support", "Support"),
group_name = "Gifted Support")
#> OV
#> 0.6162483As we can see, there is substantial common support between the two
groups. After these initial checks, we can conduct the matching. This
process is based on the respective propensity score,
ps_gifted. We first address standard matching before
introducing a case of exact matching.
Standard Matching
For standard nearest neighbor matching we use the MAGMA
function. This function has four arguments. The first argument specifies
the dataset (MAGMA_sim_data). The second argument specifies
the name of the grouping variable as a character
("gifted_support"). The third argument specifies the name
of the distance variable (in our case the propensity score) as a
character ("ps_gifted"). The fourth argument is optional.
Since MAGMA has some computational load, it includes the
option of parallel computation. By default, only one CPU core is used.
The usage of more cores may reduce the computation time of matching.
While this reduction is rarely necessary in two-group matching, it
becomes more helpful for many group matching. If parallel computation is
desired, an integer larger than one can be specified for the
cores argument. If the specified value exceeds the number
of available cores, MAGMA will set this argument to the
maximum number of available cores.
Applying this function results in a dataset that extends the original
dataset by three variables. The first, weight, indicates
whether this case was matched or not. The second, step,
indicates the iterative step in which a case was matched. More
specifically, the cases with the smallest Mahalanobis distance are
matched first and, thus, receive 1 in the step variable.
The matched cases with the second smallest distance receive the
2 and so on. The step variable is essential to
compare and extract different matching solutions. The third variable,
distance indicates the propensity score distance of the
matched cases.
# Conducting matching for gifted support
MAGMA_sim_data_gifted <- MAGMA(Data = MAGMA_sim_data,
group = "gifted_support",
dist = "ps_gifted",
cores = 2)
#>
#> Input correctly identified.
#> Distance computation finished. Starting matching.
#> Matching complete!
str(MAGMA_sim_data_gifted)
#> 'data.frame': 800 obs. of 20 variables:
#> $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ gender : int 1 0 1 1 1 0 0 0 1 0 ...
#> $ gifted_support : int 1 1 1 1 1 1 0 1 0 1 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 2 2 1 1 ...
#> $ enrichment : int 0 1 0 1 0 0 0 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 1 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 1 3 1 4 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ step_gifted : num 56 140 249 117 52 231 266 304 NA 29 ...
#> $ weight_gifted : num 1 1 1 1 1 1 1 1 NA 1 ...
#> $ distance_gifted : num 4.55e-08 3.88e-06 5.06e-01 1.43e-06 3.27e-08 ...
#> $ step : num 56 140 249 117 52 231 266 304 NA 29 ...
#> $ weight : num 1 1 1 1 1 1 1 1 NA 1 ...
#> $ distance : num 4.55e-08 3.88e-06 5.06e-01 1.43e-06 3.27e-08 ...Balance Estimation
MAGMA matches cases iteratively until all cases of the smallest group
are matched. Thus, it is necessary to find the optimal sample size after
matching. To do so, Balance_MAGMA estimates the four
balance criteria for each possible sample size. Note that MAGMA uses a
lower limit of at least n = 20 per group. To use the function,
you need to specify at least your (matched) dataset
(MAGMA_sim_data_gifted), the name of the grouping variable
as a character ("gifted_support"), and your covariates as a
character vector (covariates_gifted). Moreover,
Balance_MAGMA needs the name of the variable that indicates
the iterative step, in which a case was matched. Since the
MAGMA function names this variable "step" this
name is the default of the step argument.
Balance_MAGMA returns a list of length four. In the case
of a univariate grouping variable, this list contains three numeric
vectors and one list of length two. The three vectors comprise Pillai’s
Trace, mean g, and adjusted d-ratio over the
iteratively increased sample size. The list of length two includes one
numeric vector, d-ratio, and one matrix. The matrix gives all
pairwise d’s for all covariates. Both d-ratio and
pairwise d’s are also estimated iteratively over the increasing
sample sizes. The example below estimates the balance for the matching
solution and shows the balance criteria and pairwise effects for a group
sample size of n = 100.
This function is augmented by the function
Balance_extract. This function extracts the balance
criteria (effects = FALSE) or the pairwise effects
(effects = TRUE) for a specific sample size per group
(samplesize) from the results of
Balance_MAGMA. It returns a vector containing the balance
criteria or the pairwise effects respectively.
# Estimating the four balance criteria iteratively over possible sample sizes
Balance_gifted <- Balance_MAGMA(Data = MAGMA_sim_data_gifted,
group = "gifted_support",
covariates = covariates_gifted,
step = "step")
#>
#> Start estimating Pillai's Trace.
#> Pillai's Trace finished. Starting to compute d-ratio.
#> d-ratio finished. Starting to compute mean-g.
#>
#> Mean g was computed using random effects meta-analysis with metafor.
#> Mean g finished. Starting to compute adjusted d-ratio.
#> Adjusted d-ratio finished.
#> Balance estimation finished.
# Extracting balance criteria for 100 cases per group
Balance_100_criteria <- Balance_extract(Balance = Balance_gifted,
samplesize = 100,
effects = FALSE)
Balance_100_criteria
#> Pillai's Trace d-ratio mean g adj. d-ratio
#> 0.02 1.00 0.10 0.74
# Extracting pairwise effects for 100 cases per group
Balance_100_effects <- Balance_extract(Balance = Balance_gifted,
samplesize = 100,
effects = TRUE)
Balance_100_effects
#> GPA_school IQ_score Motivation parents_academic
#> 0.07 0.20 -0.01 -0.12
#> gender
#> 0.10Balance Evaluation
After balance estimation, the “optimal” sample size to extract remains veiled. Which sample size is “optimal” strongly depends on the specific research aim. For example, the sample size requirements of some statistical procedures may narrow the selection, as may the need to eliminate a specific pairwise effect. We therefore encourage to adapt the selection of optimal sample size to the specific research aim.
A first indicator of such an optimal sample size is the trend of the
balance criteria across the iteratively increased sample size. The
function Plot_MAGMA visualizes this trend. The one
mandatory argument, Balance, specifies the object of the
balance estimation results. Note that this object must be the result of
a Balance_MAGMA function. Per default,
Plot_MAGMA creates plots for all four balance criteria.
However, the criterion argument enables the plotting of
specific balance criteria. Note that running this code will result in a
warning. This warning is a result of the specified minimum sample size
for balance estimation and can be ignored.
The trend of these plots depends strongly on the data. Nonetheless,
they can be a good indicator of breakpoints in the sample size-balance
associations. Plot_MAGMA uses fixed limits for the y-axis.
This ensures comparability of these plots and prevents misinterpretation
due to a restricted y-axis range. Both d-ratio and adjusted
d-ratio vary between 0 and 1. Although Pillai’s Trace has
theoretically the same range, we set the limits to 0 and 0.5. Larger
values are unlikely and indicate a poor balance. A stricter limit might
restrict a comprehensive trend evaluation. For mean g we set
the limits to 0 and 1. The lower limit 0 corresponds to the theoretical
lower limit. The rationale for the upper limit is comparable to that of
Pillai’s Trace.
# Plotting balance trend over sample size
Plot_MAGMA(Balance = Balance_gifted,
criterion = c("Pillai", "d_ratio", "mean_g", "Adj_d_ratio"))
Besides the trend over sample size, the “optimized” values for each
criterion are of interest. We interpret a criterions as optimized as it
takes its lowest (Pillai’s Trace and mean g) or highest
(d-ratio and adjusted d-ratio) value and having the
highest sample size with this value. The function
Table_MAGMA extracts these values. With the optional
argument filename the function can return these values in a
table in an extra file. More specifically it returns a 4x5 table. In
each of the four rows one balance criterion has its “optimal” value. The
five columns present the sample size per group for the matching solution
where the respective criterion is optimal, and the four balance criteria
values for this sample size per group. In addition to creating a file
with a table, the table is printed to the console, too.
Similar to Plot_MAGMA the first argument,
Balance, of Table_MAGMA specifies the object
of the balance estimation results. The second argument,
filename, specifies the desired name of the extra file
containing the table. This argument is a character.
Table_MAGMA(Balance = Balance_gifted,
filename = "Balance_gifted.docx")
#> Balance table successfully created!
#> # A tibble: 4 × 6
#> Criterion_optimized Pillai_Trace d_ratio mean_g adjusted_d_ratio n_per_group
#> <chr> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 Best Pillai 0 1 0.03 0.97 276
#> 2 Best mean g 0 1 0.03 0.97 276
#> 3 Best adj. d-ratio 0 1 0.03 0.97 276
#> 4 Best d-ratio 0.01 1 0.06 0.88 296Post-Matching Statistics
After the decision for a final matching model, we suggest reporting
the descriptive statistics and pairwise effects of the selected model.
To do this, we can use the same function as we used for reporting
pre-matching statistics (MAGMA_desc). Additional to the
parameters we specified above (Data,
covariates, and group), we need to name the
step variable (step_var) and the desired sample size per
group (step_num). As above, this function results in a
table with descriptive statistics and pairwise effects (Cohen’s d) of
all covariates.
# Computing descriptive statistics and pairwise effects for 100 cases per group
descs_gifted_post <- MAGMA_desc(Data = MAGMA_sim_data_gifted,
group = "gifted_support",
covariates = covariates_gifted,
step_num = 100,
step_var = "step",
filename = "stats_gifted_post.docx")
# Displaying the table with defined colum names
descs_gifted_post %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"No Support N", "No Support Mean", "No Support SD",
"Support N", "Support Mean", "Support SD",
"d"))
#> Overall N Overall Mean Overall SD No Support N No Support Mean
#> gifted_support 200 0.50 0.50 100 0.00
#> GPA_school 200 3.38 0.76 100 3.41
#> IQ_score 200 102.82 12.50 100 104.05
#> Motivation 200 3.87 0.84 100 3.87
#> parents_academic 200 0.42 0.49 100 0.39
#> gender 200 0.47 0.50 100 0.49
#> No Support SD Support N Support Mean Support SD d
#> gifted_support 0.00 100 1.00 0.00 -Inf
#> GPA_school 0.76 100 3.36 0.77 0.07
#> IQ_score 12.13 100 101.59 12.80 0.20
#> Motivation 0.84 100 3.88 0.84 -0.01
#> parents_academic 0.49 100 0.45 0.50 -0.12
#> gender 0.50 100 0.44 0.50 0.10Exact Matching
Unlike in standard matching, only cases sharing the same value on a
variable can be matched in exact matching. In this example, the variable
for exact matching is enrichment. This means that cases
that participated in afternoon enrichment can only be matched with other
cases that participated. Consequentially, non-participants can only be
matched with other non-participants. Reasons for choosing exact matching
instead of standard matching may be, for instance, a nested data
structure. For example, students may come from different schools and
only students from the same school should be matched together. Except
for this preselection of possible matches, the process of matching stays
the same as in the previous example. The main difference is that we need
to use the function MAGMA_exact instead of
MAGMA to match the data. This function has an additional
argument, where the name of the exact variable must be defined as a
character ("enrichment"). Note that the initial unbalance,
descriptive statistics, and density overlap are the same as for standard
matching.
MAGMA_sim_data_gifted_exact <- MAGMA_exact(Data = MAGMA_sim_data,
group = "gifted_support",
dist = "ps_gifted",
exact = "enrichment",
cores = 2)
#>
#> Input correctly identified!
#> Distance computation finished. Starting matching
#> Matching complete!
str(MAGMA_sim_data_gifted_exact)
#> 'data.frame': 800 obs. of 20 variables:
#> $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ gender : int 1 0 1 1 1 0 0 0 1 0 ...
#> $ gifted_support : int 1 1 1 1 1 1 0 1 0 1 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 2 2 1 1 ...
#> $ enrichment : int 0 1 0 1 0 0 0 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 1 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 1 3 1 4 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ step_gifted : num 56 140 249 117 52 231 266 304 NA 29 ...
#> $ weight_gifted : num 1 1 1 1 1 1 1 1 NA 1 ...
#> $ distance_gifted : num 4.55e-08 3.88e-06 5.06e-01 1.43e-06 3.27e-08 ...
#> $ step : num 156 136 125 167 32 230 265 303 NA 18 ...
#> $ weight : num 1 1 1 1 1 1 1 1 NA 1 ...
#> $ distance : num 3.92e-05 1.91e-05 1.10e-05 9.41e-05 3.27e-08 ...Balance Estimation and Visualization
The steps of balance estimation, visualization, and post matching
statistics are the same as for standard matching. Therefore, we only
shortly summarize these steps in this example. The only change we need
to consider is the different data frame that contains the result of the
exact matching solution MAGMA_sim_data_gifted_exact.
Balance_gifted_exact <- Balance_MAGMA(Data = MAGMA_sim_data_gifted_exact,
group = "gifted_support",
covariates = covariates_gifted,
step = "step")
#>
#> Start estimating Pillai's Trace.
#> Pillai's Trace finished. Starting to compute d-ratio.
#> d-ratio finished. Starting to compute mean-g.
#>
#> Mean g was computed using random effects meta-analysis with metafor.
#> Mean g finished. Starting to compute adjusted d-ratio.
#> Adjusted d-ratio finished.
#> Balance estimation finished.
# Extracting balance criteria for 100 cases per group
Balance_100_criteria_exact <- Balance_extract(Balance = Balance_gifted_exact,
samplesize = 100,
effects = FALSE)
Balance_100_criteria_exact
#> Pillai's Trace d-ratio mean g adj. d-ratio
#> 0.02 1.00 0.12 0.70
# Extracting pairwise effects for 100 cases per group
Balance_100_effects_exact <- Balance_extract(Balance = Balance_gifted_exact,
samplesize = 100,
effects = TRUE)
Balance_100_effects_exact
#> GPA_school IQ_score Motivation parents_academic
#> 0.16 0.17 0.05 -0.16
#> gender
#> 0.06
# Plotting trend over increasing sample size
Plot_MAGMA(Balance = Balance_gifted_exact,
criterion = c("Pillai", "d_ratio", "mean_g", "Adj_d_ratio")) 
# Creating table
Table_MAGMA(Balance = Balance_gifted_exact,
filename = "Balance_gifted_exact.docx")
#> Balance table successfully created!
#> # A tibble: 4 × 6
#> Criterion_optimized Pillai_Trace d_ratio mean_g adjusted_d_ratio n_per_group
#> <chr> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 Best mean g 0 1 0.03 0.97 274
#> 2 Best Pillai 0 1 0.03 0.97 279
#> 3 Best adj. d-ratio 0 1 0.03 0.97 279
#> 4 Best d-ratio 0.01 1 0.07 0.88 296
# Computing descriptive statistics and pairwise effects for 100 cases per group
descs_gifted_post_exact <- MAGMA_desc(Data = MAGMA_sim_data_gifted_exact,
group = "gifted_support",
covariates = covariates_gifted,
step_num = 100,
step_var = "step",
filename = "stats_gifted_post_exact.docx")
# Displaying the table with defined colum names
descs_gifted_post_exact %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"No Support N", "No Support Mean", "No Support SD",
"Support N", "Support Mean", "Support SD",
"d"))
#> Overall N Overall Mean Overall SD No Support N No Support Mean
#> gifted_support 200 0.50 0.50 100 0.00
#> GPA_school 200 3.39 0.79 100 3.45
#> IQ_score 200 101.70 13.25 100 102.79
#> Motivation 200 3.88 0.88 100 3.90
#> parents_academic 200 0.38 0.49 100 0.34
#> gender 200 0.47 0.50 100 0.48
#> No Support SD Support N Support Mean Support SD d
#> gifted_support 0.00 100 1.00 0.00 -Inf
#> GPA_school 0.79 100 3.32 0.80 0.16
#> IQ_score 13.24 100 100.61 13.23 0.16
#> Motivation 0.92 100 3.86 0.85 0.05
#> parents_academic 0.48 100 0.42 0.50 -0.16
#> gender 0.50 100 0.45 0.50 0.06Three-Group Example
This is a three-group example. The independent variable (i.e., the
treatment) is teacher_ability_rating. We are interested in
how teacher-rated student ability, which ranged from below average (BA)
to average (A) and above average (AA), affects the college GPA. While
the process of matching and balance evaluation is the same as in the
two-group example, the grouping variable
(teacher_ability_rating), and consequently the distance
variable/propensity score changes (ps_tar). Moreover, in
this example we change the covariates by substituting
gender with gifted_support.
Due to these changes in grouping variable and covariates, we need to
re-estimate the initial unbalance, descriptive statistics, and area of
common support for teacher-rated ability. Note that due to having three
groups, the estimation of mean g slightly changes. Instead of
estimating the mean effect using metafor (Viechtbauer,
2010) we use robumeta (Fischer et al., 2023). However, this
only affects the backend computation, while the code and the display of
results are not affected. For more details, see Feuchter et
al. (2022).
# Computing descriptive statistics and all pairwise effects for three groups
descs_tar_pre <- MAGMA_desc(Data = MAGMA_sim_data,
group = "teacher_ability_rating",
covariates = covariates_tar,
filename = "stats_tar_pre.docx")
descs_tar_pre %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"BA N", "BA Support Mean", "BA Support SD",
"A N", "A Mean", "A SD",
"AA N", "AA Mean", "AA SD",
"d BA-A", "d BA-AA", "d A-AA"))
#> Overall N Overall Mean Overall SD BA N BA Support Mean
#> teacher_ability_rating 800 2.02 0.80 243 1.00
#> GPA_school 800 3.29 0.93 243 2.76
#> IQ_score 800 100.51 15.00 243 92.93
#> Motivation 800 3.82 0.89 243 3.77
#> parents_academic 800 0.40 0.49 243 0.39
#> gifted_support 800 0.38 0.49 243 0.35
#> BA Support SD A N A Mean A SD AA N AA Mean AA SD d BA-A
#> teacher_ability_rating 0.00 294 2.00 0.00 263 3.00 0.00 -Inf
#> GPA_school 0.82 294 3.33 0.85 263 3.73 0.86 -0.68
#> IQ_score 14.63 294 101.20 12.84 263 106.73 14.54 -0.60
#> Motivation 0.88 294 3.72 0.88 263 3.98 0.90 0.06
#> parents_academic 0.49 294 0.38 0.49 263 0.42 0.49 0.02
#> gifted_support 0.48 294 0.40 0.49 263 0.39 0.49 -0.10
#> d BA-AA d A-AA
#> teacher_ability_rating -Inf -Inf
#> GPA_school -1.15 -0.47
#> IQ_score -0.95 -0.40
#> Motivation -0.24 -0.29
#> parents_academic -0.06 -0.08
#> gifted_support -0.08 0.02
# Estimating and printing initial unbalance for teacher rated ability
unbalance_tar <- initial_unbalance(Data = MAGMA_sim_data,
group = "teacher_ability_rating",
covariates = covariates_tar)
#> Mean g was computed using robust variance meta-analysis with robumeta.
unbalance_tar
#> Pillai's Trace d-ratio Mean g adj. d-ratio
#> Unbalance 0.24 0.47 0.35 0.47
# Estimating and plotting density overlap in teacher rated ability propensity scores
# Returns vector for each pairwise overlap
Density_overlap(Data = MAGMA_sim_data,
variable = "ps_tar",
group = "teacher_ability_rating",
variable_name = "Propensity Score",
group_labels = c("Below Average", "Average", "Above Average"),
group_name = "Teacher Rated Ability")
#> OV OVPairs.Below Average-Average
#> 0.7387731 0.5164718
#> OVPairs.Below Average-Above Average OVPairs.Average-Above Average
#> 0.3483660 0.7658709Standard Matching
After this check of initial unbalance, descriptive statistics, and common support, we can conduct the standard matching. As mentioned above, we need to adapt the grouping variable and the distance variable/propensity score for this example.
MAGMA_sim_data_tar <- MAGMA(Data = MAGMA_sim_data,
group = "teacher_ability_rating",
dist = "ps_tar",
cores = 2)str(MAGMA_sim_data_tar)
#> 'data.frame': 800 obs. of 17 variables:
#> $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ gender : int 1 0 1 1 1 0 0 0 1 0 ...
#> $ gifted_support : int 1 1 1 1 1 1 0 1 0 1 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 2 2 1 1 ...
#> $ enrichment : int 0 1 0 1 0 0 0 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 1 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 1 3 1 4 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ step : num 222 237 28 106 66 62 182 NA 236 38 ...
#> $ weight : num 1 1 1 1 1 1 1 NA 1 1 ...
#> $ distance : num 2.53 3.35 2.96e-06 1.88e-01 5.22e-05 ...Balance Estimation and Visualization
We only need to change the specific object names, covariates, grouping variable, and the filename for the table. Otherwise, the process of balance estimation and visualization remains the same.
Balance_tar <- Balance_MAGMA(Data = MAGMA_sim_data_tar,
group = "teacher_ability_rating",
covariates = covariates_tar,
step = "step")# Balance criteria for 100 cases per group
Balance_100_tar_criteria <- Balance_extract(Balance = Balance_tar,
samplesize = 100,
effects = FALSE)
Balance_100_tar_criteria
#> Pillai's Trace d-ratio mean g adj. d-ratio
#> 0.03 0.67 0.14 0.64
# Extracting pairwise effects for 100 cases per group
Balance_100_tar_effects <- Balance_extract(Balance = Balance_tar,
samplesize = 100,
effects = TRUE)
Balance_100_tar_effects
#> GPA_school_1_2 IQ_score_1_2 Motivation_1_2
#> 0.32 0.22 0.20
#> parents_academic_1_2 gifted_support_1_2 GPA_school_1_3
#> 0.04 0.00 0.37
#> IQ_score_1_3 Motivation_1_3 parents_academic_1_3
#> 0.23 0.07 0.14
#> gifted_support_1_3 GPA_school_2_3 IQ_score_2_3
#> 0.04 0.03 0.05
#> Motivation_2_3 parents_academic_2_3 gifted_support_2_3
#> 0.26 0.10 0.04
# Plotting trend over increasing sample size
Plot_MAGMA(Balance = Balance_tar,
criterion = c("Pillai", "d_ratio", "mean_g", "Adj_d_ratio")) 
# Creating table
Table_MAGMA(Balance = Balance_tar,
filename = "Balance_tar.docx")
#> Balance table successfully created!
#> # A tibble: 4 × 6
#> Criterion_optimized Pillai_Trace d_ratio mean_g adjusted_d_ratio n_per_group
#> <chr> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 Best mean g 0.01 0.8 0.1 0.75 116
#> 2 Best adj. d-ratio 0.01 0.87 0.1 0.75 117
#> 3 Best Pillai 0.01 0.93 0.11 0.74 121
#> 4 Best d-ratio 0.01 0.93 0.12 0.72 123Post-matching statistics
After selecting a final solution we can report the post-matching descriptive statistics. To this end, we need to adapt our arguments to this three-group example.
# Computing descriptive statistics and pairwise effects for 100 cases per group
descs_tar_post <- MAGMA_desc(Data = MAGMA_sim_data_tar,
group = "teacher_ability_rating",
covariates = covariates_tar,
step_num = 100,
step_var = "step",
filename = "stats_tar_post.docx")
# Displaying the table with defined colum names
descs_tar_post %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"BA N", "BA Support Mean", "BA Support SD",
"A N", "A Mean", "A SD",
"AA N", "AA Mean", "AA SD",
"d BA-A", "d BA-AA", "d A-AA"))
#> Overall N Overall Mean Overall SD BA N BA Support Mean
#> teacher_ability_rating 300 2.00 0.82 100 1.00
#> GPA_school 300 3.21 0.66 100 3.36
#> IQ_score 300 99.29 12.99 100 101.19
#> Motivation 300 3.86 0.87 100 3.89
#> parents_academic 300 0.39 0.49 100 0.42
#> gifted_support 300 0.41 0.49 100 0.42
#> BA Support SD A N A Mean A SD AA N AA Mean AA SD d BA-A
#> teacher_ability_rating 0.00 100 2.00 0.00 100 3.00 0.00 -Inf
#> GPA_school 0.68 100 3.14 0.70 100 3.12 0.59 0.32
#> IQ_score 11.71 100 98.67 11.74 100 98.02 15.13 0.21
#> Motivation 0.86 100 3.73 0.82 100 3.96 0.93 0.19
#> parents_academic 0.50 100 0.40 0.49 100 0.35 0.48 0.04
#> gifted_support 0.50 100 0.42 0.50 100 0.40 0.49 0.00
#> d BA-AA d A-AA
#> teacher_ability_rating -Inf -Inf
#> GPA_school 0.38 0.03
#> IQ_score 0.23 0.05
#> Motivation -0.08 -0.26
#> parents_academic 0.14 0.10
#> gifted_support 0.04 0.04Exact Matching
We use the same covariates as we used for the standard three-group
matching. The only change is the use of gender as exact
variable. Thus, girls will only be matched to girls, and boys will only
be matched to other boys across the three groups. Initial unbalance,
descriptive statistics, and common support are the same as for standard
matching. Below you find the entire matching, balance estimation, and
visualization process.
MAGMA_sim_data_tar_exact <- MAGMA_exact(Data = MAGMA_sim_data,
group = "teacher_ability_rating",
dist = "ps_tar",
exact = "gender",
cores = 2)str(MAGMA_sim_data_tar_exact)
#> 'data.frame': 800 obs. of 17 variables:
#> $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ gender : int 1 0 1 1 1 0 0 0 1 0 ...
#> $ gifted_support : int 1 1 1 1 1 1 0 1 0 1 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 2 2 1 1 ...
#> $ enrichment : int 0 1 0 1 0 0 0 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 1 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 1 3 1 4 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ step : num 226 NA 165 108 71 43 188 NA 241 54 ...
#> $ weight : num 1 NA 1 1 1 1 1 NA 1 1 ...
#> $ distance : num 2.787944 NA 1.040494 0.231533 0.000273 ...Balance_tar_exact <- Balance_MAGMA(Data = MAGMA_sim_data_tar_exact,
group = "teacher_ability_rating",
covariates = covariates_tar,
step = "step") # Balance criteria for 100 cases per group
Balance_100_tar_criteria_exact <- Balance_extract(Balance = Balance_tar_exact,
samplesize = 100,
effects = FALSE)
Balance_100_tar_criteria_exact
#> Pillai's Trace d-ratio mean g adj. d-ratio
#> 0.03 0.67 0.16 0.61
# Extracting pairwise effects for 100 cases per group
Balance_100_tar_effects_exact <- Balance_extract(Balance = Balance_tar_exact,
samplesize = 100,
effects = TRUE)
Balance_100_tar_effects_exact
#> GPA_school_1_2 IQ_score_1_2 Motivation_1_2
#> 0.36 0.34 0.10
#> parents_academic_1_2 gifted_support_1_2 GPA_school_1_3
#> 0.12 0.12 0.37
#> IQ_score_1_3 Motivation_1_3 parents_academic_1_3
#> 0.24 0.14 0.12
#> gifted_support_1_3 GPA_school_2_3 IQ_score_2_3
#> 0.04 0.00 0.05
#> Motivation_2_3 parents_academic_2_3 gifted_support_2_3
#> 0.24 0.00 0.08
# Plotting trend over increasing sample size
Plot_MAGMA(Balance = Balance_tar_exact,
criterion = c("Pillai", "d_ratio", "mean_g", "Adj_d_ratio")) #Could be omitted
# Creating table
Table_MAGMA(Balance = Balance_tar_exact,
filename = "Balance_tar_exact.docx")
#> Balance table successfully created!
#> # A tibble: 4 × 6
#> Criterion_optimized Pillai_Trace d_ratio mean_g adjusted_d_ratio n_per_group
#> <chr> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 Best mean g 0.01 0.8 0.12 0.7 113
#> 2 Best adj. d-ratio 0.01 0.8 0.12 0.7 114
#> 3 Best Pillai 0.01 0.73 0.12 0.69 116
#> 4 Best d-ratio 0.01 0.8 0.14 0.65 127
# Computing descriptive statistics and pairwise effects for 100 cases per group
descs_tar_post_exact <- MAGMA_desc(Data = MAGMA_sim_data_tar_exact,
group = "teacher_ability_rating",
covariates = covariates_tar,
step_num = 100,
step_var = "step",
filename = "stats_tar_post_exact.docx")
# Displaying the table with defined column names
descs_tar_post_exact %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"BA N", "BA Support Mean", "BA Support SD",
"A N", "A Mean", "A SD",
"AA N", "AA Mean", "AA SD",
"d BA-A", "d BA-AA", "d A-AA"))
#> Overall N Overall Mean Overall SD BA N BA Support Mean
#> teacher_ability_rating 300 2.00 0.82 100 1.00
#> GPA_school 300 3.21 0.65 100 3.37
#> IQ_score 300 98.95 13.11 100 101.40
#> Motivation 300 3.83 0.91 100 3.82
#> parents_academic 300 0.38 0.49 100 0.42
#> gifted_support 300 0.39 0.49 100 0.42
#> BA Support SD A N A Mean A SD AA N AA Mean AA SD d BA-A
#> teacher_ability_rating 0.00 100 2.00 0.00 100 3.00 0.00 -Inf
#> GPA_school 0.68 100 3.13 0.64 100 3.13 0.60 0.36
#> IQ_score 11.93 100 97.36 11.53 100 98.07 15.30 0.34
#> Motivation 0.91 100 3.73 0.91 100 3.95 0.90 0.10
#> parents_academic 0.50 100 0.36 0.48 100 0.36 0.48 0.12
#> gifted_support 0.50 100 0.36 0.48 100 0.40 0.49 0.12
#> d BA-AA d A-AA
#> teacher_ability_rating -Inf -Inf
#> GPA_school 0.37 0.00
#> IQ_score 0.24 -0.05
#> Motivation -0.14 -0.24
#> parents_academic 0.12 0.00
#> gifted_support 0.04 -0.082x2/four-group Example
In this 2x2-design example the variables gifted_support
and enrichment are the independent (i.e., the treatment)
variables. We are interested in how receiving giftedness support and the
participation in afternoon enrichment affect the college GPA. The
process of matching and balance evaluation is the same as in the
previous examples. The grouping variable consists of two variables, so
we use a character vector as input for the grouping variable
(c("gifted_support", "enrichment")). Consequently, we also
adapt the distance variable/propensity score (ps_2x2).
Moreover, we use the same covariates as in the two-group example. Note
that this 2x2 matching is largely equivalent to a 1x4 matching. Thus,
using the multinominal variable support_enrichment as
grouping variable would lead to similar results. In this example, we
focus on the 2x2 case only.
Major distinctions between such a multinominal four-group matching
and a 2x2 matching occur only in the balance estimation. Because of the
changes in grouping variable and covariates, we need to estimate the
initial unbalance for the two independent variables
gifted_support and enrichment. The result of
the initial_unbalance function changes slightly in this
two-factorial design. Pillai’s Trace is estimated separately for the two
main effects and the interaction. Thus, d-ratio, mean
g, and adjusted d-ratio are the same for a 2x2 or a
four-group matching, while changes occur only for Pillai’s Trace. In a
first step, we define the covariates and estimate initial unbalance,
descriptive statistics, and area of common support.
# Defining the covariates
covariates_2x2 <- c("GPA_school",
"IQ_score",
"Motivation",
"parents_academic",
"gender")
# Computing descriptive statistics and all pairwise effects
descs_2x2_pre <- MAGMA_desc(Data = MAGMA_sim_data,
group = c("gifted_support", "enrichment"),
covariates = covariates_2x2,
filename = "stats_2x2_pre.docx")
#> 2x2 groups are represented as 4 groups.
descs_2x2_pre %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"Sup & No En N", "Sup & No En Mean",
"Sup & No En SD",
"Sup & En N", "Sup & En Mean", "Sup & En SD",
"No Sup & No En N", "No Sup & No En Mean", "No Sup & No En SD",
"No Sup & En N", "No Sup & En Mean", "No Sup & En SD",
"d YesNo-YesYes", "d YesNo-NoNo", "d YesNo-NoYes",
"d YesYes-NoNo", "d YesYes-YNoYes",
"d NoNo-NoYes"))
#> Overall N Overall Mean Overall SD Sup & No En N
#> group_long 800 2.63 1.07 175
#> GPA_school 800 3.29 0.93 175
#> IQ_score 800 100.51 15.00 175
#> Motivation 800 3.82 0.89 175
#> parents_academic 800 0.40 0.49 175
#> gender 800 0.50 0.50 175
#> Sup & No En Mean Sup & No En SD Sup & En N Sup & En Mean
#> group_long 1.00 0.00 130 2.00
#> GPA_school 3.44 0.88 130 3.71
#> IQ_score 105.09 14.09 130 110.26
#> Motivation 3.79 0.90 130 3.89
#> parents_academic 0.43 0.50 130 0.53
#> gender 0.49 0.50 130 0.48
#> Sup & En SD No Sup & No En N No Sup & No En Mean
#> group_long 0.00 309 3.00
#> GPA_school 0.88 309 3.01
#> IQ_score 12.74 309 94.27
#> Motivation 0.86 309 3.75
#> parents_academic 0.50 309 0.37
#> gender 0.50 309 0.50
#> No Sup & No En SD No Sup & En N No Sup & En Mean
#> group_long 0.00 186 4.00
#> GPA_school 0.93 186 3.31
#> IQ_score 13.30 186 99.74
#> Motivation 0.89 186 3.92
#> parents_academic 0.48 186 0.32
#> gender 0.50 186 0.51
#> No Sup & En SD d YesNo-YesYes d YesNo-NoNo d YesNo-NoYes
#> group_long 0.00 -Inf -Inf -Inf
#> GPA_school 0.86 -0.31 0.47 0.15
#> IQ_score 15.01 -0.38 0.80 0.37
#> Motivation 0.90 -0.11 0.04 -0.14
#> parents_academic 0.47 -0.20 0.12 0.23
#> gender 0.50 0.02 -0.02 -0.04
#> d YesYes-NoNo d YesYes-YNoYes d NoNo-NoYes
#> group_long -Inf -Inf -Inf
#> GPA_school 0.76 0.46 -0.33
#> IQ_score 1.22 0.74 -0.39
#> Motivation 0.16 -0.03 -0.19
#> parents_academic 0.33 0.44 0.10
#> gender -0.04 -0.06 -0.02
# Estimating initial unbalance
unbalance_2x2 <- initial_unbalance(Data = MAGMA_sim_data,
group = c("gifted_support", "enrichment"),
covariates = covariates_2x2)
#> Mean g was computed using robust variance meta-analysis with robumeta.
unbalance_2x2
#> Pillai's Trace gifted_support Pillai's Trace enrichment
#> Unbalance 0.13 0.05
#> Pillai's Trace IA d-ratio Mean g adj. d-ratio
#> Unbalance 0.01 0.53 0.29 0.46
# Estimating and plotting density overlap in gifted support & enrichment propensity score
Density_overlap(Data = MAGMA_sim_data,
variable = "ps_2x2",
group = c("gifted_support", "enrichment"),
variable_name = "Propensity Score",
group_labels = c("No Support & No Enrichment",
"No Support & Enrichment",
"Support & No Enrichment",
"Support & Enrichment"),
group_name = "Gifted Support & Enrichment")
#> 2x2 groups are represented as 4 groups.
#> OV
#> 0.8692445
#> OVPairs.No Support & No Enrichment-No Support & Enrichment
#> 0.8264201
#> OVPairs.No Support & No Enrichment-Support & No Enrichment
#> 0.6062272
#> OVPairs.No Support & No Enrichment-Support & Enrichment
#> 0.8951000
#> OVPairs.No Support & Enrichment-Support & No Enrichment
#> 0.4477958
#> OVPairs.No Support & Enrichment-Support & Enrichment
#> 0.7450246
#> OVPairs.Support & No Enrichment-Support & Enrichment
#> 0.6573381Standard Matching
After this check of initial unbalance, descriptive statistics, and
area of common support, we can conduct the matching. As mentioned above,
we need to adapt the grouping variable and the distance
variable/propensity score for this example. Moreover, conducting a
2x2/four-group matching has some computational load. If a RAM threshold
is exceeded, MAGMA computes quasi-systematic matching. This
means that the systematic matching algorithm of MAGMA is
applied to random subgroups of the sample. Again, this only affects the
computational backend and does not affect the user application. Note
that in the case of a multifactorial matching, MAGMA
returns a fourth variable. This fourth variable is a multinominal
version of the two independent variables (group_long).
MAGMA_sim_data_2x2 <- MAGMA(Data = MAGMA_sim_data,
group = c("gifted_support", "enrichment"),
dist = "ps_2x2",
cores = 2)str(MAGMA_sim_data_2x2)
#> 'data.frame': 800 obs. of 18 variables:
#> $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ gender : int 1 0 1 1 1 0 0 0 1 0 ...
#> $ gifted_support : int 1 1 1 1 1 1 0 1 0 1 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 2 2 1 1 ...
#> $ enrichment : int 0 1 0 1 0 0 0 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 1 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 1 3 1 4 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ group_long : num 1 2 1 2 1 1 3 1 3 2 ...
#> $ step : num 71 41 35 19 NA 12 NA NA NA 105 ...
#> $ weight : num 1 1 1 1 NA 1 NA NA NA 1 ...
#> $ distance : num 3.61e-03 2.15e-04 1.24e-04 4.69e-05 NA ...Balance Estimation and Visualization
The only difference to the previous examples in the process is, that
the grouping variable is a character vector of length two. As a result
of this, Plot_MAGMA displays three Pillai’s Trace plots
corresponding to the two main effects and the interaction.
Balance_2x2 <- Balance_MAGMA(Data = MAGMA_sim_data_2x2,
group = c("gifted_support", "enrichment"),
covariates = covariates_2x2,
step = "step")
# Balance criteria for 100 cases per group
Balance_100_2x2_criteria <- Balance_extract(Balance = Balance_2x2,
samplesize = 100,
effects = FALSE)
Balance_100_2x2_criteria
#> Pillai's Trace ME 1 Pillai's Trace ME 2 Pillai's Trace IA d-ratio
#> 0.02 0.01 0.02 0.73
#> mean g adj. d-ratio
#> 0.16 0.60
# Extracting pairwise effects for 100 cases per group
Balance_100_2x2_effects <- Balance_extract(Balance = Balance_2x2,
samplesize = 100,
effects = TRUE)
Balance_100_2x2_effects
#> GPA_school_1_2 IQ_score_1_2 Motivation_1_2
#> 0.17 0.12 0.17
#> parents_academic_1_2 gender_1_2 GPA_school_1_3
#> 0.22 0.06 0.10
#> IQ_score_1_3 Motivation_1_3 parents_academic_1_3
#> 0.03 0.21 0.10
#> gender_1_3 GPA_school_1_4 IQ_score_1_4
#> 0.12 0.01 0.27
#> Motivation_1_4 parents_academic_1_4 gender_1_4
#> 0.34 0.19 0.06
#> GPA_school_2_3 IQ_score_2_3 Motivation_2_3
#> 0.07 0.09 0.03
#> parents_academic_2_3 gender_2_3 GPA_school_2_4
#> 0.12 0.18 0.19
#> IQ_score_2_4 Motivation_2_4 parents_academic_2_4
#> 0.38 0.17 0.41
#> gender_2_4 GPA_school_3_4 IQ_score_3_4
#> 0.12 0.12 0.29
#> Motivation_3_4 parents_academic_3_4 gender_3_4
#> 0.14 0.29 0.06
# Plotting trend over increasing sample size
Plot_MAGMA(Balance = Balance_2x2,
criterion = c("Pillai", "d_ratio", "mean_g", "Adj_d_ratio")) 
# Creating table
Table_MAGMA(Balance = Balance_2x2,
filename = "Balance_2x2.docx")
#> Balance table successfully created!
#> # A tibble: 6 × 8
#> Criterion_optimized Pillai_Trace_ME1 Pillai_Trace_ME2 Pillai_Trace_IA d_ratio
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 Best Pillai ME 2 0.02 0 0.04 0.7
#> 2 Best Pillai IA 0.02 0.02 0.01 0.8
#> 3 Best mean g 0.02 0.02 0.01 0.8
#> 4 Best d-ratio 0.02 0.01 0.02 0.83
#> 5 Best Pillai ME 1 0.01 0.01 0.02 0.8
#> 6 Best adj. d-ratio 0.02 0.01 0.02 0.8
#> # ℹ 3 more variables: mean_g <dbl>, adjusted_d_ratio <dbl>, n_per_group <int>
# Computing descriptive statistics and all pairwise effects after matching
descs_2x2_post <- MAGMA_desc(Data = MAGMA_sim_data_2x2,
group = c("gifted_support", "enrichment"),
covariates = covariates_2x2,
step_num = 100,
step_var = "step",
filename = "stats_post_pre.docx")
#> 2x2 groups are represented as 4 groups.
descs_2x2_post %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"Sup & No En N", "Sup & No En Mean",
"Sup & No En SD",
"Sup & En N", "Sup & En Mean", "Sup & En SD",
"No Sup & No En N", "No Sup & No En Mean", "No Sup & No En SD",
"No Sup & En N", "No Sup & En Mean", "No Sup & En SD",
"d YesNo-YesYes", "d YesNo-NoNo", "d YesNo-NoYes",
"d YesYes-NoNo", "d YesYes-YNoYes",
"d NoNo-NoYes"))
#> Overall N Overall Mean Overall SD Sup & No En N
#> group_long 400 2.50 1.12 100
#> GPA_school 400 3.52 0.86 100
#> IQ_score 400 104.15 11.20 100
#> Motivation 400 3.90 0.85 100
#> parents_academic 400 0.44 0.50 100
#> gender 400 0.52 0.50 100
#> Sup & No En Mean Sup & No En SD Sup & En N Sup & En Mean
#> group_long 1.00 0.00 100 2.00
#> GPA_school 3.46 0.90 100 3.61
#> IQ_score 104.55 10.59 100 105.81
#> Motivation 3.74 0.88 100 3.89
#> parents_academic 0.42 0.50 100 0.53
#> gender 0.50 0.50 100 0.47
#> Sup & En SD No Sup & No En N No Sup & No En Mean
#> group_long 0.00 100 3.00
#> GPA_school 0.86 100 3.55
#> IQ_score 10.29 100 104.85
#> Motivation 0.85 100 3.92
#> parents_academic 0.50 100 0.47
#> gender 0.50 100 0.56
#> No Sup & No En SD No Sup & En N No Sup & En Mean
#> group_long 0.00 100 4.00
#> GPA_school 0.89 100 3.46
#> IQ_score 10.79 100 101.40
#> Motivation 0.83 100 4.03
#> parents_academic 0.50 100 0.33
#> gender 0.50 100 0.53
#> No Sup & En SD d YesNo-YesYes d YesNo-NoNo d YesNo-NoYes
#> group_long 0.00 -Inf -Inf -Inf
#> GPA_school 0.78 -0.17 -0.10 0.00
#> IQ_score 12.66 -0.12 -0.03 0.27
#> Motivation 0.85 -0.17 -0.21 -0.34
#> parents_academic 0.47 -0.22 -0.10 0.19
#> gender 0.50 0.06 -0.12 -0.06
#> d YesYes-NoNo d YesYes-YNoYes d NoNo-NoYes
#> group_long -Inf -Inf -Inf
#> GPA_school 0.07 0.18 0.11
#> IQ_score 0.09 0.38 0.29
#> Motivation -0.04 -0.16 -0.13
#> parents_academic 0.12 0.41 0.29
#> gender -0.18 -0.12 0.06Exact Matching
We use the same covariates as we used for the standard 2x2 matching.
The only change constitutes the use of
teacher_ability_rating as an exact variable. This means
that only students with same teacher rated ability can be matched (e.g.,
above average with above average). Below is the entire matching, balance
estimation, and visualization process. Initial unbalance, descriptive
statistics, and area of common support are the same as for standard
matching.
MAGMA_sim_data_2x2_exact <- MAGMA_exact(Data = MAGMA_sim_data,
group = c("gifted_support", "enrichment"),
dist = "ps_2x2",
exact = "teacher_ability_rating",
cores = 2)str(MAGMA_sim_data_2x2_exact)
#> 'data.frame': 520 obs. of 18 variables:
#> $ ID : int 1 2 3 4 5 6 10 13 14 16 ...
#> $ gender : int 1 0 1 1 1 0 0 1 0 1 ...
#> $ gifted_support : int 1 1 1 1 1 1 1 0 0 0 ...
#> $ teacher_ability_rating: int 3 2 2 1 3 3 1 3 2 1 ...
#> $ enrichment : int 0 1 0 1 0 0 1 0 0 1 ...
#> $ parents_academic : int 1 1 1 0 1 0 0 0 0 0 ...
#> $ GPA_school : num 4.73 4.41 3.23 2.88 2.98 ...
#> $ IQ_score : num 111.6 97.8 120.1 97.9 74.4 ...
#> $ Motivation : num 5.36 4.41 3.57 4.34 3.54 ...
#> $ college_GPA : num 4.01 3.41 2.73 4.02 4.09 ...
#> $ support_enrichment : int 3 4 3 4 3 3 4 1 1 2 ...
#> $ ps_tar : num 0.0657 0.1085 0.1899 0.4261 0.4782 ...
#> $ ps_2x2 : num 0.3 0.32 0.21 0.237 0.571 ...
#> $ ps_gifted : num 0.431 0.364 0.626 0.353 0.187 ...
#> $ group_long : num 1 2 1 2 1 1 2 3 3 4 ...
#> $ step : num 58 7 116 88 48 19 62 101 82 114 ...
#> $ weight : num 1 1 1 1 1 1 1 1 1 1 ...
#> $ distance : num 1.50e-03 1.76e-05 5.02e-01 9.73e-02 6.57e-04 ...# Estimating Balance
Balance_2x2_exact <- Balance_MAGMA(Data = MAGMA_sim_data_2x2_exact,
group = c("gifted_support", "enrichment"),
covariates = covariates_2x2,
step = "step") #Not necessary to define here# Balance criteria for 100 cases per group
Balance_100_2x2_criteria_exact <- Balance_extract(Balance = Balance_2x2_exact,
samplesize = 100,
effects = FALSE)
Balance_100_2x2_criteria_exact
#> Pillai's Trace ME 1 Pillai's Trace ME 2 Pillai's Trace IA d-ratio
#> 0.01 0.01 0.02 0.77
#> mean g adj. d-ratio
#> 0.12 0.69
# Extracting pairwise effects for 100 cases per group
Balance_100_2x2_effects_exact <- Balance_extract(Balance = Balance_2x2_exact,
samplesize = 100,
effects = TRUE)
Balance_100_2x2_effects_exact
#> GPA_school_1_2 IQ_score_1_2 Motivation_1_2
#> 0.12 0.16 0.17
#> parents_academic_1_2 gender_1_2 GPA_school_1_3
#> 0.04 0.10 0.10
#> IQ_score_1_3 Motivation_1_3 parents_academic_1_3
#> 0.06 0.23 0.08
#> gender_1_3 GPA_school_1_4 IQ_score_1_4
#> 0.02 0.01 0.18
#> Motivation_1_4 parents_academic_1_4 gender_1_4
#> 0.21 0.22 0.02
#> GPA_school_2_3 IQ_score_2_3 Motivation_2_3
#> 0.02 0.10 0.05
#> parents_academic_2_3 gender_2_3 GPA_school_2_4
#> 0.04 0.12 0.14
#> IQ_score_2_4 Motivation_2_4 parents_academic_2_4
#> 0.34 0.04 0.26
#> gender_2_4 GPA_school_3_4 IQ_score_3_4
#> 0.12 0.12 0.24
#> Motivation_3_4 parents_academic_3_4 gender_3_4
#> 0.01 0.30 0.00
# Plotting trend over increasing sample size
Plot_MAGMA(Balance = Balance_2x2_exact,
criterion = c("Pillai", "d_ratio", "mean_g", "Adj_d_ratio")) #Could be omitted
# Creating table
Table_MAGMA(Balance = Balance_2x2_exact,
filename = "Balance_2x2_exact.docx")
#> Balance table successfully created!
#> # A tibble: 6 × 8
#> Criterion_optimized Pillai_Trace_ME1 Pillai_Trace_ME2 Pillai_Trace_IA d_ratio
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 Best Pillai ME 1 0 0.04 0.07 0.6
#> 2 Best Pillai IA 0.01 0.01 0.02 0.83
#> 3 Best mean g 0.01 0.01 0.02 0.83
#> 4 Best d-ratio 0.01 0 0.02 0.87
#> 5 Best adj. d-ratio 0.01 0 0.02 0.8
#> 6 Best Pillai ME 2 0.01 0 0.02 0.8
#> # ℹ 3 more variables: mean_g <dbl>, adjusted_d_ratio <dbl>, n_per_group <int>
# Computing descriptive statistics and all pairwise effects post matching
descs_2x2_post <- MAGMA_desc(Data = MAGMA_sim_data_2x2_exact,
group = c("gifted_support", "enrichment"),
covariates = covariates_2x2,
step_num = 100,
step_var = "step",
filename = "stats_2x2_post.docx")
#> 2x2 groups are represented as 4 groups.
descs_2x2_post %>%
purrr::set_names(c("Overall N", "Overall Mean", "Overall SD",
"Sup & No En N", "Sup & No En Mean",
"Sup & No En SD",
"Sup & En N", "Sup & En Mean", "Sup & En SD",
"No Sup & No En N", "No Sup & No En Mean", "No Sup & No En SD",
"No Sup & En N", "No Sup & En Mean", "No Sup & En SD",
"d YesNo-YesYes", "d YesNo-NoNo", "d YesNo-NoYes",
"d YesYes-NoNo", "d YesYes-YNoYes",
"d NoNo-NoYes"))
#> Overall N Overall Mean Overall SD Sup & No En N
#> group_long 400 2.50 1.12 100
#> GPA_school 400 3.54 0.86 100
#> IQ_score 400 104.14 11.18 100
#> Motivation 400 3.90 0.85 100
#> parents_academic 400 0.46 0.50 100
#> gender 400 0.50 0.50 100
#> Sup & No En Mean Sup & No En SD Sup & En N Sup & En Mean
#> group_long 1.00 0.00 100 2.00
#> GPA_school 3.50 0.92 100 3.60
#> IQ_score 104.05 11.57 100 105.81
#> Motivation 3.77 0.86 100 3.92
#> parents_academic 0.47 0.50 100 0.49
#> gender 0.51 0.50 100 0.46
#> Sup & En SD No Sup & No En N No Sup & No En Mean
#> group_long 0.00 100 3.00
#> GPA_school 0.86 100 3.59
#> IQ_score 10.35 100 104.74
#> Motivation 0.85 100 3.96
#> parents_academic 0.50 100 0.51
#> gender 0.50 100 0.52
#> No Sup & No En SD No Sup & En N No Sup & En Mean
#> group_long 0.00 100 4.00
#> GPA_school 0.89 100 3.49
#> IQ_score 10.56 100 101.97
#> Motivation 0.84 100 3.95
#> parents_academic 0.50 100 0.36
#> gender 0.50 100 0.52
#> No Sup & En SD d YesNo-YesYes d YesNo-NoNo d YesNo-NoYes
#> group_long 0.00 -Inf -Inf -Inf
#> GPA_school 0.76 -0.11 -0.10 0.01
#> IQ_score 11.99 -0.16 -0.06 0.18
#> Motivation 0.85 -0.18 -0.22 -0.21
#> parents_academic 0.48 -0.04 -0.08 0.22
#> gender 0.50 0.10 -0.02 -0.02
#> d YesYes-NoNo d YesYes-YNoYes d NoNo-NoYes
#> group_long -Inf -Inf -Inf
#> GPA_school 0.01 0.14 0.12
#> IQ_score 0.10 0.34 0.25
#> Motivation -0.05 -0.04 0.01
#> parents_academic -0.04 0.27 0.31
#> gender -0.12 -0.12 0.00References
Austin, P. C. (2014). A comparison of 12 algorithms for matching on the propensity score. Statistics in Medicine, 33(6), 1057–1069. https://doi.org/10.1002/sim.6004
Feuchter, M. D., Urban, J., Scherrer, V., Breit, M. L., & Preckel, F. (2022). Introduction and Demonstration of the Many-Group Matching (MAGMA)-Algorithm: Matching Solutions for Two or More Groups. https://doi.org/10.17605/OSF.IO/AEDXB
Fisher, Z., Tipton, E., & Zhipeng, H. (2023). robumeta: Robust Variance Meta-Regression. R package version 2.1, https://CRAN.R-project.org/package=robumeta.
Imai, K., & van Dyk, D. A. (2004). Causal Inference With General Treatment Regimes. Journal of the American Statistical Association, 99(467), 854–866. https://doi.org/10.1198/016214504000001187
Jacovidis, J. N. (2017). Evaluating the performance of propensity score matching methods: A simulation study [Doctoral Dissertation]. James Madison University.
Li, M. (2013). Using the Propensity Score Method to Estimate Causal Effects. Organizational Research Methods, 16(2), 188–226. https://doi.org/10.1177/1094428112447816
McCaffrey, D. F., Griffin, B. A., Almirall, D., Slaughter, M. E., Ramchand, R., & Burgette, L. F. (2013). A tutorial on propensity score estimation for multiple treatments using generalized boosted models. Statistics in Medicine, 32(19), 3388–3414. https://doi.org/10.1002/sim.5753
Pastore, M., Loro, P.A.D., Mingione, M., Calcagni, A. (2022). overlapping: Estimation of Overlapping in Empirical Distributions. R package version 2.1. https://CRAN.R-project.org/package=overlapping
Powell, M. G., Hull, D. M., & Beaujean, A. A. (2020). Propensity Score Matching for Education Data: Worked Examples. The Journal of Experimental Education, 88(1), 145–164. https://doi.org/10.1080/00220973.2018.1541850
Ridgeway, G., McCaffrey, D. F., Morral, A. R., Cefalu, M., Burgette, L. F., Pane, J. D., & Griffin, B. A. (2015). Toolkit for Weighting and Analysis of Nonequivalent Groups: A tutorial for the twang package. Rand.
Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41–55. https://doi.org/10.1093/biomet/70.1.41
Rosenbaum, P. R., & Rubin, D. B. (1985). Constructing a Control Group Using Multivariate Matched Sampling Methods That Incorporate the Propensity Score. The American Statistician, 39(1), 33–38.
Stuart, E. A. (2010). Matching methods for causal inference: A review and a look forward. Statistical Science : A Review Journal of the Institute of Mathematical Statistics, 25(1), 1–21. https://doi.org/10.1214/09-STS313
Thoemmes, F. J., & Kim, E. S. (2011). A Systematic Review of Propensity Score Methods in the Social Sciences. Multivariate Behavioral Research, 46(1), 90–118. https://doi.org/10.1080/00273171.2011.540475
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1-48. https://doi.org/10.18637/jss.v036.i03