The function PSweight_SW is used to estimate the average potential outcomes corresponding to
each treatment group among the target population. The function currently implements
the following types of weights: the inverse probability of treatment weights (IPW: target population is the combined population),
average treatment effect among the treated weights (treated: target population is the population receiving a specified treatment),
overlap weights (overlap: target population is the overlap population at clinical equipoise), matching weights (matching: target population
is population obtained under 1:1 matching), entropy weights (entropy: target population is the population weighted by the entropy function).
Augmented propensity score weighting estimators are also allowed, with propensity scores and outcome model estimates either estimated
within the function, or supplied by external routines. The function now includes support for survey designs that include specific survey weights,
mainly focusing on binary treatments for now by incorporating survey weights into propensity score estimation and both point and augmented estimators
for the outcome estimation.
PSweight_SW(
ps.formula = NULL,
ps.estimate = NULL,
trtgrp = NULL,
zname = NULL,
yname,
data,
weight = "overlap",
survey.indicator = FALSE,
survey.design = "Independent",
svywtname = NULL,
delta = 0,
augmentation = FALSE,
augmentation.type = "WET",
bootstrap = FALSE,
R = 50,
out.formula = NULL,
out.estimate = NULL,
family = "gaussian",
ps.method = "glm",
ps.control = list(),
out.method = "glm",
out.control = list()
)PSweight_SW returns a PSweight_SW object containing a list of the following values:
estimated propensity scores for both population and sample levels, average potential outcomes corresponding to each treatment,
variance-covariance matrix of the point estimates, the label for each treatment group,
and estimates in each bootstrap replicate if bootstrap = TRUE.
A summary of PSweight_SW can be obtained with summary.PSweight.
trtgrpa character indicating the treatment group.
propensitya data frame of estimated propensity scores. When survey.indicator = TRUE, it is population level propensity score estimated by survey-weighted regression.
propensity.samplea data frame of estimated sample level propensity scores. This element is included when survey.indicator = TRUE.
muhataverage potential outcomes by treatment groups, with reference to specific target populations.
covmuvariance-covariance matrix of muhat.
mubootan optional list of point estimates in each bootstrap replicate bootstrap = TRUE.
groupa table of treatment group labels corresponding to the output point estimates muhat.
an object of class formula (or one that can be coerced to that class):
a symbolic description of the propensity score model to be fitted. Additional details of model specification
are given under "Details". This argument is optional if ps.estimate is not NULL.
an optional matrix or data frame containing estimated (generalized) propensity scores for
each observation. Typically, this is an N by J matrix, where N is the number of observations and J is the
total number of treatment levels. Preferably, the column name of this matrix should match the name of treatment level,
if column name is missing or there is a mismatch, the column names would be assigned according to alphabatic order
of the treatment levels. A vector of propensity score estimates is also allowed in ps.estimate, in which
case a binary treatment is implied and the input is regarded as the propensity to receive the last category of
treatment by alphabatic order, unless otherwise stated by trtgrp.
an optional character defining the "treated" population for estimating the average treatment
effect among the treated (ATT). Only necessary if weight = "treated". This option can also be used to specify
the treatment (in a two-treatment setting) when a vector argument is supplied for ps.estimate.
Default value is the last group in the alphebatic order.
an optional character specifying the name of the treatment variable in data.
an optional character specifying the name of the outcome variable in data.
an optional data frame containing the variables in the propensity score model
and outcome model (if augmented estimator is used). If not found in data, the variables are
taken from environment(formula).
a character or vector of characters including the types of weights to be used.
"IPW" specifies the inverse probability of treatment weights for estimating the average treatment
effect among the combined population. "treated" specifies the weights for estimating the
average treatment effect among the treated. "overlap" specifies the (generalized) overlap weights
for estimating the average treatment effect among the overlap population, or population at
clinical equipoise. "matching" specifies the matching weights for estimating the average treatment effect
among the matched population (ATM). "entropy" specifies the entropy weights for the average treatment effect
of entropy weighted population (ATEN). Default is "overlap".
logical. Indicates whether survey weights are used in the estimation.
Default is FALSE. If TRUE, survey weights specified in svywtname will be used.
character. Specifies the survey design scenario for estimation. Acceptable values are "Retrospective", "Independent", and "Prospective". "Retrospective" indicates that the sampling process depends on both treatment assignment and covariates, "Independent" (the default) means that the sampling process is independent of treatment assignment, and "Prospective" signifies that sampling is conducted prior to treatment assignment, although treatment may later be influenced by the sampling process.
an optional character specifying the name of the survey weight variable in data.
Default is NULL. Only required if survey.indicator is TRUE. If survey.indicator is TRUE
and svywtname is not provided, a default survey weight of 1 will be applied to all samples.
trimming threshold for estimated (generalized) propensity scores. Should be no larger than 1 / number of treatment groups. Default is 0, corresponding to no trimming.
logical. Indicate whether augmented weighting estimators should be used.
Default is FALSE.
a character specifying the type of augmentation to use when augmentation = TRUE. Supported values are
"WET" (weighted regression estimator), "MOM" (moment estimator), and "CVR" (clever covariate estimator). The default is
"WET".
logical. Indaicate whether bootstrap is used to estimate the standard error
of the point estimates. Default is FALSE.
an optional integer indicating number of bootstrap replicates. Default is R = 50.
an object of class formula (or one that can be coerced to that class):
a symbolic description of the outcome model to be fitted. Additional details of model specification
are given under "Details". This argument is optional if out.estimate is not NULL.
an optional matrix or data frame containing estimated potential outcomes
for each observation. Typically, this is an N by J matrix, where N is the number of observations
and J is the total number of treatment levels. Preferably, the column name of this matrix should
match the name of treatment level, if column name is missing or there is a mismatch,
the column names would be assigned according to alphabatic order of the treatment levels, with a
similar mechanism as in ps.estimate.
a description of the error distribution and link function to be used in the outcome model.
Only required if out.formula is provided. Supported distributional families include
"gaussian" (link = identity), "binomial" (link = logit) and "poisson" (link = log).
See family in glm for more details. Default is "gaussian".
a character to specify the method for estimating propensity scores. "glm" is default, and "gbm" and "SuperLearner" are also allowed.
a list to specify additional options when method is set to "gbm" or "SuperLearner".
a character to specify the method for estimating the outcome regression model. "glm" is default, and "gbm" and "SuperLearner" are also allowed.
a list to specify additional options when out.method is set to "gbm" or "SuperLearner".
A typical form for ps.formula is treatment ~ terms where treatment is the treatment
variable (identical to the variable name used to specify zname) and terms is a series of terms
which specifies a linear predictor for treatment. Similarly, a typical form for out.formula is
outcome ~ terms where outcome is the outcome variable (identical to the variable name
used to specify yname) and terms is a series of terms which specifies a linear
predictor for outcome. Both ps.formula and out.formula by default specify generalized
linear models when ps.estimate and/or out.estimate is NULL. The argument ps.method and out.method allow users to choose
models other than glm to fit the propensity score and outcome regression models for augmentation. Additional arguments in the gbm() function can be supplied through the ps.control and out.control arguments. Please refer to the user manual of the gbm package for all the
allowed arguments. "SuperLearner" is also allowed in the ps.method and out.method arguments. Currently, the SuperLearner method only supports binary treatment with the default method set to "SL.glm". The estimation approach is default to "method.NNLS" for both propensity and outcome regression models.
Prediction algorithms and other tuning parameters can also be passed through ps.control and out.control. Please refer to the user manual of the SuperLearner package for all the allowed specifications.
The function now includes support for the survey setting, mainly focusing on binary treatments by incorporating survey weights into propensity score estimation and both point and augmented estimators for the outcome stage. In survey settings,
external propensity score estimates are not supported; both population-level and sample-level propensity scores are estimated using the internal routines.
When augmentation = TRUE in the survey setting, three augmented estimators are supported: the moment estimator(MOM),
the clever covariate regression estimator(CVR), and the weighted regression estimator (WET, which is the
default). The user can select the desired estimator by setting the augmentation.type parameter accordingly.
When comparing two treatments, ps.estimate can either be a vector or a two-column matrix of estimated
propensity scores. If a vector is supplied, it is assumed to be the propensity scores to receive the treatment, and
the treatment group corresponds to the last group in the alphabetical order, unless otherwise specified by trtgrp.
When comparing multiple (J>=3) treatments, ps.estimate needs to be specified as an N by J matrix,
where N indicates the number of observations, and J indicates the total number of treatments.
This matrix specifies the estimated generalized propensity scores to receive each of the J treatments.
In general, ps.estimate should have column names that indicate the level of the treatment variable,
which should match the levels given in Z.
If column names are empty or there is a mismatch, the column names will be created following
the alphabetical order of values in Z, and the rightmost column of ps.estimate is assumed
to be the treatment group when estimating ATT. trtgrp can also be used to specify the treatment
group for estimating ATT. The same mechanism applies to out.estimate, except that the input for out.estimate
must be an N by J matrix, where each row corresponds to the estimated potential outcomes (corresponding to each treatment)
for each observation.
The argument zname and/or yname is required when ps.estimate
and/or out.estimate is not NULL.
In survey settings, when survey.indicator is TRUE, the argument svywtname (which specifies the survey weight variable in data) is required;
if svywtname is not provided, a default survey weight of 1 is applied to all observations. The argument
survey.design must be specified to reflect the sampling mechanism: for example, "Retrospective" indicates
that the sampling process depends on both treatment assignment and covariates, "Independent" assumes that sampling
is independent of treatment assignment, and "Prospective" signifies that sampling is conducted prior to treatment assignment,
although treatment may later be influenced by the sampling results.
Current version of PSweight allows for five types of propensity score weights used to estimate population level ATE (IPW), ATT (treated) and
ATO (overlap), ATM (matching) and ATEN (entropy) under survey settings. These weights are members of larger class of balancing weights defined in Li, Morgan, and Zaslavsky (2018).
Specific definitions of these weights are provided in Li, Morgan, and Zaslavsky (2018), Li and Greene (2013), Zhou, Matsouaka and Thomas (2020), Zeng, Li and Tong (2025).
When there is a practical violation of the positivity assumption, delta defines the symmetric
propensity score trimming rule following Crump et al. (2009). With multiple treatments, delta defines the
multinomial trimming rule introduced in Yoshida et al. (2019). The overlap weights can also be considered as
a data-driven continuous trimming strategy without specifying trimming rules, see Li, Thomas and Li (2019).
Additional details on balancing weights and generalized overlap weights for multiple treatment groups are provided in
Li and Li (2019). Zeng, Li, and Tong (2025) further specify how the survey weights can be incorporated into propensity score weighting under both retrospective and prospective scenarios.
Their approach supports both a weighting-only estimator and all three augmented estimators(MOM, CVR and WET), with corresponding sandwich variance estimators developed.
These enhancements are implemented in the current version of PSweight.
If augmentation = TRUE, an augmented weighting estimator will be implemented. For binary treatments, the augmented
weighting estimator is presented in Mao, Li and Greene (2018). For multiple treatments, the augmented weighting estimator is
mentioned in Li and Li (2019), and additional details will appear in our ongoing work (Zhou et al. 2020+). When
weight = "IPW", the augmented estimator is also referred to as a doubly-robust (DR) estimator.
In survey settings, the augmented estimator is further extended to support three variants: the moment estimator (MOM),
the clever covariate estimator (CVR), and the weighted regression estimator (WET); the default choice is WET.
Users can select the desired variant by specifying the augmentation.type parameter.
When bootstrap = TRUE, the variance will be calculated by nonparametric bootstrap, with R bootstrap
replications. The default of R is 50. Otherwise, the variance will be calculated using the sandwich variance
formula obtained in the M-estimation framework. In survey settings, however, bootstrapping is currently not supported;
we recommend that users employ the sandwich variance estimator instead.
Crump, R. K., Hotz, V. J., Imbens, G. W., Mitnik, O. A. (2009). Dealing with limited overlap in estimation of average treatment effects. Biometrika, 96(1), 187-199.
Li, L., Greene, T. (2013). A weighting analogue to pair matching in propensity score analysis. The International Journal of Biostatistics, 9(2), 215-234.
Li, F., Morgan, K. L., Zaslavsky, A. M. (2018). Balancing covariates via propensity score weighting. Journal of the American Statistical Association, 113(521), 390-400.
Mao, H., Li, L., Greene, T. (2019). Propensity score weighting analysis and treatment effect discovery. Statistical Methods in Medical Research, 28(8), 2439-2454.
Li, F., Thomas, L. E., Li, F. (2019). Addressing extreme propensity scores via the overlap weights. American Journal of Epidemiology, 188(1), 250-257.
Yoshida, K., Solomon, D.H., Haneuse, S., Kim, S.C., Patorno, E., Tedeschi, S.K., Lyu, H., Franklin, J.M., Stürmer, T., Hernández-Díaz, S. and Glynn, R.J. (2019). Multinomial extension of propensity score trimming methods: A simulation study. American Journal of Epidemiology, 188(3), 609-616.
Li, F., Li, F. (2019). Propensity score weighting for causal inference with multiple treatments. The Annals of Applied Statistics, 13(4), 2389-2415.
Zhou, Y., Matsouaka, R. A., Thomas, L. (2020). Propensity score weighting under limited overlap and model misspecification. Statistical Methods in Medical Research, 29(12), 3721-3756.
Zeng, Y., Li, F., & Tong, G. (2025). Moving toward best practice when using propensity score weighting in survey observational studies. arXiv preprint arXiv:2501.16156.
data("psdata")
data("psdata_bin_prospective_fp")
data("psdata_bin_retrospective_fp")
# Define the formulas
ps.formula <- trt ~ cov1 + cov2 + cov3 + cov4 + cov5 + cov6
out.formula <- Y ~ cov1 + cov2 + cov3 + cov4 + cov5 + cov6
# Prospective design without augmentation
pato1_sw <- PSweight_SW(ps.formula = ps.formula, yname = "Y",
data = psdata_bin_prospective_fp, weight = "overlap",
survey.indicator = TRUE, survey.design = "Prospective",
svywtname = "survey_weight",
delta = 0.1, augmentation = FALSE, bootstrap = FALSE, R = 50,
out.formula = NULL, out.estimate = NULL, family = "gaussian",
ps.method = "glm", ps.control = list(),
out.method = "glm", out.control = list())
summary(pato1_sw)
# Retrospective design with augmentation using the Weighted Regression Estimator(WET) estimator
pato2_sw <- PSweight_SW(ps.formula = ps.formula, yname = "Y",
data = psdata_bin_retrospective_fp, weight = "overlap",
survey.indicator = TRUE, survey.design = "Retrospective",
svywtname = "survey_weight",
delta = 0.1, augmentation = TRUE, augmentation.type = "WET",
bootstrap = FALSE, R = 50,
out.formula = out.formula, out.estimate = NULL, family = "gaussian",
ps.method = "glm", ps.control = list(),
out.method = "glm", out.control = list())
summary(pato2_sw)
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