  In observational study, the propensity score has the central role to estimate
causal effects. Since the propensity score is usually unknown, estimating by
appropriate procedures is an indispensable step. A point to note that a causal
effect estimator might have some bias if a propensity score model was
misspecified; valid model construction is important. To overcome the problem, a
variety of interesting methods has been proposed. In this paper, we review four
methods: using ordinary logistic regression approach; CBPS proposed by Imai and
Ratkovic; boosted CART proposed by McCaffrey and colleagues; a semiparametric
strategy proposed by Liu and colleagues. Also, we propose the novel robust two
step strategy: estimating each candidate model in the first step and
integrating them in the second step. We confirm the performance of these
methods through simulation examples by estimating the ATE and ATO proposed by
Li and colleagues. From the results of the simulation examples, the boosted
CART and CBPS with higher-order balancing condition have good properties; both
the estimate of the ATE and ATO has the small variance and the absolute value
of bias. The boosted CART and CBPS are useful for a variety of estimands and
estimating procedures.
