Using a monotone single‐index model to stabilize the propensity score in missing data problems and causal inference

Abstract

The augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems. An important component of this method is the propensity score.Popular parametric models for the propensity score include the logistic, probit, and complementary log-log models. A common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables. To avoid the need to choose a model, we model the propensity score via a semiparametric single-index model, in which the score is an unknown monotonic nondecreasing function of the given single index. Under this new model, the augmented inverse weighting estimator (AIWE) of the mean of the response is asymptotically linear, semiparametrically efficient, and more robust than existing estimators. Moreover, we have made a surprising observation. The inverse probability weighting and AIWEs based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model. A heuristic explanation of this phenomenon is provided. A real-data example is used to illustrate the proposed methods.KEYWORDS causal inference, empirical process, inverse weighting, missing data, pool adjacent violation algorithm, single-index model 1442

Kernel density estimation in mixture models with known mixture proportions

“…We consider the AIDS Clinical Trials Group Study 175 (ACTG 175) data, which has also been analyzed by Qin et al 25 ACTG 175 is a randomized clinical trial comparing monotherapy (zidovudine or didanosine) with combination therapy (zidovudine and didanosine or zidovudine and zalcitabine) in adults infected with HIV type I whose CD4 T cell counts were between 200 and 500 per cubic millimeter. The CD4 T cell counts were measured at baseline, week 20, and week 96.…”

Parametric versus Semi and Nonparametric Regression Models

“…al 2016Mahmoud et. al , 2019Toma and Fulga 2018;Li, et al 2017;Qin et al 2018). SIM is more flexible compared to parametric models and does not lack from curse of dimentionality problem compared to nonparametric models.…”

Behaviour of the Monotone Single Index Model Under Repeated Measurements

“…In addition, Foster et al (2013) argue that, whenever appropriate, the assumption of monotonicity "noticeably improves performance". The advantages of the monotone single index model have also been investigated in Qin et al (2019), Wan et al (2017) where applications to, respectively, causal inference and drug interaction modelling were considered. Qin et al (2019) considered a similar estimator as we do here assuming continuous design, while Wan et al (2017) studied a spline approach, which has the drawback of having to use cross-validation in order to select to optimal knot positions.…”

“…Our main contributions in this paper are to give a precise description of the asymptotic behavior of the maximum likelihood estimator (MLE) in the monotone single index model under discrete design and develop an asymptotic goodness-of-fit test which could be used before applying our approach. Similarly to Qin et al (2019), Chen and Samworth (2016), we study the case where the sample size n is allowed to grow while the dimension d of the predictors is held fixed. One of our main findings is that, as opposed to the classical monotone single index models where the nonparametric LSE or MLE of the link function converges at the n 1/3 -rate, the asymptotics of the MLE under the discrete design exhibit the faster rate of n 1/2 .…”