Ensemble Approaches to Estimating the Population Mean with Missing Response

Abstract

We propose new ensemble approaches to estimate the population mean for missing response data with fully observed auxiliary variables. We first compress the working models according to their categories through a weighted average, where the weights are proportional to the square of the least-squares coefficients of model refitting. Based on the compressed values, we develop two ensemble frameworks, under which one is to adjust weights in the inverse probability weighting procedure and the other is built upon an additive structure by reformulating the augmented inverse probability weighting function. The asymptotic normality property is established for the proposed estimators through the theory of estimating functions with plugged-in nuisance parameter estimates. Simulation studies show that the new proposals have substantial advantages over existing ones for small sample sizes, and an acquired immune deficiency syndrome data example is used for illustration. Scand J Statist 44 doubly robust point estimator, Kim & Haziza (2014) also considered a doubly robust variance estimator and extended them to the context of survey sampling. From the empirical likelihood (EL) perspective, Qin & Zhang (2007) and Qin et al. (2008Qin et al. ( , 2009) studied missing response problems and also developed doubly robust imputation. Tsiatis (2006) and Kang & Schafer (2007) provided comprehensive overviews of the semiparametric methodology on missing data and doubly robust estimators.Beyond double robustness, multiple robustness refers to the property that the estimator is consistent as long as one of the multiple working propensity score models or one of the multiple working outcome regression models is correctly specified. Towards this goal, Han & Wang (2013) proposed to maximize an EL function subject to multiple moment calibration equations. Chan (2013) developed a least-squares (LS) procedure that still maintains the multiple robustness property. Han (2014, 2016) further extended the multiply robust estimation to regression models, and because of the EL calibration, their proposals are robust against the extreme values of the fitted propensity score. Chan & Yam (2014) studied a class of calibration estimators with multiple working regression models based on the generalized EL. By incorporating multiple working models into a single constrained optimization procedure, these methods enjoy the desirable multiple robustness property.A common feature of the aforementioned multiply robust methods is that they all rely on a constrained optimization procedure, and the number of constraints is equal to the number of employed working models. Recognizing the computational burden induced by a large number of constraints in the EL procedure and also the estimation instability of LS with small observational probabilities, we propose new ensemble approaches for robust inference in the presence of missing response data. Rather than directly pooling all working models into a single constrained optimization procedure, we compress them first throu...

Multiply robust nonparametric multiple imputation for the treatment of missing data

“…To reduce the computational burden, we adopt the approach of Duan and Yin (2017), who proposed to compress the working models through weighted averages. Let…”

“…To provide additional protection, Han and Wang (2013) introduced the concept of multiple robustness; see also Han (2014aHan ( , 2014bHan ( , 2016aHan ( , 2016b, Chan and Yam (2014), Chen and Haziza (2017) and Duan and Yin (2017).…”

Multiply robust imputation procedures for zero-inflated distributions in surveys

“…Using the normalized scores rather than the customary prediction ${\hat{U}}_{qi}^{\top }{\hat{\eta }}_q$ ensures that the weighted average lies in the appropriate range; see Duan and Yin (2017). Note that η̂ q is the weighted least square estimator obtained regressing δ i on Û qi based on the responding units.…”

“…Suppose that the first model, q 1 ( x i ; γ 1 ) in 𝒞 3 is correctly specified. Then, it can be shown that q̂ i = q 1 ( x i ; γ̂ 1 )+ O p ( n −1/2 ); see Duan and Yin (2017).…”

Recent Developments in Dealing with Item Non‐response in Surveys: A Critical Review

“…Here, if a D .a 1 ; ; a h / is a h-vector, a 2 denotes the column vector .a 2 1 ; ; a 2 h /. Using the normalised scores ensures that the weighted average lies in the appropriate range; see Duan & Yin (2017). The scores O p i and O m i compress respectively the information contained in the J non-response models and the L imputation models.…”