Discussions

Abstract

International Statistical Review (2011), 79, 2, 221–230 doi:10.1111/j.1751-5823.2011.00144.x Discussions Anastasios A. Tsiatis and Marie Davidian Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203, USA E-mail: tsiatis@ncsu.edu We congratulate the authors (henceforth LSD) on a long overdue, detailed review of the connection between the AIPW estimators derived in the incomplete data context via semipara- metric theory by Robins, Rotnitzky, and colleagues and the survey calibration estimators used widely in survey sampling. Although this connection has been noted previously (e.g. Robins & Rotnitzky, 1998; Rotnitzky, 2009), the present article appears to be the first in the statistical literature to offer a more comprehensive account. We had only a passing familiarity with this connection, and we are grateful to the editors for the opportunity to offer this discussion, whose preparation required us to acquire a deeper understanding. In what follows, we hope to complement the presentation of LSD by highlighting some further relationships and differences between the two perspectives. We adopt the notation used by the authors and consider estimation of the population total r eg , which, as in Section 2.1 of LSD may be written T. We focus on the regression estimator T ), where equivalently as T N R i i (β) = T T x i β, π i i=1 a representation that may be more familiar to statisticians well-versed in the AIPW literature. We reiterate and expand upon some important differences between the missing data and survey sampling contexts noted by LSD. In survey sampling, the realizations (x 1 , y 1 ), . . . , (x N , y N ) that comprise the population are regarded as fixed, and inference on T = Ni=1 y i , or, equivalently, the population mean N −1 T, is the goal. This is based on data (x i , R i , R i y i ), i = 1, . . . , N, drawn from the population according to a fixed, known design, where n = Ni=1 R i , and Pr(R i = 1) = π i and Pr(R i = 1, R j = 1) = π ij for π i known and π ij known or unknown, i, j = 1, . . . , N. The (x i , y i ) may be viewed as realizations of random variables (X i , Y i ), i = 1, . . . , N, representing an independent and identically distributed (iid) sample from some super-population; however interest focuses on the fixed quantity T (S¨arndal et al., 2003). In contrast, in the incomplete data context, interest is in estimation of μ = E(Y ), a parameter associated with the super-population. Instead of observing a realization of i.i.d. (X i , Y i ), i = 1, . . . , N, we observe a realization of i.i.d. (X i , R i , R i Y i ), i = 1, . . . , N, where R i is an indicator of whether or not the value of Y i is observed or missing. Ordinarily, the probabilities of observing Y i for each i are not fixed by design; rather, missingness arises according to some unknown mechanism about which some assumption is made. A common assumption is that R i is conditionally independent of Y i given X i , the so-called “missing at random” (MAR) assumption, under which π(X i ) = Pr(R i = 1|X i , Y i ) = Pr(R i = 1|X i ). MAR cannot be verified from the observed data, so the C 2011 The Authors. International Statistical Review C 2011 International Statistical Institute. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.