Correction to: Semiparametric Inference for Non-monotone Missing-Not-at-Random Data: the No Self-Censoring Model

This article refers to:

Semiparametric Inference for Nonmonotone Missing-Not-at-Random Data: The No Self-Censoring Model

Daniel Malinsky^a, Ilya Shpitser^b, and Eric J. Tchetgen Tchetgen^c

^aDepartment of Biostatistics, Columbia University, New York City, NY

^bDepartment of Computer Science, Johns Hopkins University, Baltimore, MD

^cDepartment of Statistics, The Wharton School of the University of Pennsylvania, Philadelphia, PA

CONTACT Daniel Malinsky [email protected] Department of Biostatistics, Columbia University, New York, NY 10027-6902.

Our Corollary 1 mistakenly omits a part of the $\text{OR}(R,L)$ expression. The corollary should say

Corollary 1.

Under Assumptions 1 and 2,$$\text{OR}(R,L)=\hspace{0.17em}\text{exp}\hspace{0.17em}\{(1-R_1)\delta h_1(L_{-1})+\dots +(1-R_K)\delta h_K(L_{-K})\}$$where $\delta h_i(L_{-i})=\hspace{0.17em}\text{log}\hspace{0.17em}(\frac{p(R_i=0|R_{-i}=1,L_{-i})}{p(R_i=1|R_{-i}=1,L_{-i})})-\hspace{0.17em}\text{log}\hspace{0.17em}(\frac{p(R_i=0|R_{-i}=1,L_{-i}=0)}{p(R_i=1|R_{-i}=1,L_{-i}=0)})$ $+\hspace{0.17em}\text{log}\hspace{0.17em}(\frac{\text{OR}(R_i,(R_1,\dots ,R_{i-1})|(R_{i+1},\dots ,R_K)=1,L_{-i})}{\text{OR}(R_i,(R_1,\dots ,R_{i-1})|(R_{i+1},\dots ,R_K)=1,L_{-i}=0)})$ for $i=1,\dots ,K$ (and the last term is defined to be zero for i = 1).

We are grateful to Wang Miao and Yilin Li for bringing this to our attention. We note that in the K = 2 case, the ratio $\text{OR}(·,·|L)/\text{OR}(·,·|L=0)=1$ and so the omitted last term vanishes, but that is not true for $K\ge 3$. From the proof of Theorem 1, we know that the product of odds ratios in the factorization can be expressed as a product of every pairwise odds ratio and all 3-way, …, K-way interaction terms, each of which is a function of only observed data. So these additional terms in the expression above are functions of only observed data. For notational simplicity and for minimal deviation from the way, the corollary is expressed in the article, we can leave it expressed this way.

This leads to a correction of Theorem 2, in order to account for the addition contribution to the influence function from taking the derivative ${\nabla }_t\hspace{0.17em}\text{log}\hspace{0.17em}\frac{{\text{OR}}_{\mathrm{t}}(R_i,(R_1,\dots ,R_{i-1})|(R_{i+1},\dots ,R_K)=1,L_{-i})}{\text{OR}(R_i,(R_1,\dots ,R_{i-1})|(R_{i+1},\dots ,R_K)=1,L_{-i}=0)}$.

Theorem 2.

In ${\mathcal{M}}_{\text{nsc}}$, the efficient influence function for β is ${\varphi }_{\text{nsc}}(\beta )=-\mathbb{E}{\left[\frac{\partial }{\partial \beta }{\varphi }_{\text{odds}}(\beta )\right]}^{-1}\times ({\varphi }_{\text{odds}}(\beta )+{\varphi }_{\text{adj}}(\beta ))$, with ${\varphi }_{\text{odds}}(\beta )$ from Lemma 1 and$$\begin{array}{c}{\varphi }_{\text{adj}}(\beta )=-\sum _{i=1}^K\mathbb{E}[(1-R_i)|L_{-i}]\frac{\mathbb{I}(R_{-i}=1)}{p(R_{-i}=1|L_{-i})}\\ \times \left(\frac{R_i}{p(R_i=1|R_{-i},L_{-i})}-1\right)\frac{p(R_i=1|R_{-i},L_{-i})}{p(R_i=0|R_{-i},L_{-i})}\\ \times \left(1-\frac{p(R_i=1|R_{(1:i-1)},R_{(i+1:K)}=1,L)}{p(R_i=0|R_{(1:i-1)},R_{(i+1:K)}=1,L)}\right)\\ \times \mathbb{E}[\Delta (R,L)|R_i=0,L_{-i}]-\sum _{i=1}^K\mathbb{E}[(1-R_i)\left|L_{-i}\right]\\ \times \frac{\mathbb{I}(R_{(i+1:K)}=1)}{p(R_{(i+1:K)}=1|L_{-i})}\\ \times \left(\frac{R_i}{p(R_i=1|R_{(1:i-1)},R_{(i+1:K)}=1,L_{-i})}-1\right)\\ \times \frac{1}{\text{OR}(R_i,R_{(1:i-1)}|R_{(i+1:K)}=1,L_{-i})}\\ \times \mathbb{E}[\Delta (R,L)|R_i=0,L_{-i}]\end{array}$$

with $\Delta (R,L)\equiv {\varphi }_{\text{full}}(\beta )-\mathbb{E}[{\varphi }_{\text{full}}(\beta )|R,L_{(R)}].$

All other results and analyses, including our proposed AIPW estimator, are not affected by this error.