Corrigendum to: Generalizing evidence from randomized trials using inverse probability of sampling weights (Journal of the Royal Statistical Society: Series A (Statistics in Society), (2018), 181, 4, (1193-1209), 10.1111/rssa.12357)

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Regarding the paper Buchanan, A. L., Hudgens, M. G., Cole, S. R., Mollan, K. R., Sax, P. E., Daar, E. S., … & Mugavero, M. J. (2018). Generalizing evidence from randomized trials using inverse probability of sampling weights. Journal of the Royal Statistical Society. Series A, (Statistics in Society), 181(4), 1193-1209. The expression on page 1196 for the weight for each individual in the cohort should be (Formula presented.). Corrections to online appendix B appear below. Following those corrections, the paragraph on page 1198 should be revised as follows: A comparison of equations (3) and (5) shows that the variance is typically smaller in this setting when the sampling scores are estimated (see the online appendix B). Therefore, even if the correct sampling scores are known, estimation of the sampling scores is often preferable because of improved efficiency. This is analogous to a well-known result for inverse-treatment-weighted estimators (Hirano et al., 2003; Robins et al., 1992; Wooldridge, 2007). These changes do not affect the simulation results in Section 4 or the data analysis in Section 5. We thank Michael Jetsupphasuk, Fan Li, and George Papandonatos for their comments. REVISED APPENDIX B First consider the case when β is known. The asymptotic variance of (Formula presented.) can be expressed as (Formula presented.) where τ = (1,−1) and (Formula presented.). Next consider the case when β is estimated. Let (Formula presented.) and define (Formula presented.) and let (Formula presented.). Then, using block matrix notation note (Formula presented.) where in general (Formula presented.) is a r×c matrix of zeros. It follows that (Formula presented.) where * denotes quantities not expressed explicitly and (Formula presented.) Therefore, (Formula presented.) where (Formula presented.). If (Formula presented.) is semi-positive definite, then (Formula presented.). Below we derive sufficient conditions for this to be the case. Assume as m, n, N→∞ that n/N → Pr[S=1], (m+n)/N → f ∈ (0,1), and m/(N−n) → g ∈ (0,1). Then it is straightforward to show (Formula presented.) and (Formula presented.) Because (Formula presented.) it follows that (Formula presented.) Suppose w = w(Z) < m/N for all Z. That is, the conditional probability for any individual of participating in a trial is less than the probability of participating in a cohort study. Often cohort studies are larger than trials, so this assumption may be reasonable in many settings. It follows that (Formula presented.), implying that (Formula presented.)

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Journal of the Royal Statistical Society. Series A: Statistics in Society