Get access
Advertisement

Two-stage instrumental variable methods for estimating the causal odds ratio: Analysis of bias

Authors

  • Bing Cai,

    Corresponding author
    1. Merck Research Laboratories, UG1D-60, P.O. Box 1000, North Wales, PA 19454-1099, U.S.A.
    2. Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine, Blockley Hall, 6th FLR 423 Guardian Dr., Philadelphia, PA 19104-6021, U.S.A.
    • Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine, Blockley Hall, Guardian Dr., Philadelphia, PA 19104-6021, U.S.A.
    Search for more papers by this author
  • Dylan S. Small,

    1. Department of Statistics, Wharton School, 464 JMHH/6340, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.
    Search for more papers by this author
  • Thomas R. Ten Have

    1. Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine, Blockley Hall, 6th FLR 423 Guardian Dr., Philadelphia, PA 19104-6021, U.S.A.
    Search for more papers by this author

Abstract

We present closed-form expressions of asymptotic bias for the causal odds ratio from two estimation approaches of instrumental variable logistic regression: (i) the two-stage predictor substitution (2SPS) method and (ii) the two-stage residual inclusion (2SRI) approach. Under the 2SPS approach, the first stage model yields the predicted value of treatment as a function of an instrument and covariates, and in the second stage model for the outcome, this predicted value replaces the observed value of treatment as a covariate. Under the 2SRI approach, the first stage is the same, but the residual term of the first stage regression is included in the second stage regression, retaining the observed treatment as a covariate. Our bias assessment is for a different context from that of Terza (J. Health Econ. 2008; 27(3):531–543), who focused on the causal odds ratio conditional on the unmeasured confounder, whereas we focus on the causal odds ratio among compliers under the principal stratification framework. Our closed-form bias results show that the 2SPS logistic regression generates asymptotically biased estimates of this causal odds ratio when there is no unmeasured confounding and that this bias increases with increasing unmeasured confounding. The 2SRI logistic regression is asymptotically unbiased when there is no unmeasured confounding, but when there is unmeasured confounding, there is bias and it increases with increasing unmeasured confounding. The closed-form bias results provide guidance for using these IV logistic regression methods. Our simulation results are consistent with our closed-form analytic results under different combinations of parameter settings. Copyright © 2011 John Wiley & Sons, Ltd.

Get access to the full text of this article

Ancillary