Effects of an unobserved confounder on a system with an intermediate outcome

In the theory of graphical Markov models in which relations between many variables are simplified via conditional independencies a special role is played by directed acyclic graphs. They can be used to represent statistical models in which data are generated in a stepwise fashion. Responses and intermediate variables may be event histories. We discuss such a system with sequentially administered treatments and a confounder, that is a variable which affects both the final outcome and one of its explanatory variables. The effect of not observing the confounder is to obtain the final and an intermediate outcome as joint responses and leads to the important observation by Robins and Wasserman (1997) that any univariate conditional distribution for the final outcome will be inappropriate for analysis no matter whether the intermediate outcome is conditioned on or not. It means in particular that the independence structure of the observed variables can no longer be fully described by a directed acyclic graph, that criteria for reading indepencencies off graphs have to be modified and that joint instead of univariate regression models are needed. These modifications resolve directly the puzzling situation which has been discussed by the above authors for randomized clinical trials as a case in which a true hypothesis of no treatment effect is always falsely rejected. Joint response models provide an alternative route for avoiding this unpleasant situation.

[1]  N. Wermuth,et al.  Linear Dependencies Represented by Chain Graphs , 1993 .

[2]  Nanny Wermuth,et al.  On association models defined over independence graphs , 1998 .

[3]  N. Wermuth,et al.  Graphical Models for Associations between Variables, some of which are Qualitative and some Quantitative , 1989 .

[4]  T. Haavelmo.  The Statistical Implications of a System of Simultaneous Equations , 1943 .

[5]  P. Spirtes,et al.  Using Path Diagrams as a Structural Equation Modeling Tool , 1998 .

[6]  A. Dawid.  Conditional Independence in Statistical Theory , 1979 .

[7]  T. W. Anderson.  Asymptotically Efficient Estimation of Covariance Matrices with Linear Structure , 1973 .

[8]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[9]  Nanny Wermuth,et al.  Multivariate Dependencies: Models, Analysis and Interpretation , 1996 .

[10]  D. A. Kenny,et al.  Correlation and Causation. , 1982 .

[11]  N. Wermuth,et al.  On Substantive Research Hypotheses, Conditional Independence Graphs and Graphical Chain Models , 1990 .

[12]  J. Koster.  On the Validity of the Markov Interpretation of Path Diagrams of Gaussian Structural Equations Systems with Correlated Errors , 1999 .

[13]  Nanny Wermuth,et al.  Likelihood Factorizations for Mixed Discrete and Continuous Variables , 1999 .

[14]  D. A. Kenny,et al.  Correlation and Causation , 1937, Wilmott.

[15]  N. Wermuth.  Linear Recursive Equations, Covariance Selection, and Path Analysis , 1980 .

[16]  M. Frydenberg.  Marginalization and Collapsibility in Graphical Interaction Models , 1990 .

[17]  Nanny Wermuth,et al.  Response models for mixed binary and quantitative variables , 1992 .

[18]  James M. Robins,et al.  Estimation of Effects of Sequential Treatments by Reparameterizing Directed Acyclic Graphs , 1997, UAI.

[19]  S. S. Wilks.  The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses , 1938 .