Censoring Due to Death 

 D. James Greiner 

 I'm interested in the problem of "censoring due to death" within the framework of the Rubin Causal Model ("RCM"). 

 As readers will know, the RCM is a framework for studying the effects of causes in which the science is represented via a set of potential outcomes for each unit.  (A potential outcome is the value the dependent variable would take on if the treatment variable had a certain value, whether or not the treatment variable actually had that value).  An assignment mechanism decides what treatment (e.g., active treatment or control) a unit receives and thus which potential outcome will be observed.  Unit-level causal effects are defined as the difference in the potential outcomes of some quantity of interest.  The fundamental problem of causal inference is that we can observe at most one potential outcome for each unit.  Unobserved potential outcomes are treated as missing data.  Observational studies are analyzed as "broken" randomized experiments, broken in the sense that the assignment mechanism was not recorded and therefore must be reconstructed in some approximate way.  For a more complete discussion, see Holland, P.W. (1986). Statistics and Causal Inference.  Journal of the American Statistical Association 81: 945--960. 

 Censoring or truncation due to death occurs when some units' failure to comply with a post-treatment condition renders their values of the quantity of interest undefined.  Consider for example a medical study designed to assess the effect of a new cancer treatment on the percentage of patients who survive cancer-free for ten years.  Suppose some individuals die from car accidents or drug overdoses or other causes clearly unrelated to cancer before the ten-year time period has elapsed.  Such individuals do not have a value for ten-year cancer-free survival, so their values of the quantity of interest are undefined.  (The problem here is not that these individuals' values for cancer-free survival are missing data; rather, the problem is that they have no such values.)  Under such circumstances, some quantitative analysts simply remove such individuals from the study and analyze the remainder.  This course of action can bias results in several different ways.  To illustrate one such way, it could be that individuals who die from non-cancer related causes might smoke, have less healthy diets, refuse to wear seat belts, or otherwise engage in more risky behavior than many of the other individuals in the study.  If the treatment is effective in warding off cancer, there could be more deaths unrelated to cancer in the treated group than the control group, because some treated group members survive cancer that would otherwise have killed them long enough to be felled by, for example, car accidents, before ten years are up.  This difference could render comparison of the units remaining in the treated and control groups an inappropriate method of assessing the effect of the treatment. 

 The key is to realize that a comparison of ten-year cancer-free survival rates only makes sense for units who would not die from causes unrelated to cancer if assigned treatment AND who would not die from causes unrelated to cancer if assigned control.  Thus, removing individuals who died from causes unrelated to cancer is not enough. 

 The remaining group actually assigned control may include some units who would have died from non-cancer causes if they had been assigned treatment, and the remaining group actually assigned treatment may have some units who would have died from non-cancer causes had they been assigned control.  The researcher must take appropriate steps to remove both sets of people from the study, so as to isolate the set of individuals who would not die from causes unrelated to cancer regardless of treatment assignment.  Junni Zhang (Peking University) and Don Rubin (Harvard University) discuss these issues in "Estimation of Causal Effects Via Principal Stratification when Some Outcomes Are Truncated by 'Death,'" (2003).  Journal of Educational and Behavioral Statistics 28:353-368.  They extend them in a forthcoming paper with Fabrizia Mealli (University of Florence) currently entitled "Evaluating Causal Effects in the Presence of 'Truncation by Death' -Likelihood-based Analysis via Principal Stratification."