The idea of characterizing units by their response function, rather than their baseline features has several advantages, stemming primarily from the parsimony achieved by the former. Whereas each unit may have thousands of features, standing in unknown relationships to X and Y, the number of functions that those features can induce is limited by the cardinality of X and Y, and each such function defines the response of Y unequivocally.
Robins and Greenland were the first to capitalize on this advantage and have used classification by response type as a cornerstone in many of their works, including confounding (1986) attribution (1988, 1989a,b) and effect decomposition (1992).
Chickering and Pearl (1997)
as well as Imbens and Rubin (1997)
used the parsimony of response type classification in a Bayesian framework, to obtain posterior distributions of causal effects in noncompliance settings. It is obviously easier to assign meaningful priors to a 16-dimensional polytope than to a space of the many features that characterize each unit (see Pearl, 2009a
, Ch. 8).
Baker and Lindeman (1994)
and Imbens and Angrist (1994)
introduced a new element into this analysis. Realizing that the population averaged treatment effect (ATE) is not identifiable in experiments marred by noncompliance, they have shifted attention to a specific response type (i.e., compliers) for which the causal effect was identifiable, and presented the latter as an approximation
for ATE. This came to be known as LATE (Local Average Treatment Effect) and has spawned a rich literature with many variants (Angrist, Imbens, and Rubin, 1996
, Heckman and Vytlacil, 2001
, Heckman, 2005
) all focusing on a specific stratum or a subset of strata for which the causal effect could be identified under various combinations of assumptions and designs. However, most authors in this category do not state explicitly whether their focus on a specific stratum is motivated by mathematical convenience, mathematical necessity (to achieve identification) or a genuine interest in the stratum under analysis.
Though membership in response-type classes is generally not identifiable and is vulnerable to unpredictable changes,1
such membership may occasionally be at the center of a research question. For example, the effect of treatment on subjects who would have survived regardless of treatment
may become the center of interest in the context of censorship by death (Robins, 1986
). Likewise, survival in cancer cases caused by
hormone replacement therapy need be distinguished from survival in cancer cases caused by
other factors (Sjölander, Humphreys, and Vansteelandt, 2010
). In such applications, expressions of the form
emerge organically as the appropriate research questions, where Z
is some post-treatment variable, and the condition (Zx
) specifies the response-type stratum of interest.