of Identification causal effects Statistical inference
Identification (statistical) of parameter: the parameter can be calculated (correctly) from the joint distribution of observable quantities Here: we would like to be able to learn something about the potential outcomes : Distributions and expectations of Comparisons (e.g., differences or ratios of expectations; causal risk differences or ratios)
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Problem: potential outcomes population
,
not observed for every subject in
What can be stated about these sometimes unobserved quantities? 1. 2. 3. No additional assumptions: bounds-introduce briefly Additional assumptions: identification-discuss at length Intermediate approaches: Bayesian, etc.
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Bounds: consider binary Y What is largest value for What is smallest? What about ? ? that is consistent with data?
What is maximum value for causal risk difference
(Consider inference either for finite population (no sampling) or sample of infinite size)
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Can expand probability:
because we observe (and so Y) for unexposed subjects (A=0) (consistency assumption) What quantities above are observable, estimable from data?
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All quantities known or estimable except What are bounds on ?
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Limits for
: :
Thus, limits for entire population; i.e., on
For binary outcome, can learn something about potential outcomes without making any further assumptions about unobserved quantities Not dependent on study design For other potential outcome
How can one construct bounds on risk difference?
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For lower bound, take largest value for value for
consistent with data, smallest
Similarly, for upper bound:
Applicable cohort to studies, both randomized and not randomized
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Difference between upper and lower bounds on
is 1
So half of theoretical range (-1 to 1; range of 2) excluded by observable variables with no other assumptions No matter how large study, how much data obtained, this approach cannot get narrower range for causal risk difference Can never exclude null hypothesis of no treatment effect
For what other outcome types (non-binary) can one derive nonparametric bounds?
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Requires boundedness of the potential outcomes
For whom are bounds as formulated above derived?
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Either for large populations or for specific sample under study How would one derive inference from sample about population?
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One could calculate sampling variances of bounds Will generally lead to even wider intervals
Nonparametric bounds are generally fairly broad More precise inference desired
Suggest methods (very general) for obtaining inferences
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To get more pre...