TECHNICAL APPENDIX: DEFINITIONS OF PARAMETERS 1
2
For a set of N individuals indexed by i, consider a dichotomous (0,1) exposure Xi, dichotomous 3
(0,1) outcome Yi. Assume a closed cohort with no loss to follow-up, all subjects followed for the 4
same time period. Potential outcomes are 𝑌𝑖𝑋=𝑥, abbreviated as 𝑌𝑖𝑥; in this case there are two 5
potential outcomes 𝑌𝑖1 and 𝑌𝑖0. We define the average risk difference as 6
7
∑𝑌𝑖1 𝑁
𝑁
𝑖=1
− ∑𝑌𝑖0 𝑁
𝑁
𝑖=1
= 𝜇(𝑌1) − 𝜇(𝑌0), 8
9
which compares the mean potential outcome (which in this case is a risk) had all N individuals 10
been exposed (X=1) to the mean potential outcome had all N been unexposed (X=0). For 11
definitional simplicity, we hereafter assume the target population is the set of N individuals, and 12
thus that the sample- and population average risk differences are the same.
13 14
We define a population attributable risk difference as 15
16
𝜇(𝑌) − 𝜇(𝑌0), 17
18
where 𝜇(𝑌) = ∑𝑁𝑖=1𝑌𝑁𝑖. That is, the population attributable risk difference equals the risk of the 19
outcome given the observed exposure, minus the risk if all had been unexposed. (We can define 20
a complementary population attributable risk difference analogously as 𝜇(𝑌1) − 𝜇(𝑌).) If we 21
assume that 𝑌𝑖1 ≥ 𝑌𝑖0 for all i (for a harmful outcome, no one is harmed by exposure; that is, 22
monotonicity), then the magnitude of the population attributable risk difference will in general be 23
smaller than that of the population average risk difference because (again in general) 𝜇(𝑌1) ≥ 24
𝜇(𝑌) ≥ 𝜇(𝑌0).
25 26
We define a generalized intervention risk difference as 27
28
𝜇(𝑌) − 𝜇(𝑌𝑥∗), 29
30
where x* is a function of the observed exposure x, that is x*=f(x). In the special case where 31
x*=f(x)=0, that is the function removes all exposure, then the generalized intervention risk 32
difference coincides with the population attributable risk difference.
33 34
We define a dynamic intervention risk difference as 35
36
𝜇(𝑌) − 𝜇(𝑌𝑥∗∗), 37
38
where x** is a function of both the observed exposure x and the observed covariate z, that is 39
x**=g(x, z). In the special case where x**=g(x,z) is invariant to z, the dynamic intervention risk 40
difference coincides with the generalized intervention risk difference.
41 42
Others have presented additional useful and related taxonomies of contrasts. In particular, Young 43
et al. describe treatment regimes as dynamic or static (does treatment depend on covariate values 44
or not?), and as deterministic or stochastic (does a participant’s history fully determine their 45
treatment or not?) (1). A mapping to our taxonomy is straightforward. Population (and sample) 46
average causal effects and population attributable effects are both static and deterministic.
47
Generalized intervention effects are static and (generally) stochastic, but may be deterministic 48
(including when the generalized intervention effect coincides with the population attributable 49
effect). Dynamic intervention effects are dynamic (hence the name) and often stochastic, but 50
could likewise be deterministic: in addition to the same special case noted for generalized 51
intervention effects, dynamic intervention effects might be deterministic if they removed 52
exposure deterministically from only exposed individuals with a particular set of characteristics.
53 54 55
REFERENCE 56
57
1. Young JG, Hernán MA, Robins JM. Identification, Estimation and Approximation of 58
Risk under Interventions that Depend on the Natural Value of Treatment Using Observational 59
Data. In: Epidemiol Methods: De Gruyter; 2014. p. 1-19.
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