SUPPLEMENTAL MATERIAL
eAppendix 1 - RELATIONSHIPS OF DEFINITIONS 1 AND 1A WITH 2.
We showed in the main text how Definition 2 includes Definition 1 as a special case because a functional defines a population-level parameter1 and how Definition 2 is more general because the distributions can be multivariate, rather than univariate, for vector-valued rather than a scalar counterfactual outcomes (e.g., Example 1, Equation 7) Furthermore, in some situations – such as if stable unit treatment value assumption (SUTVA) doesn’t hold (Appendix, main text) – it may be more natural to consider population parameters which might not easily or naturally be defined using individual-level counterfactuals like Di(e).
Definition 2 extends Definition 1a. To see how Definition 2 includes 1a as a special case, suppose m(e) is the counterfactual value of a population measure m if exposure of all in the target had been set to e. Then the causal effect under Definition 1a is m(1) – m(0). To apply our general Definition (2), consider the (degenerate)
distribution of counterfactual values of m, where m takes on value m(e) if exposure had been set to e with probability 1. Then m(e) trivially parameterizes each distribution, for e= 0,1. We could also consider the
counterfactual value of m to have a non-degenerate probabilistic distribution, in which case our generalization still applies (as a contrast of counterfactual parameters for the two distributions of m).
Definition 2 also generalizes Definition 1a because it can be explicitly based on underlying individual-level outcomes and it explicitly recognizes the possibility that outcomes can be vector-valued (have many components), such as disease status and survival (Di(e), Si(e)) at specified times. See also the main text and, for discussion of individual-level and population-average effects, reference (10) and commentaries.
Note e1: Although not normally applied in this way, Definition 1a can also be used to define a causal prevalence ratio as in Example 3 (main text). Here, we must carefully define the counterfactual population measure m(e).
We obtain the cPR if we define m(e) by setting exposure to e at time t=0, following the cohort until time t=1, then calculating m(e) as the number alive with disease at t=1 divided by the number alive. As required for causal effects, m(e) involves the identical baseline population under two counterfactual conditions, but calculation is multi-step: 1) set exposure at time t=0; 2) follow-up through t=1 to identify the subgroup of survivors, and 3)
calculate the prevalence in that subgroup. The exposure can affect survival, outcome occurrence or disease duration. The underlying, multi-component (vector-valued) outcome used in Definition 2 for the cPR is reflected in Definition 1a by the multi-step procedure for calculating the measure. If m(e) were viewed as applying to the survivors, the basic principle of contrasting outcomes in the same population under two sets of conditions could be violated making such an interpretation inappropriate. If m(e) were viewed as an estimator of exposure effect on incidence (only), that interpretation could also be inappropriate. As in Example 3, m(e) is the effect on prevalence at t=1 in the target.
Note e2. If the counterfactual measures in Definition 1a are viewed as potentially complex with calculation involving several steps as in Note e1 and applied to the same population under different exposures (even though one step is identification of a subgroup), then it overlaps even more with Definition 2. Our contribution is then largely to point out how this complexity is still compatible with the principle that the effects be on the same target population under different exposure conditions. Also note that we refer to “population causal effects”, but perhaps should instead refer to a “measure of a population causal effect” since a particular choice of contrast, parameter, outcome, etcetera in Definitions 1, 1a, and 2 yields just one measure of the exposure’s effects.
Note e3. Although time-varying confounding is not our focus, we note that Definition 2 is consistent with the causal effects often of interest in g-computation and g-estimation2. For example, if a contrast of the quantity on the right side of Equation A2.16 of reference3 is viewed as defining the causal effect of interest for two regimens under correct specification, then the effect is consistent with Definition 2 and is an important, early example of that formulation. Simplifying slightly and using our notation, the multivariate outcome is the vector (𝐿̅ (e), 𝑇𝑖 𝑖(e)) where for individual i 𝐿̅𝑖(e) is the counterfactual covariate experience over time and 𝑇𝑖(e) is the failure time, if exposure (regimen) were set to e (vector-valued). The integral on the right of A2.16 defines a functional mapping the distribution of (𝐿̅ (e), 𝑇𝑖 𝑖(e)) to parameter θ(e) of Definition 2; the tth coordinate of θ(e) is P(Ti(e) > t).
eAppendix 2 – SUPPLEMENTAL EXAMPLES
In the main text, we defined a causal effect as a contrast of parameters for the population distributions of counterfactuals. We illustrated how these parameters can be the average of individual-level, deterministic outcomes (Di(e)) as in Equation 2, or another parameter of their distributions and briefly noted some of the generalizations (Appendix), including how to define effects wherein the exposure effects are reflected by individual-level parameters,4-6 and outcomes can be non-deterministic. We now further discuss and illustrate some of the generalizations.
To illustrate application with non-deterministic outcomes, θ(e) could parameterize the distribution of probabilities pi(e) = P(Di(e) =1), for E set to e. Each individual is conceptualized as having an outcome parameter or
parameters that determines his/her probability or distribution of experiencing certain values of a non-deterministic outcome Di. The probabilities reflect unknown factors and other uncertainties under an assumed model. We should specify, either implicitly or explicitly, the probability distribution or structural equation model that the parameter characterizes (examples below). Functionals of the distributions of pi(e) define population-level parameters, leading naturally to a population causal effect (PCE; Equation 5).
Example e1(causal risk odds ratios): Once we consider Definitions from the both the individual and population
perspectives and those based on outcomes and parameters, the range of potential Definitions increases.
Interestingly, the different effects do not necessarily coincide. Here, we consider four of the possible ways to define a causal risk odds ratio (cROR). Letting pi(e) be P(Di(e)=1) if E were set to e and expectations Ep[.] be for the target population, possible Definitions of the causal ROR include:
1A) cROR = EP[Di(1)]/ (1–EP[Di(1)] ))/ (EP[Di(0)]/ (1–EP[Di(0)])) 2A) cROR = EP[pi(1)]/(1–EP[pi(1)] ))/(EP[pi(0)]/(1–EP[pi(0)])) = r(1)/(1– r(1))/ r(0)/(1– r(0))
3A) cROR = EP[βi(1)]/ EP[βi(0)]
4A) cROR = EP[ψi]
where βi(e) = pi(e)/((1–pi(e)), ψi = (pi(1)/(1–pi(1)))/(pi(0)/(1– pi(0))) and r(e) = EP[pi(e)]. With the individual-level Bernoulli model, the effect in Equation (1A) equals that in (2A), as seen by first taking the expectation E[Di(e)|
pi(e)] which equals pi(e) under that model, and then averaging over the target. If a probabilistic model or structural equation isn’t involved, then summation over the target can replace the expectation. (With a finite target
population and an equal-probability-of-selection model, EP[pi(e)] = ∑𝑁𝑖=1𝑝𝑖(e)/𝑁) and EP[Di(e)] =
∑𝑁𝑖=1𝐷𝑖(e)/𝑁).)
Equation (1A) or (2A) probably corresponds to the way many epidemiologists7,8 conceptualize causal effects involving the ROR measure. Equation (1A) is a PCE defined using individual-level counterfactual outcome events Di(e); Equation (2A) is a PCE defined using individual-level counterfactual parameters pi(e). Under SUTVA, exchangeability and consistency – the corresponding contrasts of observed risks among exposed and unexposed are statistically consistent estimators of the effects.
Equations (3A) and (4A) represent additional, possible ways to define cRORs using alternative, individual-level parameters using the individual-level risk odds (βi(e)) or risk odds ratio (ψi(e)). Both are well-defined if pi(e) is never 0 or 1. For example, if an aspirin regimen (compared to no aspirin), changed individual i’s 5-year
probability of MI occurrence from 0.1 (pi(0) = 0.1) to 0.05 (pi(1) = 0.05), then the causal ROR for this individual (ψi) would be 2.11. Equation (3A) would define a PCE as the contrast of the average βi(1) with the average βi(0), averaged over individuals in the target population. Equation (4A) represents a parameter of the distribution of ICEs. Of course, causal effects defined by Equations (3A) or (4A) will not necessarily equal those defined by Equations (1A) and (2A), depending on the distribution of pi(e).
These examples illustrate how seemingly-similar Definitions of PCEs and average ICEs can differ, even though all involve risk odds ratios in some way. Similar considerations apply to the causal risk ratio. Explicitly defining effects can therefore be important.
Example e2 (networks): We are interested in the effect of a behavioral intervention on sexual networks. Our specific interest centers on effects on certain network measures, such as “betweenness centrality” which is the
mean number of shortest possible paths between a pair of individuals.9,10 A causal effect can be defined as the value, averaged over all pairs, of this measure after implementation of the intervention compared to what its value would have been, had the program not been implemented. The measure is defined in terms of pairs; it not readily calculated, if at all, from a single, scalar-valued, individual-level counterfactual outcome. Rather, it depends in a complex way on the joint distribution of vectors of individual-level counterfactual outcomes (contacts); the causal effect is definable under Definition 2; it’s also definable under Definition 1a using complex, counterfactual measures.
Example e3 – (conditional risk ratios). In this example, we quantify the possible magnitude difference between
the conditional risks similar to those labeled cCRRs and cCRRf in Example 1b. We assume that exposure is never beneficial.
As we did previously11, we consider death in each of three intervals, from baseline (time 0) to time 1, from time 1 to time 2, or time 2 or later. (The causal effect cCRRs here corresponds to the causal effect under Definition 2 of our previous paper11). We use the potential outcome model with potential outcome types as summarized in eTable 1. To simplify the example, we assume that exposure effects with this model are monotonic – exposure is never beneficial. However, this simplification is not necessary and is not used in the main text. Rather, it
simplifies the example and makes quantification easier and more transparent.
As derived previously,11 we can obtain the conditional risk ratio cCRRs in the target of survivors by first
determining the conditional risk for the exposed survivors; this is the proportion of them who would die in period 2: (pD +pF) /(pD +pE +pF). The counterfactual condition risk, had they been unexposed, is the proportion of these same exposed survivors who would have died in period 2 had they been unexposed: pD/(pD+ pE +pF). Thus, cCRRs = (pD+pF)/pD.
On the other hand, we can obtain the conditional risk ratio cCRRf, by first determining the conditional risk for the exposed baseline population. This conditional risk is the proportion of survivors who would die in period 2: (pD
+pF) /(pD +pE +pF), in agreement with the above. The counterfactual conditional risk for the target (exposed
baseline population) had they been unexposed is the proportion of them who would die in period 2 had they been unexposed: (pB+pD)/(pB+pC+pD+pE+pF). Thus, cCRRf = [(pD +pF) /(pD +pE +pF)]/[(pB+pD )/(pB+pC+pD+pE+pF)].
Comparing the expression for cCRRs with that for cCRRf shows that these causal effects can be substantially different. For example, substitution of pB = pC = pD = pE =pF = 0.1 gives cCRRs = 2.0 and cCRRf = 1.67, a modest but potentially meaningful difference.
eTable 1, Potential Outcome Types, and frequencies in Population (modified slightly from Flanders and Klein11).
Potential Outcome Type
Population Proportion, among those with E=1a
Population Proportion, among those with E=0b
Counterfactual time to Death:
(Ti,e)c
Comment
Ti,1 Ti,0
A pA qA 1 1 E has no effect
B pB qB 1 2 E advances death from time 2 to 1
C pC qC 1 3 E advances death from time 3 to 1
D pD qD 2 2 E has no effect
E pE qE 3 3 E has no effect
F pF qF 2 3 E advances death from time 3 to 2
G pG qG 2 1 Excluded, E beneficial
H pH qH 3 1 Excluded, E beneficial
I pI qI 3 2 Excluded, E beneficial
aproportion of the exposed population (E=1) with each counterfactual response type; pA is the proportion with response type A; pB is the proportion with type B, and so on. pA+ ...+ pF =1.
bproportion of the unexposed population (E=0) with each counterfactual response type; qA is the proportion with response type A; etc. However, the q’s aren’t needed since the target is the exposed baseline population.
c Ti,e is the counterfactual period at death for individual i, if exposure were set to e.
Example e4 (prevalence): This example provides another illustration of using prevalence in defining a causal
effect measure. One of a town’s two drinking-water plants was inadvertently cross-contaminated by sewage, after which 25% of the residents served by each plant drank bottled, rather than tap water. Public health officials want to estimate the effect of the contamination on prevalence of gastrointestinal illness (GII) among the town’s residents seven days after the contamination (prevalence is defined as the proportion of residents who, on day 7,
have symptoms meeting their Definition of GII). This effect is a PCE, defined as the prevalence of GII among all residents, compared to what the prevalence would have been absent the contamination. Assuming exchangeability and no ‘interference’, it’s naturally estimated by comparing the observed prevalence among all residents with that among residents served by the uncontaminated plant. Recovery time and fatalities affect prevalence, but aren’t sources of bias: they are part of the causal process. However, if the contrast of observed prevalences were used to estimate the effect of contamination on GII occurrence (i.e., an occurrence effect defined using appropriate contrasts of counterfactual risks), then recoveries and fatalities might be sources of bias. Here the prevalence and risk contrasts (cRR) provide overlapping information, but that information can differ because illness can resolve.
For example, if GII resolved completely the prevalence contrast would correctly indicate no residual effect whereas the risk contrast, which addresses a different effect, might not. Parenthetically, if being served by the contaminated plant was a surrogate for drinking contaminated water, measurement error would be present because some residents drank bottled water, whereas this wouldn’t introduce measurement error for estimating the effect of being served by the contaminated plant – it’s part of the causal pathway.
eAPPENDIX 3 – ADDITIONAL DEFINITIONS, STRUCTURE, CONSIDERATIONS
In eAppendix 3, we consider additional statistical structure that may be required; define bias and “statistically consistent”; and, discuss certain implications of these concepts.
Definitions of expectations and probabilities require a statistical framework or model. Use of parameters in causal effect Definitions (effects on parameters) also requires a distribution or perhaps structural equations that the parameter characterizes. In particular, we can define EP[Di(e)], EP[Di|Ei=e], EP[ri(e)] and associated probabilities as simple averages for the target or specified population unless otherwise specified, viewing individuals as selected randomly from that population. If Di(e), Di and ri(e) are viewed as random we can first take expectations using the assumed probabilistic structure (as in Example e1), then average over the population.
Statistical inferences in observational studies may require extending this view to include additional, assumed structure12. For example, individuals in the study could be viewed as being sampled from a near-infinite super- population in which exposure was randomly assigned.6 Alternatively if Di, Di(e) and ri(e) are viewed as
probabilistic, ∑Di for example might be viewed as a binomial random variable. Similarly, if the target population is viewed as a sample from a larger group of populations, then we could define a PCE using contrasts of θ(e), where θ(e) parameterizes the distribution of a population counterfactual outcome, measure or parameter of interest. Our main goals concern effect definitions, but our discussion remains relevant with additional structure.
An estimator 𝜃̂𝑛 of a parameter θ based on a sample of size n is statistically consistent if and only if, for every ε >
0, lim
𝑛→∞𝑃 (|𝜃̂ − 𝜃| > 𝜀) =0. We also refer to statistically inconsistent estimators as “biased”. An estimator of a 𝑛 counterfactual, 𝜃(e)̂ (e.g., the observed risk among those with E=e), is an unbiased estimator of the
counterfactual mean 𝐸𝑃[𝜃𝑖(e)] if and only if: 𝐸𝑃[𝜃(e)̂] = 𝐸𝑃[𝜃𝑖(e)], for e =0,1. This usage is compatible with that of Hernán and Robins7 whose definition of no (systematic) bias of an estimator of a CE using observed risks is that: EP[Di|Ei=1] – EP[Di|Ei=0] = EP[Di(1)] – EP[Di(0)] and that EP[Di|Ei=1] / EP[Di|Ei=1] = EP[Di(1)] / EP[Di(0)].
Assuming no division by 0, these two requirements are equivalent to E𝑃[D(e)] = E𝑃[D|E = e] for e = 0,1.13 Both bias and “statistically consistent” remain well-defined even if individuals, their counterfactual outcomes and parameters are viewed (the default here) as being selected randomly from a finite cohort (with equal probability, without replacement, and 𝑛 → ∞ taken to mean n =N), rather than from an infinite super-population. We reserve
“consistent” for statistically consistent, and “consistency” for counterfactual-model consistency (i.e., Di(e) = Di if Ei=e).
Certain estimators like the observed ROR may not have a well-defined expectation (e.g., the estimator’s denominator can be 0 especially if the sample is small fraction of the target). However, these estimators can be statistically consistent since the denominator need not be 0 for large n. For example, if the exposure-specific observed risk in the target, EP(D|E=e), equals EP[D(e)] for e= 0,1 for n = N (e.g., the entire cohort or large N), then the ROR-estimator calculated as PP(D|E=1)/(1-PP(D|E=1))/ PP(D|E=0)/(1-PP(D|E=0)), is a consistent estimator of the corresponding causal contrast. Here and throughout, we assume no misclassification or non- ignorable missingness and, if a stratified analysis is used positivity.7
We now provide conditions under which Equation (8), counterfactual-model consistency and SUTVA are sufficient for observed contrasts to be statistically consistent estimators of causal effects; we assume the target is the study population at baseline and no misclassification, missingness or lost to follow-up (when applicable).
Others7,13 have shown that consistency, exchangeability and SUTVA imply contrasts of population-averages of observed, individual-level outcomes in the target are statistically consistent estimators of the corresponding contrasts of population-averages of counterfactual individual-level outcomes. To show conditions under which a PCE estimator is statistically consistent, we suppose that known, continuous functions f(.,.) and g(.) express the PCE as a contrast of the average (expectation) of counterfactual, individual-level outcomes (e.g., as in Equation 7) and that observed averages are used to estimate the counterfactual ones. If exchangeability, consistency and SUTVA hold for the estimators of each argument of g(.), then the contrast based on this function evaluated using the average of observed outcomes is a consistent estimator of the PCE. In particular, we suppose the causal effect f(𝜽(1), 𝜽(0)) = θ(1) – θ(0), where θ(e) is defined as g(EP[Ai(e)], EP[Bi(e)],..., EP[Ni(e)]) and where Ep[.] is the average or expectation for the target. If the stated assumptions hold for each outcome Ai(e), Bi(e), ..., Ni(e), then g(𝐴̅1, 𝐵̅1, ..., 𝑁̅1 – g(𝐴̅0, 𝐵̅0, ..., 𝑁̅0) is a consistent estimator of θ(1) – θ(0), where 𝐴̅e, 𝐵̅e, .., 𝑁̅e, are the observed averages among those with E = e. The proof follows essentially by definition: exchangeability, consistency, and SUTVA imply 𝐴̅e, 𝐵̅e, .., 𝑁̅e are consistent, and so these estimators converge in probability to EP[Ai(e)], EP[Bi(e)], ..., EP[Ni(e)], respectively for e= 0,1. (For an appropriate, infinite super-population, the law of large numbers would imply convergence in probability.) This implies g(𝐴̅e, 𝐵̅e, .., 𝑁̅e) converges in probability to g(EP[Ai(e)], EP[Bi(e), ..., EP[Ni(e)]) = θ(e). By subtraction, g(𝐴̅1, 𝐵̅1, ..., 𝑁̅1) – g(𝐴̅0, 𝐵̅0, ..., 𝑁̅0) converges in probability to the PCE, proving the claim. (Here and throughout, a bold font indicates that the term is a vector.)
eAPPENDIX 4 - SUPPLEMENTAL PROOF
Claim: the observed conditional risk ratio (CRR) is a statistically consistent estimator of the cCRRf in a randomized clinical trial.
Proof: By definition cCRRf = 𝜃(1)/𝜃(0), where 𝜃(e) = ∑𝑖∈𝑃𝑓D𝑖(e)/ ∑𝑖∈𝑃𝑓𝑆𝑖(e) and Di(e) and Si(e) are counterfactual occurrence and survival, respectively. Here, 𝜃(e) is a parameter of the empiric distribution of (D𝑖(e), S𝑖(e)), but we could also conceptualize a probabilistic model for subject selection). In an RCT, we expect exchangeability7 (e.g., no confounding) to hold for D𝑖(e) and S𝑖(e). We can write exchangeability7,14 as D𝑖(e) ∐ 𝐸 and S𝑖(e) ∐ 𝐸, for E = 0,1,where ∐ refers to independence, that is assumed to hold under
hypothetical repetitions of random exposure assignment and perfect compliance.
By independence under exchangeability, plus model consistency7,15 of the counterfactual model:
1) 𝐸[∑𝑖∈𝑃𝑓D𝑖(1)] =E[∑𝑖∈𝑃𝑓𝐷𝑖(1)|𝐸 = 1] = E[∑𝑖∈𝑃𝑓𝐷𝑖|𝐸 = 1]
2) 𝐸[∑𝑖∈𝑃𝑓D𝑖(0)] = 𝐸[∑𝑖∈𝑃𝑓D𝑖(0)|𝐸 = 1] =𝐸[∑𝑖∈𝑃𝑓D𝑖(0)|𝐸 = 0] = E[∑𝑖∈𝑃𝑓𝐷𝑖|𝐸 = 0]
3) 𝐸[∑𝑖∈𝑃𝑓S𝑖(1)] =E[∑𝑖∈𝑃𝑓𝑆𝑖(1)|𝐸 = 1] = E[∑𝑖∈𝑃𝑓𝑆𝑖|𝐸 = 1]
4) 𝐸[∑𝑖∈𝑃𝑓S𝑖(0)] = 𝐸[∑𝑖∈𝑃𝑓S𝑖(0)|𝐸 = 1] =𝐸[∑𝑖∈𝑃𝑓S𝑖(0)|𝐸 = 0] = E[∑𝑖∈𝑃𝑓𝑆𝑖|𝐸 = 0]
In Equations 1-4, the expectation is over hypothetical repetitions of treatment assignment and N is the size of the groups assigned to receive exposure (and no exposure). In Equation 1, the first equality follows by
exchangeability (by randomization, E is independent of the counterfactual value of D); the second equality follows by counterfactual-model consistency. Equalities in the other equations follow for similar reasons. For large N, the observed risks and proportions surviving: ∑𝑖∈𝐸+D𝑖/𝑁, ∑𝑖∈𝐸−D𝑖/𝑁, ∑𝑖∈𝐸+S𝑖/𝑁, and ∑𝑖∈𝐸−S𝑖/𝑁 will be close (with high probability; i.e. statistically consistent) to their respective expectations
E[∑𝑖∈𝑃𝑓𝐷𝑖|𝐸 = 1], E[∑𝑖∈𝑃𝑓𝐷𝑖|𝐸 = 0], E[∑𝑖∈𝑃𝑓𝑆𝑖|𝐸 = 1], and E[∑𝑖∈𝑃𝑓𝑆𝑖|𝐸 = 0] respectively. Therefore, substituting these consistent estimators into the expression for cCRRf (after explicitly including the expectation) shows that the ratio of observed conditional risks ∑𝑖∈𝐸+D𝑖/𝑁 ÷ ∑𝑖∈𝐸+S𝑖/𝑁, and ∑𝑖∈𝐸−D𝑖/𝑁 ÷ ∑𝑖∈𝐸−S𝑖/𝑁 is
a consistent estimator of cCRRf, where summation over “𝑖 ∈ 𝐸 +“ and “𝑖 ∈ 𝐸 −“ means summation over those randomized to exposure or non-exposure.
SUPPLEMENTAL EQUATIONS FOR EFFECTS IN TABLE OF MAIN TEXT eTable 2. Supplemental Equations for Effects in Table of Main Text
Label (Abbreviation) Equation Defining the Effect causal risk ratio (cRR) 1 cRR = ∑𝑖∈𝑇𝐷𝑖(1)/ ∑𝑖∈𝑇𝐷𝑖(0) causal conditional risk ratio
(cCRRs)
2 cCRRs = ∑𝑖∈𝑃𝑠D𝑖(1)/ ∑𝑖∈𝑃𝑠D𝑖(0)
causal conditional risk ratio (cCRRf) 3 cCRRf = (∑𝑖∈𝑃𝑓D𝑖(1)/ ∑𝑖∈𝑃𝑓S𝑖(1)) / (∑𝑖∈𝑃𝑓D𝑖(0)/ ∑𝑖∈𝑃𝑓S𝑖(0)) causal prevalence ratio (cPR) 4 cPR = (∑𝑖∈𝑇B𝑖(1)/∑𝑖∈𝑇S𝑖(1))/(∑𝑖∈𝑇B𝑖(0)/∑𝑖∈𝑇S𝑖(0)) causal effects on time patterns -CEt 5 CEt = (𝐸[𝐶𝑖,1(1)]/𝐸[𝐶𝑖.1(0)], … , 𝐸[𝐶𝑖,6(1)]/𝐸[𝐶𝑖,6(0)]) total causal effect (TCE) 6 TCE = (∑𝑖∈𝑇D𝑖(1)/ ∑𝑖∈𝑇S𝑖(1)) – (∑𝑖∈𝑇D𝑖(0)/ ∑𝑖∈𝑇S𝑖(0)) causal risk odds ratio (cROR) 7 cROR = EP[Di(1)]/ (1–EP[Di(1)] ))/ (EP[Di(0)]/ (1–EP[Di(0)])) causal effect on betweenness
centrality measure5
8 betweenness centrality measure if all in target were exposed compared to that measures if unexposed5
1the target T is the cohort of interest, defined at baseline (time 0).
2the summation is over all in the target Ps (defined in text); Di(e) denotes the counterfactual individual-level outcomes- developing the outcome D in the interval 6 to 12 months if E had been set to e.
3the summation is over all in the target Pf (defined in text); Di(e) and Si(e) denote the counterfactual individual- level outcomes: developing the outcome D in the interval 6 to 12 months and remaining alive at 6 months, respectively, if E had been set to e.
4the summation is over all in the target T defined at baseline (time 0); Bi(e) and Si(e) denote the counterfactual individual-level outcomes: being alive with an ulcer, and remaining alive, respectively, if E had been set to e.
5this effect is measured as a vector of contrasts of average counterfactual outcomes at each time
6TCE approximates a causal conditional risk difference, analogous to cCRRf , but using a difference contrast; T is the target, defined at baseline; Di(e) and Si(e) denote the counterfactual individual-level outcomes: developing the outcome D in the period of interest and remaining alive until the beginning of that period, respectively.
7EP[.] is the expectation over the target T for a presumed statistical model; Di(e) is the counterfactual outcome for individual i if E were set to e; see also eAppendix 1 for alternative definitions that are possible and consistent with Definition 2, Equation 5.
8Betweenness centrality measure as described in Example A2.
REFERENCES
1. Bickel PJ, Klaassen CAJ, Ritov Y, Wellner JA. Efficient and Adaptive Estimation of Semiparametric Models.
New York: Springer-Verlag; 1993.
2. Robins J. A new approach to causal inference in mortality studies with sustained exposure periods- application to control of the health worker survivor effect. Mathematical Modeling 1986;7:1393-515.
3. Robins JM, Blevins D, Ritter G, Wulfsohn M. G-estimation of the effect of prophylaxis therapy for Pneumocystis carinii pneumonia on the survival of AIDS patients. Epidemiology 1992:319-36.
4. Greenland S, Brumback B. An overview of relations among causal modelling methods. Int J Epidemiol 2002;31:1030-7.
5. Greenland S, Robins J, Pearl J. Confounding and collapsibility in causal inference. Stat Sci 1999;14:29-46.
6. Hernán MA. A Definition of causal effect for epidemiology. J Epidemiol Community Health 2004;58:265- 71.
7. Hernan MA, Robins J. Causal Inference. 2013 ed2013.
8. Rothman KJ, Greenland S, Lash TL. Modern epidemiology. 3rd ed. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins; 2008.
9. De P, Singh AE, Wong T, Yacoub W, Jolly A. Sexual network analysis of a gonorrhoea outbreak. Sexually transmitted infections 2004;80:280-5.
10. Stephenson K, Zelen M. Rethinking centrality: Methods and examples. Social Networks 1989;11:1-37.
11. Flanders WD, Klein M. Properties of 2 Counterfactual Effect Definitions of a Point Exposure.
Epidemiology 2007;18:453-60 10.1097/01.ede.0000261472.07150.4f.
12. Greenland S. Randomization, statistics, and causal inference. Epidemiology 1990;1:421-9.
13. Flanders W, Eldridge R. Summary of Relationships between exchangeability, biasing paths and bias.
European J Epidemiol 2014;accepted.
14. Flanders WD, Eldridge RC. Summary of relationships between exchangeability, biasing paths and bias.
European Journal of Epidemiology 2014:1-11.
15. Hernán MA, VanderWeele TJ. Compound treatments and transportability of causal inference.
Epidemiology (Cambridge, Mass) 2011;22:368.
