# Completion Potentials of Sufficient Component Causes

Many epidemiologists are familiar with Rothman's sufficient component cause model. In this paper, I propose a new index for this model, the completion potential index I show that, with proper assumptions (monotonicity, independent competing causes, proportional hazards), completion potentials for various classes of sufficient causes are estimable from routine epidemiologic data (cohort, case-control or time-to-event data). I discuss the advantage of the completion potential index over indices of rate ratio, rate difference, causal-pie weight, population attributable fraction, and attributable fraction within the exposed population. Hypothetical and real data examples are used. The completion potential index proposed here allows better characterization of complex interactive effects of multiple monotonic risk factors.

Supplemental Digital Content is available in the text.

From the Research Center for Genes, Environment and Human Health and Institute of Epidemiology and Preventive Medicine, College of Public Health, National Taiwan University, Taipei, Taiwan.

Submitted 10 January 2011; accepted 29 November 2011.

Supported by a grant from the National Science Council, Taiwan (NSC 99-2628-B-002-061-MY3). The author reported no other financial interests related to this research.

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article (www.epidem.com). This content is not peer-reviewed or copy-edited; it is the sole responsibility of the author.

Correspondence: Wen-Chung Lee, Rm. 536, No. 17, Xuzhou Rd., Taipei 100, Taiwan. E-mail: wenchung@ntu.edu.tw.

Rothman's sufficient component cause model^{1},^{2} ^{p. 5–9, 80–82} is familiar to most epidemiologists. A sufficient cause contains a combination of component causes. The model is deterministic in nature, positing that the disease is destined to occur once a sufficient cause is completed. The model also has a stochastic part. The presence of a single risk factor is not sufficient for the disease—all the complement causes (the other component causes in the same sufficient cause) must also appear to complete a sufficient cause. The occurrence of complement causes is a stochastic process that adds uncertainty to the model. Furthermore, a particular risk factor need not be present—there can be many sufficient causes for the same disease, some of which do not require the presence of that risk factor.

Recently, Hoffmann et al^{3} and Liao and Lee^{4} showed how to quantify sufficient causes using real data. In this paper, I propose a new index for the sufficient component cause model, the completion potential index. I show that with proper assumptions, completion potentials for various classes of sufficient causes are estimable from routine epidemiologic data (cohort, case-control or time-to-event data). In addition, I discuss the advantages of the completion potential index over other indices, using hypothetical and real data examples.

## METHODS

### Two Risk Factors

Consider the relation of 2 dichotomous risk factors (*X* _{1} = *x* _{1}, *X* _{2} = *x* _{2}, with *x* _{1}, *x* _{2} ∈ {0,1}) and a disease in a follow-up of a population. I assume that people do not change their risk-factor status during the follow-up. Let Rate^{profile=x1,x2}(*t*) denote the (instantaneous) disease rate of people in the population with a risk-factor profile of *X* _{1} = *x* _{1}, *X* _{2} = *x* _{2} at follow-up time *t*, and assume no confounding^{2} ^{p. 67–68, 385–386}: Rate^{SET[profile=x1,x2]}(*t*) = Rate^{profile=x1,x2}(*t*). This assumption dictates that Rate^{SET[profile=x1,x2]}(*t*), which is the “counterfactual” disease rate in the population if all of its people assumed the risk-factor profile of *X* _{1} = *x* _{1}, *X* _{2} = *x* _{2}, can be estimated directly from Rate^{profile=x1,x2}(*t*), an observable “factual” disease rate. Assume also that risk factors have a negligible impact on person-time in this population. With these 2 assumptions, an association measure such as a rate ratio or a rate difference can be interpreted as a causal effect measure.^{2} ^{p. 52–53}

Furthermore, I propose 2 assumptions on Rothman's sufficient component cause model per se, namely, monotonicity and independent competing causes. The first assumes that the effects of *X* _{1} and *X* _{2} are increasing monotonically such that neither of them can ever be preventive to disease or interact with other factors antagonistically (Rate^{profile=x1,x2}(*t*) ≥ Rate^{profile=x′1,x′2}(*t*) whenever *x* _{1} ≥ *x*′_{1} and *x* _{2} ≥ *x*′_{2}).^{2},^{4},^{5} In other words, component causes such as *X* _{1} = 0 and *X* _{2} = 0 cannot be present in any class of sufficient causes. (For a monotonic preventive factor, such as exercise, one can reverse the coding [lack of exercise] and treat it as a monotonic risk factor.) With 2 risk factors, there are a total of 2^{number of risk factors} = 2^{2} = 4 possible classes of sufficient causes (Fig. 1). Without the monotonicity assumption, the number of possible classes of sufficient causes (3^{number of risk factors} = 3^{2} = 9) would be larger than the degrees of freedom (with 2 risk factors, there are at most 4 types of risk-factor profiles for people in the population), creating a nonidentifiability problem.

The 4 classes of sufficient causes are referred to here as (1) the all-*U* (all-unknown) class: containing neither *X* _{1} and *X* _{2} as its component causes; (2) the *X* _{1} class: containing *X* _{1} but not *X* _{2} as its component causes; (3) the *X* _{2} class: containing *X* _{2} but not *X* _{1} as its component causes; and (4) the *X* _{1} and *X* _{2} interactive (synergistic) class: containing both *X* _{1} and *X* _{2} as its component causes (Fig. 1). Here, the *X* _{1} and *X* _{2} are known risk factors, and there are 4 distinct unknowns in the 4 classes of sufficient causes, each of which may contain a multitude of unknown constituent factors of its own. “Interaction” or “synergism” means that 2 (or more) known risk factors are present in the same sufficient cause.

As mentioned earlier, *X* _{1} and *X* _{2} are treated as fixed values. The arrivals of unknown complement causes are treated as random events. The completion rate for a class of sufficient causes is defined as the arrival rate of the unknown complement causes in that particular class. Let Rate_{class=0,0}(*t*) denote the (instantaneous) completion rate at follow-up time *t* for the all-*U* class; Rate_{class=1,0}(*t*) for the *X* _{1} class; Rate_{class=0,1}(*t*) for the *X* _{2} class; and Rate_{class=1,1}(*t*) for the *X* _{1} and *X* _{2} interactive class.

The second assumption on Rothman's model is that the arrivals of the 4 classes of unknowns are independent of one another: this is the independent competing-cause model^{6} (or a rare-disease version of the simple independent action model^{7}). This assumption implies a simple linear relation between the completion rates and the disease rates. First, note that people exposed to neither of the 2 risk factors can develop the disease only through the completion of the all-*U* class:

People exposed to *X* _{1} but not to *X* _{2} can develop the disease through the completion of either the all-*U* class or the *X* _{1} class:

In a similar vein,

and

Equations (1)–(4) can be cast into a single equation:

where the 2 risk factors are indexed by *i*.

The rate ratio (RR) for people with a particular risk-factor profile is defined as the ratio of the disease rate for those people and the background disease rate (disease rate for people exposed to neither of the risk factors), that is,

The completion potential (CP) index for a class of sufficient causes is defined as the ratio of the completion rate of that particular class and that of the all-*U* class, that is,

From Equations (5)–(7), there is also a simple linear relation between completion potential indices and rate ratio indices:

Table 1 presents a hypothetical population in 1 year. At that particular time in that population, the completion rates (10^{−5}) are 12.0 (from Eq1) for the all-*U* class; 13.2 − 12.0 = 1.2 (from Eq2 − Eq1) for the *X* _{1} class; 72.0 − 12.0 = 60.0 (from Eq3 − Eq1) for the *X* _{2} class; and 85.2 − 12.0 − (13.2 − 12.0) − (72.0 − 12.0) = 12.0 (from Eq4 − Eq1 – [Eq2 – Eq1] – [Eq3 – Eq1]) for the *X* _{1} and *X* _{2} interactive class. The completion potentials are therefore (from Eq7): 12.0/12.0 = 1.0 for the all-*U* class; 1.2/12.0 = 0.1 for the *X* _{1} class; 60.0/12.0 = 5.0 for the *X* _{2} class; and 12.0/12.0 = 1.0 for the *X* _{1} and *X* _{2} interactive class (also referred to as the “relative excess rate due to interaction” [RERI]).^{8} One can also check the linear relation between completion potentials and rate ratios in Equation (8) for this example.

Although *X* _{1} is a risk factor for the disease, its completion potential is rather low (CP = 0.1). This implies that the *X* _{1} factor in itself plays a minor role in disease causation: the *X* _{1} class of sufficient causes (which requires the presence of *X* _{1}) is only one-tenth as likely to cause disease as the all-*U* class (which does not require the presence of *X* _{1}). The *X* _{2} factor plays a bigger role by comparison. It has a completion potential of 5.0, implying that the *X* _{2} class of sufficient causes is 5 times as likely to cause disease as the all-*U* class. We also see that synergism of *X* _{1} and *X* _{2} (simultaneous presence of both factors) represents yet another pathway to disease. Such an *X* _{1} and *X* _{2} interactive class of sufficient causes is as likely to cause disease as the all-*U* class.

### General Formulation

eAppendix 1 (http://links.lww.com/EDE/A579) shows that, under the assumptions of monotonicity and independent competing causes, Equation (5) earlier in the text can be extended to the general situation of *K* risk factors (indexed by *i* = 1, …, *K*) and becomes

Note that the classes of sufficient causes are represented by the binary strings, *c* _{1}, …, *cK* ∈ {0, 1}, such that a class contains the *i*th risk factor as one of its component causes if *ci* ≠ 0.

Next, assume proportional hazards (rates):

where the disease rates themselves depend on *t*, but the rate ratios are time invariant. This assumption is commonly invoked in analyzing epidemiologic and biomedical data. Based on this assumption, the ratio scale (ie, rate ratios) is a natural scale to gauge exposure-disease relations. In this context, a linear model means that the rate ratios themselves follow a linear relation, and not that a rate difference scale is being used to gauge exposure-disease relations.

Comparing Equations (9) and (10), note that the completion potentials must also be time-invariant and correspond to the beta coefficients of the following Cox-type model:

with CP_{class=c1,…,cK}(*t*)=CP_{class=c1,…,cK}=β_{c1,…,cK}, for *c* _{1},…,*c* _{K}∈{0,1}.The regressand in Equation (11), the disease rate, is in a linear rather than the usual log-linear scale. The regressors in Equation (11)

represent the main-effect terms (if containing only one of *x* _{1},…,*x* _{K}) or the interaction terms (if being a product of at least two of *x* _{1},…,*x* _{K}). To fit Equation (11), a nonlog-linear Cox model, one can use the SAS codes provided by Langholz and Richardson.^{9} Because a negative completion potential (and a negative completion rate) is inadmissible, one should further constrain all instances of β_{c1,…,cK} in Equation (11) to be nonnegative. This can be achieved through a model-building procedure described in a study by Liao and Lee.^{4} Note, however, that we do not need to follow the hierarchy principle (that an interaction term can be included in the model only if all its corresponding main-effect terms are included).

The model can also be extended to deal with risk factors in an ordinal scale with an arbitrary number of categories and a continuous scale, if it can be properly categorized (see eAppendix 2, http://links.lww.com/EDE/A579).

### Estimation in Case-Control Studies

Note that Equation (11) earlier in the text is a special case of the multiplicative-intercept model.^{2} ^{p. 430–431} The intercept part, Rate^{profile=0,…,0}(*t*), is time dependent and is nuisance from the perspective of sufficient component cause modeling. For a case-control study with time matching (a nested case-control study in a closed cohort or a case-control study with density sampling in an open cohort), one can use the conditional likelihood approach (also see Langholz and Richardson^{9}) to eliminate the nuisance Rate^{profile=0,…,0}(*t*) and obtain estimates of the β_{c1,…,cK}'s (and hence completion potentials).

If the controls are a random sample of all noncases in the population, rate ratios can be approximated by odds ratios under the rare-disease assumption.^{2} ^{p. 114, 125} In that case, Equation (11) becomes a linear model for odds ratios:

This is also a multiplicative-intercept model, and thus the completion potentials are again estimable as the beta parameters in Equation (12) (see Liao and Lee^{4}).

### Relation With Causal-pie Weights and Attributable Fractions

The individual-based causal-pie weights (CPW) index, CPW_{class=c1,…,cK} ^{profile=x1,…,xK}, can be defined as the probability that a diseased person with a risk-factor profile of *x* _{1}, …, *xK* had developed the disease because of the completion of the class of sufficient causes with a binary string of *c* _{1}, …, *cK*. By Bayes theorem, the individual-based causal-pie weight indices can be calculated from the completion potential indices or from the beta coefficients:

where *I*(statement), an indicator function, has a value of 1 if the statement is true, and a value of 0 if otherwise. (The statement, “*xi* ≥ *ci* for all *i*,” in Equation (13) is for defining a “completable class,” as presented in eAppendix 1, http://links.lww.com/EDE/A579.) For any diseased person with *x* _{1}, …, *xK* ∈ {0, 1}, the total 2^{K} individual-based causal-pie weights sum to one, that is,

I define the population-wide causal-pie weight, CPW_{class=c1,…,cK} ^{population}, as the probability that a randomly chosen diseased person in the population had developed the disease because of the completion of the class of sufficient causes with a binary string of *c* _{1}, …, *cK*. Assume that a population has a total of *n* diseased persons (indexed by *j*). For each and every class of sufficient causes with *c* _{1}, …, *cK* ∈ {0, 1}, the population-wide causal-pie weight is the average of the *n* individual-based causal-pie weights of the same class, that is:

The total 2^{K} population-wide causal-pie weights also sum to one, that is,

The attributable fractions can be calculated from causal-pie weights. A population attributable fraction (PAF) associated with a target profile of *t* _{1}, …, *tK* ∈ {0, 1} is

An attributable fraction within the exposed population (AFE) with a risk-factor profile of *x* _{1}, …, *xK* is

In Table 2, I present completion potentials, causal-pie weights and population attributable fractions for the hypothetical population in Table 1.

### Comparison With Other Indices

The completion potential index allows the effects of multiple risk factors to be characterized from a new perspective—in terms of relative likelihoods of causing disease for various classes of sufficient causes. As these classes of sufficient causes represent either the independent effect of a risk factor or the interaction effect of a group of risk factors, completion potentials thus help disentangle the complex relation between multiple risk factors and the disease. In contrast, the traditional indices of rate ratios and rate difference (such as in Table 1) by themselves do not provide such information. (An inspection of rate ratios and rate differences in Table 1 could misleadingly suggest that it is *X* _{2} that really counts, and that *X* _{1} by contrast can practically be ignored, either for its independent effect or its interaction effect.)

As defined before, the completion potential index for a class of sufficient causes is the ratio of the completion rate of that particular class and that of the all-*U* class. Therefore, if the completion rates (arrival rates of the unknown complement causes) in a different population are not the same as (and also not of a certain constant multiple of) the corresponding completion rates in the study population, the completion potentials for the corresponding classes of sufficient causes in the 2 populations will also be different. By comparing completion potentials across populations, one can thus glean important information about the relative abundances of the unknown complement causes in different populations. By contrast, the traditional indices of rate ratios and rate differences do not readily lead to inferences about the unknown complement causes.

Individual-based causal-pie weights are simple functions of completion potentials, and therefore inherit all the aforementioned desirable properties of completion potentials. Moreover, an individual-based causal-pie weight has a straightforward probabilistic interpretation: it is the probability that a diseased person in the study population with a certain risk-factor profile had developed the disease because of the completion of a certain class of sufficient causes. However, the set of the individual-based causal-pie weights for any diseased person cannot provide a complete picture about disease potentials for all possible classes of sufficient causes: an individual-based causal-pie weight for a particular class is necessarily zero, if the class contains as one of its complement causes a risk factor that the person is not exposed to (the class is not a completable class for that individual, as presented in eAppendix 1, http://links.lww.com/EDE/A579). For example, the following individual-based causal-pie weights in Table 2 are zero: CPW_{class=1,0} ^{profile=0,0}, CPW_{class=0,1} ^{profile=0,0}, CPW_{class=1,1} ^{profile=0,0}, CPW_{class=0,1} ^{profile=1,0},CPW_{class=1,1} ^{profile=1,0}, CPW_{class=1,0} ^{profile=0,1}, and CPW_{class=1,1} ^{profile=0,1}.

Population-wide causal-pie weights are the averages of the individual-based causal-pie weights of all the diseased persons in a population. A population-wide causal-pie weight represents the proportion of diseased persons in the population that had developed the disease because of the completion of a certain class of sufficient causes. eAppendix 3 (http://links.lww.com/EDE/A579) shows that the value of a population-wide causal-pie weight for a particular class depends on 2 factors: (i) the proportion of the population exposed to no fewer than those risk factors required by that class, and (ii) the completion potential for that class in the population. The (i), mentioned earlier, defines those people in the population who are susceptible to that particular class. With (ii), mentioned earlier, the disease-causing potential of that class is factored in. The population-wide causal-pie weight is thus a composite index. When one encounters a low population-wide causal-pie weight, it may imply that the class referred to by the index has a low likelihood of causing disease. Alternatively, it may simply be that the study population has few people exposed to those (or more) risk factors required by the class. For example, in Table 2, the population-wide causal-pie weight for the *X* _{1} and *X* _{2} interactive class is rather low (3.8%). But one should not mistake this as implying a low likelihood of causing disease for the *X* _{1} and *X* _{2} interactive class (its completion potential is actually not low; it equals that of the all-*U* class). The low value of this population-wide causal-pie weight is due to the low prevalence of people exposed to both *X* _{1} and *X* _{2} (10%) in the hypothetical population (Table 1). For a typical real-world population, it is often true that only a few people are exposed to multiple risk factors simultaneously. As such, there is a real danger of underestimating the true potentials that multiple risk factors can interact in causing the disease, if one uses the population-wide causal-pie weight index.

The indices of attributable fractions quantify the impacts of hypothetical intervention programs. From the population attributable-fraction values presented in Table 2, we see that the disease burden in the hypothetical population in Table 1 will be reduced by 60.6%, if *X* _{2} can be eliminated, which is much larger than the corresponding value (5.3%), if *X* _{1} is eliminated. We thus can prioritize the intervention program targeting *X* _{2}. However, population attributable-fraction indices do not attribute a risk probabilistically to various classes of sufficient causes and are ill suited to characterize the independent and interactive effects of multiple risk factors.

### APPLICATION TO REAL DATA

I reexamined the European Prospective Investigation into Cancer and Nutrition-Potsdam data,^{10},^{11} previously analyzed by Hoffmann et al^{3} and by Liao and Lee.^{4} The study outcome is incident myocardial infarction. There are a total of 159 newly diagnosed cases and 26,813 controls. The study is actually a cohort study, but Hoffmann et al^{3} and Liao and Lee^{4} analyzed it as a case-control study (all the noncases were included as controls) without regard to time-to-event information. The risk factors evaluated are smoking (*X* _{1}), hypertension (*X* _{2}), obesity (*X* _{3}), and lack of exercise (*X* _{4}), all of which are dichotomous variables. Table 2 in the paper by Liao and Lee^{4} shows the distributions of the risk-factor profiles in cases and controls.

I fit the linear odds ratio model (12) to the data using the same covariates and interaction terms fit by Liao and Lee,^{4} and the result is Odds^{profile=x1} ^{,…,x4} = 0.00058 × (1 + 4.3*x* _{2} + 3.7*x* _{3} + 5.6*x* _{1} *x* _{4} + 5.4*x* _{2} *x* _{3} + 16.3*x* _{1} *x* _{3} *x* _{4}). From this (and with the reasonable rare-disease assumption for incident myocardial infarction), we see that the completion potentials are (approximately) 4.3 for hypertension, 3.7 for obesity, 5.6 for interaction between smoking and lack of exercise, 5.4 for interaction between hypertension and obesity, and 16.3 for interaction between smoking, obesity, and lack of exercise (Table 3 and Fig. 2). (This paper focuses on point estimates. To further quantify the uncertainties in completion potentials, one can use the bootstrapping technique described in the study by Liao and Lee.^{4})

We see that the *X* _{2} class (independent effect of hypertension) and the *X* _{3} class (independent effect of obesity) are each about 4 times as likely to cause disease as the all-*U* class. The two-factor interactive effects between smoking and lack of exercise and between hypertension and obesity are of similar magnitude with the aforementioned independent effects; either class is about 5 times as likely to cause disease as the all-*U* class. The three-factor interactive effect between smoking, obesity, and lack of exercise is much more striking by comparison; the class is about 16 times as likely to cause disease as the all-*U* class.

This example also demonstrates a severe underestimation of the interaction effects using the population-wide causal-pie weight indices. For example, the very strong three-factor interactive effect between smoking, obesity, and lack of exercise (completion potential = 16.3) would be masked with the population-wide causal-pie weight index (Table 3 and Fig. 2)—its value (21.0%) would be only as large as that of the *X* _{2} hypertension class (20.3%) and the *X* _{3} obesity class (19.5%). This is because in the population, only approximately 13% of people are exposed to smoking, obesity, and lack of exercise simultaneously (Table 2 of Liao and Lee^{4}).

As for the population attributable fractions of this example (Table 3), the disease burden will be reduced by 33.1% if smoking alone is eliminated, 37.5% if hypertension alone is eliminated, 57.7% if obesity alone is eliminated, and 33.1% if lack of exercise alone is eliminated.

## DISCUSSION

A number of researchers have previously established a link between Rothman's sufficient component cause model and the deterministic potential-outcome (counterfactual) model.^{12} ^{–} ^{14} This line of research also led to the development of empirical tests for causal mechanistic interactions.^{15} ^{–} ^{19} The tests are robust in that they can be done even without the monotonicity assumption. However, the present paper adopts a different approach. Rothman's sufficient component cause model is depicted here as part deterministic (disease is destined to occur once a sufficient cause is completed) and part stochastic (arrivals of the unknown complement causes are a random process). This mixed formulation has one important advantage: it allows the many classes of sufficient causes to be characterized by indices, such as completion potentials and causal-pie weights. This paper shows that, with proper assumptions, all these indices are estimable from routine epidemiologic data. The assumptions used in this paper are (i) monotonicity, (ii) independent competing causes, and (iii) proportional hazards (rates). (There is no need to further assume that the completion potentials themselves are constant over time; the time-invariant property of completion potentials follows directly from the 3 assumptions.)

For exposed subjects with a risk-factor profile of *x* _{1}, …, *xK* ≠ 0, …, 0, it is possible to calculate a proportionate increase in caseload due to exposures (excess fraction): AFE_{target=0,…,0} ^{profile=x1,…,xK≠0,…,0} = 1 − CPW_{class=0,…,0} ^{profile=x1,…,xK≠0,…,0} (setting *t* _{1}, …, *tK* = *c* _{1}, …, *cK* = 0, …, 0 in Equation (18)). Robins and Greenland^{6} differentiated between 2 types of fractions: the “excess fraction” (such as the one calculated using Equation (18)) and the “etiologic fraction” (fraction of cases caused by exposures, also referred to as “probability of causation”^{20}). Under the assumption of independent competing causes, excess fraction and etiologic fraction coincide, and therefore attributable fractions within the exposed population (and causal pie weights) can have probability-of-causation interpretations. If the constituent factors of the class-specific unknowns overlap, the assumption of independent competing causes will fail, and the arrivals of the class-specific unknowns will correlate positively with one another. eAppendix 4 (http://links.lww.com/EDE/A579) shows that under this condition, an attributable fraction within the exposed population is a lower bound of an etiologic fraction. This result is in line with that of Robins and Greenland^{6} p. 855: “Under certain biologically plausible assumptions, the etiologic fraction lies between that calculated under the independent competing-cause model and that computed under the rank-preserving model.” The most extreme case of dependent competing causes is that in which a particular constituent factor appears in many classes of sufficient causes, and, coincidentally, is the last factor needed in more than one class of sufficient causes. In this case, the etiologic fraction is 100% (because one can always find one or more risk factors in the classes of sufficient causes that are completed), and an attributable fraction within the exposed population calculated using Equation (18) of course serves only as a lower bound for it.

Hoffmann et al^{3} previously introduced an index, a “proportion of diseased subjects who develop the disease due to a class of sufficient causes.” However, this index has caused confusion. eAppendix 5 (http://links.lww.com/EDE/A579) shows that it is not an index associated with a particular class of sufficient causes but actually is similar to an attributable fraction (associated with a particular risk-factor profile). The index introduced by Liao and Lee^{4} is the same as the population-wide causal-pie weight in this paper. However, Liao and Lee^{4} need to calculate the population attributable fractions first, and then solve a system of population attributable-fraction equations (requiring the inversion of a large, cumbersome matrix) to obtain the population-wide causal-pie weights. In contrast, it is shown here that population-wide causal-pie weights are simple averages of individual-based causal-pie weights, which by themselves are simple functions of completion potentials.

A valid causal inference relies on the assumption of no (residual) confounding.^{2} p. 67–68, 385–386 To adjust for measured confounders (denoted by a vector **Z**), Liao and Lee^{4} proposed to use the confounder summary scores. An ordinary Cox regression containing the risk factors and the potential confounders (or the interaction terms between them) is to be fitted to the data. The confounder summary scores for those subjects with **Z** = **z**, denoted by Score^{Z=z}, are those subjects' predicted rate ratios as derived from the Cox regression, with all the risk-factor values being set to zero. Equation (11) then becomes

To fit the model, the scores can be absorbed into the regressor terms to become

Another possibility is to use the marginal structural models for interactions proposed by VanderWeele et al,^{21} which have the added benefit of being able to deal with time-dependent confounding. More work need to be done in this area.

In conclusion, the completion potential indices proposed in this paper allow better characterization of complex interactive effects of multiple monotonic risk factors.

### ACKNOWLEDGMENTS

We thank Shu-Fen Liao for helpful comments.