Assessing Causal Mechanistic Interactions: A Peril Ratio Index of Synergy Based on Multiplicativity

The assessments of interactions in epidemiology have traditionally been based on risk-ratio, odds-ratio or rate-ratio multiplicativity. However, many epidemiologists fail to recognize that this is mainly for statistical conveniences and often will misinterpret a statistically significant interaction as a genuine mechanistic interaction. The author adopts an alternative metric system for risk, the ‘peril’. A peril is an exponentiated cumulative rate, or simply, the inverse of a survival (risk complement) or one plus an odds. The author proposes a new index based on multiplicativity of peril ratios, the ‘peril ratio index of synergy based on multiplicativity’ (PRISM). Under the assumption of no redundancy, PRISM can be used to assess synergisms in sufficient cause sense, i.e., causal co-actions or causal mechanistic interactions. It has a less stringent threshold to detect a synergy as compared to a previous index of ‘relative excess risk due to interaction’. Using the new PRISM criterion, many situations in which there is not evidence of interaction judged by the traditional indices are in fact corresponding to bona fide positive or negative synergisms.


Introduction
Epidemiologists are often troubled by how to make sense of the joint exposure effects of two factors on the risk of a disease. For example, is the combined effect of tobacco smoking and asbestos exposure greater (or smaller) than what would be expected based on their individual separate effects? And, if the combined effect indeed deviates from its expectation, does it imply a mechanistic interaction between the two exposures? These apparently simple questions prove difficult to answer. First, we need a suitable scale with which to measure an effect. But which scale to use, a ratio scale (e.g., risk ratio, odds ratio and rate ratio) or a difference scale (e.g., risk difference, odds difference and rate difference)? Second, we need to define a reasonable expectation of the combined effect, against which to define an interaction. But what type of interactions: a multiplicative interaction or an additive interaction?
The assessments of interactions have traditionally been based on risk-ratio, odds-ratio or rate-ratio multiplicativity, such as using the 'synergistic index of multiplicativity' (SIM) [1]. No multiplicative interaction (SIM~1) corresponds to risk-ratio, odds-ratio or rateratio homogeneity across strata in a stratified analysis, or to absence of cross-product terms in a multiplicative model, such as logistic, Poisson or Cox regression. The use of multiplicative models is so dominating in epidemiology that a significant multiplicative interaction is often mistaken as a genuine mechanistic interaction.
Recently, an index of risk-ratio additivity, the 'relative excess risk due to interaction' (RERI) [2], has received much attention. (For two dichotomous exposures X 1 and X 2 , RERI~RR profile~1,1 {RR profile~1,0 {RR profile~0,1 z1, where RR profile~x1,x2 is the risk ratio comparing the disease risks between those with exposure profile of X 1~x1 ,X 2~x2 and those with X 1~X2~0 .) The index can assess synergisms in sufficient cause sense, i.e., causal co-actions or causal mechanistic interactions [3][4][5][6]. A sufficient cause [7] contains a combination of component causes. There may be many classes of sufficient causes for a disease. Any class with all its components completed is sufficient to cause the disease. For the aforementioned example, we may speak of synergisms if there are some lung cancer patients who had developed the disease because of the completions of the classes of sufficient causes containing both tobacco and asbestos as their components.
As the RERI above is based on 'risks', it necessarily entails follow-up of a population for a certain period, say, from time 0 to time T. The dependency on an arbitrary time point T is rather undesirable. First, it is possible that two researchers using different Ts will reach different conclusions regarding causal mechanistic interactions. And second, when T tends to infinity, RR profile~x1,x2 will tend to one for each and every x 1 ,x 2 [f0,1g, and RERI will tend to zero (perfect risk-ratio additivity). This thus masks any possible synergism between X 1 and X 2 !.
In this paper, I turn to 'rates' instead. Because all rates are defined the same way with their Ts being made to be infinitesimally small, an index of rate does not have the above T dependency problem. An alternative metric of risk is then used: the exponentiated cumulative rate, which I refer to as the 'peril'. I will show that the synergy index based on multiplicativity of peril ratios, the 'peril ratio index of synergy based on multiplicativity' (PRISM), can be used to assess synergisms in sufficient cause sense: perfect multiplicativity (PRISM~1) implying no synergism, super-multiplicativity (PRISMw1), positive synergisms, and submultiplicativity (PRISMv1), negative synergisms, respectively. I will also show that PRISM has a less stringent threshold to detect a synergy as compared to RERI, and that many situations in which there is not evidence of interaction judged by the traditional indices of SIM and RERI are in fact corresponding to bona fide positive or negative synergisms.

Methods
Consider the relation of two dichotomous exposures and a disease in a follow-up of a population in a certain time interval, (0, T). I assume that the exposure status is time-invariant and the follow-up is 100% complete (without loss to follow up and competing death). For people in the population with a risk factor profile of X 1~x1 ,X 2~x2 , let Rate profile~x1,x2 (t) denote the (instantaneous) disease rate at follow-up time t, Risk profile~x1,x2 , the cumulative disease risk (probability) in (0, T), Odds profile~x1,x2~R isk profile~x1,x2 1{Risk profile~x1,x2 , the cumulative disease odds in (0, T), and S profile~x1,x2~1 {Risk profile~x1,x2 , the survival probability at T. I assume that there is no confounding, selection bias or measurement error in this cohort study, such that the associations between the two exposures and the disease should reflect the genuine causal effects of the exposures on the disease.
As mentioned previously, a peril is simply a cumulative rate exponentiated. Without lost to follow up and competing death, a peril is also the inverse of a survival (risk complement) or one plus an odds (Exhibit S1), that is, for a 'profile~x 1 ,x 2 ' subject, and Peril class~c 1 ,c 2~e xp for a 'class~c 1 ,c 2 ' sufficient causes. A peril is dimensionless and ranges from 1 (no peril) to infinity (maximum peril). I invoke the no redundancy assumption [8,9] that in a sufficiently short time interval, (t,tzt), for each and every subject in the population there can only be at most one arrival event of the unknown components. Exhibit S2 shows that under such an assumption, a peril for a specific exposure profile is the product of four perils corresponding to the four 'completable classes' (defined in Exhibit S2) for that exposure profile, that is, and Peril profile~1,1~P eril class~Ã,Ã |Peril class~1,Ã respectively. Next, define the peril ratio (PR) for a 'profile~x 1 ,x 2 ' subject as Because perils are the inverses of survivals, a peril ratio can be interpreted as the 'fold decrease' in survival (comparing 'profile~x 1 ,x 2 ' subject with 'profile~0,0' subject). Define PRISM as PRISM is a synergy index based on multiplicativity of peril ratios, or equivalently, multiplicativity of fold decreases in survivals. Intriguingly from Equations (3),(6), we see that the above defined PRISM can alternatively be expressed using the perils of the interactive classes only:   PRISM~P eril class~1,1 |Peril class~0,0 Peril class~1,0 |Peril class~0, 1 : ð9Þ We will say there is synergism between X 1 and X 2 in sufficient cause sense, if at least one of Risk class~1,1 , Risk class~1,0 , Risk class~0,1 , and Risk class~0,0 is non-zero (or equivalently, if at least one of Peril class~1,1 , Peril class~1,0 , Peril class~0,1 , and Peril class~0,0 differs from one). Because there are a total 9 classes of sufficient causes but only a total of 4 exposure profiles, the classspecific perils (Peril class~c1,c2 ,where c 1 ,c 2 [ 1,0,Ã f g) by themselves are not identifiable (not estimable from the data). However, this non-identifiability problem does not hamper our ability to test synergisms. A two-sided test on PRISM as defined in Equation (8), is a global test for synergisms. (PRISM~1 is the condition of no multiplicative interaction on the peril scale, or equivalently, no additive interaction on the cumulative rate scale.) The significance of the test implies the presence of at least one of the following four synergy classes: 'class~1,1', 'class~1,0', 'class~0,1' and 'class~0,0'. This is because from Equation (9), PRISM=1 forbids Peril class~1,1~P eril class~1,0~P eril class~0,1~P eril class~0,0 1 and therefore also Risk class~1,1~R isk class~1,0~R isk class~0,1 Risk class~0,0~0 : (Note however that PRISM~1 does not guarantee the absence of synergisms; a perfect cancellation of the positive and negative synergisms also leads to PRISM~1:) A one-sided test on PRISM, is a test for positive synergisms of 'class~1,1' and 'class~0,0' (PRISMw1 forbids Peril class~1,1~P eril class~0,0~1 and Risk class~1,1~R isk class~0,0~0 ), and a one-sided test, a test for negative synergisms of 'class~1,0' and 'class~0,1' (PRISMv1 forbids Peril class~1,0~P eril class~0,1~1 and Risk class~1,0~R isk class~0,1~0 ). PRISM also permits a test specifically for a particular synergy class, albeit with a more stringent threshold. From Equations (3) and (9)  A comparison of the above PRISM test with the RERI test [3][4][5][6] is in order. The former is based on cumulative rates, while the latter, cumulative risks. RERI can also test for specific synergisms: RERIw1 for 'class~1,1' [corresponding to PRISM Test (13)], RERIwRR profile~1,1 for 'class~0,0' [corresponding to PRISM Test (14)], RERIv{RR profile~0,1 for 'class~1,0' [corresponding to PRISM Test (15)], and RERIv{RR profile~1,0 for 'class~0,1' [corresponding to PRISM Test (16)], respectively.
In Figure 2, the upper short-dash lines mark the threshold of RERI~1 (for 'class~1,1'), whereas the lower short-dash lines, the threshold of RERI~{RR profile~0,1 (for 'class~1,0'). We see that the RERI thresholds are considerably more stringent than the corresponding PRISM thresholds (long-dashed lines). A proof is given in Exhibit S3 showing that a synergy can always be detected by PRISM if it is detected by RERI.
It is also of interest to re-examine the situations in which there is not evidence of interaction judged, respectively, by the RERI index (in terms of risk ratios as the above, or odds ratios, OR profile~1,1 {OR profile~1,0 {OR profile~0,1 z1), and the SIM index (SIM~R R profile~1,1 RR profile~1,0 |RR profile~0,1 in terms of risk ratios, or OR profile~1,1 OR profile~1,0 |OR profile~0,1 in terms of odds ratios). From Figure 3, we see that a no-interaction line either of RERI~0 or SIM~1 can penetrate deeply into the zones of positive (PRISMw1, regions marked by the upward arrows) and negative (PRISMv1, regions marked by the downward arrows) synergisms. This suggests that many situations in which there is not evidence of interaction previously judged by the traditional indices of SIM and RERI could in fact be bona fide positive or negative synergisms. Exhibit S4 shows that it is only when the disease under study is exceedingly rare or exceedingly common that a RERI criterion for rare diseases and a SIM criterion for common diseases shall correspond to the proposed PRISM criterion.

An Example
As an example, I re-analyze a cohort study (the Example 3 in Zou's paper [15]) using the PRISM approach. The cohort study assesses the effects of age (coded as 1 if age §40 years and 0 if otherwise) and body mass index (BMI, coded as 1 if BMI §25 kg/ m 2 and 0 if otherwise) on hypertension (coded as 1 if diastolic blood pressure §90 mmHg and 0 if otherwise). Table 1 presents the perils and the 95% confidence intervals (CIs) for the four exposure profiles. Using the (young age, low BMI) as the reference, the peril ratios and the 95% CIs are also presented, which show the fold decreases in hypertension-free probabilities. The PRISM for this example is calculated as 1.0905 with a 95% CI of 1.0341,1.1499. This implies that the peril ratios in this example (fold decreases in hypertension-free probabilities) do not conform to a multiplicative model.
The test statistic of the global test for synergisms for this example is 3.1961 with a highly significant two-sided p-value of 0.0014 (for global synergisms) and a highly significant one-sided (upper tail) p-value of 0.0007 (for positive synergisms). The specific test for 'class~1,1' (the synergy between old age and high BMI in this example) has a test statistic of 1.5763 with a marginally insignificant p-value of 0.0575. (The test for the same synergy class using the RERI approach yields a comparatively much larger pvalue of 0.2399.) Note that this example is used for illustrative purposes only and should not be taken as evidence of actual synergism here since the exposures have been dichotomized. Assessing synergism or interaction for continuous exposures under dichotomization is considerably trickier [16].
If the assumption of no preventive action is deemed reasonable for this example, the cumulative completion risks and the 95% CIs can be calculated for the four classes of sufficient causes as presented in Table 2.

Discussion
To study toxic effects of two chemicals administered simultaneously, the model of simple independent action had seen a very long history of use in toxicopharmacology dating back to 1939 [17]. In recent decades, epidemiologists [11,12,[18][19][20][21][22] and researchers in other fields (infectious disease [23], genetics [24] and environmental health [25]) also began to define interactions based on deviation from independence. However, independence is a rather strong assumption. In the present context of sufficient component causes, the completions of different classes of sufficient causes are not likely to be independent events. Rather, they are more likely to be positively correlated to one another due to possible overlapping of the constituent factors of the class-specific unknowns. This paper replaces the independence assumption with a much weaker Poisson-like assumption-the no redundancy assumption. The assumption dictates that in a sufficiently short time interval, for each and every subject in the population the probability of two or more than two arrival events of the unknown components is negligible. (Suzuki et al [14] previously introduced the concept of potential completion times of sufficient causes and assumed that each potential completion time is different. This is a different way to invoke the same no redundancy assumption.) Even with strong dependency in the arrival events, the no redundancy assumption should still hold in each and every time interval that is infinitesimally small, unless one argues that the probability is non-negligible that an overlapping constituent factor happens to be the last one to arrive, and in not just one but at least two classes of sufficient causes.
The assumption of proportional hazards (rates) has often been invoked in longitudinal follow-up studies (cohort studies) [8,11,12,26,27]. The assumption is often true (or approximately so) for most situations. But occasionally, we will see a larger deviation. For example, the hazard curves for different exposure profiles [Rate profile~x1,x2 (t) for x 1 ,x 2 [ 1,0 f g] can cross each other, thus failing the assumption completely. The proposed PRISM criterion does not need the proportional hazards assumption. It is a valid synergy test irrespectively of proportionalities, nonproportionalities or crossings of the hazard curves. If the hazard curve for each and every exposure profile in population A is a constant multiple (say, c) of the corresponding hazard curve in population B, we will have PRISM A~P RISM c B (Exhibit S6) and therefore achieve the same conclusion about synergisms (apart from statistical variations) in the two populations.
In a recent paper, VanderWeele [27] studied proportional hazards models and made an interesting conclusion that ''causal interactions can disappear as time progresses, ie, whether a causal interaction is present depends on the follow-up time''. Exhibit S7 re-examines this problem using the proposed PRISM criterion. It is found that in the proportional hazards models, theoretically a synergy signal will not go away with more follow-up times. However, there does exist an optimal follow-up time for maximum power of the PRISM test: to follow up the cohort subjects until ,80% of them are diseased (assuming no lost to follow up and competing death).
In this paper, methods for assessing sufficient cause synergism for rates without the co-cause independence and monotonicity assumptions are presented. The method can be extended in several ways. First, it is worthwhile to extend the present method to deal with exposures with multiple levels. This will allow us to study dose-response exposure-disease relations as well as dosedependent causal mechanistic interactions. Second, casting the present method in a proper modeling framework should also prove useful to accommodate more than two exposures and to adjust for possible confounders. Here we need to go beyond the commonly used logistic regression (in epidemiology [7]) and probit regression (in econometrics [28]) for binary outcomes, because these two models are purely statistical in nature without a built-in causal mechanism. Third, asides from confounding, selection bias and measurement errors, a cohort study can be complicated by the problems of lost to follow up and competing deaths, etc. A valid synergy analysis for censored data also awaits further studies. Finally, in an ordinary case-control study for a rare disease, the PRISM criterion can be approximated by the RERI criterion in terms of odds ratios (Exhibit S4). Without the rare-disease assumption however, one may need to resort to alternative control sampling schemes, such as density sampling or case-base (casecohort) sampling [7]. How to test causal mechanistic interactions under such settings also deserves further studies.

Supporting Information
Exhibit S1 Relation between exponentiated cumulative rate and other indices.

(DOC)
Exhibit S2 Relation between profile-specific perils/ rates and class-specific perils/rates.

Author Contributions
Conceived and designed the experiments: WCL. Performed the experiments: WCL. Analyzed the data: WCL. Contributed reagents/materials/ analysis tools: WCL. Wrote the paper: WCL.