Counterfactual-Based Prevented and Preventable Proportions

1 Introduction

In medical science, one of important issues is to assess how much a disease can be averted by a protective exposure, in subjects that would otherwise suffer from the disease. For example, in a 1970 measles epidemic in Texarkana, Texas, USA, some diseases occurred in children who had been vaccinated against measles. At that time, the public questioned the effectiveness of the measles vaccine and Landrioan [1] reported that its efficacy was 95.9%. According to Gregg [2], this implies that vaccination prevented 96% of diseases that would have occurred in vaccinated children, had they not been vaccinated. In a 2004 varicella outbreak in Nebraska, USA, some diseases occurred in children who had been vaccinated against varicella and the Centers for Disease Control and Prevention [3] reported that the vaccine’s effectiveness was approximately 81%. According to Gregg [2], this implies that vaccination prevented approximately 80% of diseases that would have occurred in vaccinated children, had they not been vaccinated.

To measure such an impact in vaccine trials, Greenwood and Yule [4] proposed vaccine efficacy, which is defined as the proportion of undiseased subjects in the vaccinated group who, under ideal conditions, would become diseased if they had not been vaccinated. Weinberg and Szilagyi [5] stated that vaccine efficacy is best measured by double-blind, randomized, clinical controlled trials, such as those performed for the pentavalent and monovalent rotavirus vaccines. For exposures or interventions other than vaccines, vaccine efficacy is equivalent to the prevented fraction among the exposed [2]. In the field of epidemiology, Miettinen [6] formulated the prevented fraction to evaluate the effectiveness of a protective exposure. In biostatistics, Gargiullo et al. [7] proposed the variance estimator of the prevented fraction based on maximum likelihood theory for cross-sectional studies. Benichou [8] reviewed adjusted methods of estimating the prevented fraction and other ratio scales while controlling for other factors. Additionally, Laaksonen [9] discussed the calculation of the prevented fraction and its related measures based on a cohort study design. Recently, Greenland [10] discussed concepts and pitfalls in measuring and interpreting attributable fractions and prevented fractions. As observed from these vigorous discussions, the evaluation problem of vaccine efficacy and the prevented fraction is a classical topic, but remains of great interest in medical statistics even now.

When we reviewed early literature on the prevented fraction, we found that Miettinen [6] described one of the fundamental concepts of the prevented fraction that provides the motivation for this paper:

“the preventive (prevented) fraction is the proportion of cases prevented by the factor among the totality of cases that would have developed in the absence of the protective factor”[6].

As seen from Miettinen’s statement, the prevented fraction should be formulated based on the framework of causal inference, but it should not be considered within the framework of statistics routinely. In fact, Porta et al. [11, p. 224] and Boslaugh [12, p. 59] also stated

“In a study of a total population, the prevented fraction of the incidence rate is computed as (Iu−Ip)/Iu or Pe(1−Ip/Iu), where Ip is the rate of the disease in the population, Iu is what the rate would be if everyone were not exposed, and Pe is the prevalence of exposure” [11].

and

“Whereas the denominators of attributable fractions are observable numbers of cases, the denominators of prevented fractions are nonobservable (counterfactual)” [12].

However, to the best of our knowledge, most biostatistical researchers and practitioners do not appear to take this counterfactual aspect into account. Additionally, because measures in previous studies on the prevented fraction are formulated based on different target populations, they cannot be interpreted as proportions without any further assumptions and may fall outside the range [0.000,1.000]. Similar to but different from the prevented fraction, according to Porta et al. [11, p. 223], the preventable fraction is defined as the proportion of the disease in the population that would be prevented if the whole population were exposed to the factor. However, this measure is also formulated based on different target populations, and thus cannot be interpreted as a proportion.

To solve this problem, this paper defines prevented and preventable proportions based on the potential outcome model. The proposed measures are formulated based on a single target population and can be interpreted as proportions. Based on a similar motivation to ours, Suzuki et al. [13] reformulated the excess fraction, attributable fraction and etiologic fraction, and derived the bounds on these measures. However, they did not focus on the prevented and preventable fractions. Additionally, they derived their bounds under exogeneity. Conversely, we assume that both causal effects and observed probabilities are available. This assumption, which is weaker than exogeneity, enables us to evaluate the proposed measures when the causal effect is identifiable from observational studies, as well as experimental studies. In this situation, we show that the proposed measures are identifiable under the negative monotonicity assumption. When the negative monotonicity assumption is violated, we formulate the bounds within which the proposed measures must lie. The results show that the traditional prevented and preventable fractions under-evaluate the proposed measures under exogeneity assumption. Furthermore, we show that negative monotonicity together with exogeneity induces equivalence between the proposed measures and existing measures. The extensions to the case of multi-categorical exposure and the applications to experimental and observational studies are provided in the appendices.

2 Existing measures

Let X be a dichotomous protective exposure {x′(unexposed),x(exposed) and Y be a dichotomous outcome {y′(undiseased),y(diseased)}. Additionally, pr(x,y), pr(x), and pr(y|x) indicate the joint probability of (X,Y)=(x,y), the marginal probability of X=x, and the conditional probability of Y=y given X=x, respectively; similar notation is used for the other probabilities.

In this section, we introduce three representative prevented and preventable fractions: the prevented fraction in the population (PFp), the prevented fraction among the exposed (PFe), and the preventable fraction (PaFp).

In order to evaluate the prevented impact of a protective exposure on the total number of diseases that would be expected if exposure was absent, Miettinen [6] proposed the PFp as

According to Miettinen [6], PFp is interpreted as “the proportion of cases prevented by the (protective) factor among the totality of cases that would have developed in the absence of the protective factor.”

Additionally, biostatistical researchers and practitioners may want to evaluate the proportion of potential exposed cases that were prevented by the exposure [6, 14]. Miettinen [6] and Kleinbaum et al. [14] formulated PFe as

Unlike PFp, PFe focuses on the prevented impact of a protective exposure on the expected number of exposed cases that would have developed in the absence of the protective factor [14]. According to Spasoff [15], PFe is identical to the relative risk reduction that was proposed by Sackett et al. [16] in clinical epidemiology and evaluates how much of the risk is reduced in the exposed group compared with the unexposed group. It is also equivalent to vaccine efficacy, which was originally discussed by Greenwood and Yule [4] in the context of vaccine trials.

Although the prevented fraction is frequently referred to as the preventable fraction, these concepts should be distinguished. For example, Boslaugh [12] stated that

“With protective risk factors, researchers and policy analysts are often interested in how much of the current disease risk in the total target population is potentially preventable if everyone in the population were exposed. This measure, called the preventable fraction, should be distinguished from the prevented fraction. Whereas the PFp reflects the previous impact of being exposed in the population, the PaFp reflects the potential impact in the future if everyone were to become exposed” [12].

Porta et al. [11] and other researchers defined PaFp as

which is interpreted as “the proportion of the disease (in the population) that would be prevented if the whole population were exposed to the factor” [11]. To observe the difference between PFp and PaFp, Spasoff [15] considered the hypothetical epidemic in a population in which 40% of subjects had been immunized, and in which the cumulative incidence of the disease was 10% in unimmunized subjects and 2% in immunized subjects, that is, pr(x)=1−pr(x′)=0.400, pr(y|x)=0.020, pr(y|x′)=0.100 and pr(y)=pr(y|x)pr(x)+pr(y|x′)pr(x′)=0.068. Then, PFp is given by 0.320, which implies that the immunization program has reduced the incidence of the disease by 32% in the population, that is, the immunization program was 32% effective. By contrast, PaFp is given by 0.706. According to Spasoff [15], this implies that 71% of diseases could have been prevented by increasing the coverage of the immunization program from 40% to 100%.

Although Porta et al. [11] and some researchers introduced the prevented and preventable fractions based on counterfactual concepts, it is unclear how such a concept is introduced into the formulation mathematically. Additionally, it is noted that PFp, PFe and PaFp are formulated based on different target populations: the exposed and unexposed groups. Thus, the traditional prevented and preventable fractions may fall outside the range [0.000,1.000] without any further assumptions and they cannot be interpreted as proportions. To ensure that these measures are within the range [0.000,1.000], biostatistical researchers and practitioners often assume that “exposure to a given factor is believed to protect against a disease” [11], which is supported by negative monotonicity that will be introduced in Section 4.1, or that

holds [2, 6]. However, generally, the former assumption cannot be verified from observed data because the phrase “protect against a disease” conveys much more than statistical association and implies a cause-and-effect relationship between the exposure and the disease. Additionally, when unmeasured confounders exist, the former may not imply the latter and vice versa.

3 Proposed measures

In this section, we introduce the potential outcome variables that will be used to define the counterfactual-based prevented and preventable proportions.

4 Identification and bounds

5 Conclusion

The idea of prevented and preventable fractions has been widely applied in medical science to evaluate how much of a disease could be averted by a protective exposure for subjects that would have become diseased in the absence of the exposure. However, the traditional formulas used in practical situations have a drawback, which is that they cannot be interpreted as proportions without any further assumptions because they are formulated based on different target populations and may fall outside the range [0.000,1.000]. To solve this problem, when both the exposure and outcome are dichotomous, we proposed four types of new measures of potential impact based on the potential outcome model. The proposed measures are proportions: they are defined based on a single target population and are always within the range [0.000,1.000]. When both causal effects and observed probabilities are available, this paper showed that the proposed measures are identifiable under the negative monotonicity assumption. The negative monotonicity assumption can be considered as a causal assumption to justify “exposure to a given factor is believed to protect against a disease” [11]. Additionally, when negative monotonicity is violated, we formulated the bounds on the proposed measures. Furthermore, we showed that negative monotonicity together with exogeneity induces equivalence between the proposed measures and existing measures. Therefore, the proposed measures are helpful for biostatistical researchers and practitioners to assess what percentage of a disease could be averted by a protective exposure for cases that would have become diseased in the absence of the exposure.

Appendices