Choosing a Measure of Effect

2.2 MEASURES OF EFFECT

2.2.3 Choosing a Measure of Effect

We now consider which, if any, of the risk difference, risk ratio, or odds ratio is the most desirable measure of effect for closed cohort studies. One of the most con-

tentious issues revolves around the utility of RDandRR as measures of etiology (causation) on the one hand, and measures of population (public health) impact on the other. This is best illustrated with some examples. First, suppose that the probabil- ity of developing the disease is small, whether or not there is exposure; for example, π1=.0003 andπ2=.0001. ThenRD=.0002, and so exposure is associated with a small increase in the probability of disease. Unless a large segment of the popula- tion has been exposed, the impact of the disease will be small and so, from a public health perspective, this particular exposure is not of major concern. On the other hand,RR = 3 and according to usual epidemiologic practice this is large enough to warrant further investigation of the exposure as a possible cause of the disease.

Now suppose thatπ1 = .06 andπ2 = .05, so thatRD = .01 andRR = 1.2. In this example, the risk difference will be of public health importance unless expo- sure is especially infrequent, while the risk ratio is of relatively little interest from an etiologic point of view.

The above arguments have been expressed in terms of the risk difference and risk ratio, but are in essence a debate over the merits of measuring effect on an additive as opposed to a multiplicative scale. This issue has generated a protracted debate in the epidemiologic literature, with some authors preferring additive models (Roth- man, 1974; Berry, 1980) and others preferring the multiplicative approach (Walter and Holford, 1978). Statistical methods have been proposed for deciding whether an additive or multiplicative model provides a better fit to study data. One approach is to compare likelihoods based on best-fitting additive and multiplicative models (Berry, 1980; Gardner and Munford, 1980; Walker and Rothman, 1982). An alterna- tive method is to fit a general model that has additive and multiplicative models as special cases and then decide whether one or the other, or perhaps some intermediate model, fits the data best (Thomas, 1981; Guerrero and Johnson, 1982; Breslow and Storer, 1985; Moolgavkar and Venzon, 1987).

Consider a closed cohort study whereπ1 = .6 andπ2 = .2, so thatω1 = 1.5 andω2 = .25. Based on these parameters we have the following interpretations:

Exposure increases the probability of disease by an incrementRD = .4; exposure increases the probability of disease by a factorRR=3; and exposure increases the odds of disease by a factor OR = 6. This simple example illustrates that the risk difference, risk ratio, and odds ratio are three very different ways of measuring the effect of exposure on the risk of disease. It also illustrates that the risk difference and risk ratio have a straightforward and intuitive interpretation, a feature that is not shared by the odds ratio. Even ifω1=1.5 andω2=.25 are rewritten as “15 to 10”

and “1 to 4,” these quantities remain less intuitive thanπ1=.6 andπ2=.2. It seems that, from the perspective of ease of interpretation, the risk difference and risk ratio have a distinct advantage over the odds ratio.

Suppose we redefine exposure status so that subjects who were exposed according to the original definition are relabeled as unexposed, and conversely. Denoting the resulting measures of effect with a prime, we haveRD =π2−π1,RR=π2/π1, andOR= [π2(1−π1)]/[π1(1−π2)]. It follows thatRD= −RD,RR=1/RR, and OR =1/OR, and so each of the measures of effect is transformed into a reciprocal quantity on either the additive or multiplicative scale. Now suppose that we redefine disease status so that subjects who were cases according to the original definition are

relabeled as noncases, and conversely. Denoting the resulting measures of effect with a double prime, we haveRD=(1−π1)−(1−π2),RR=(1−π1)/(1−π2), and OR = [(1−π1)π2]/[(1−π2)π1]. It follows thatRD = −RDandOR=1/OR, butRR = 1/RR. The failure of the risk ratio to demonstrate a reciprocal property when disease status is redefined is a distinct shortcoming of this measure of effect.

For example, in a randomized controlled trial let “exposure” be active treatment (as compared to placebo) and let “disease” be death from a given cause. Withπ1=.01 andπ2=.02,RR=.01/.02=.5 and so treatment leads to an impressive decrease in the probability of dying. Looked at another way,RR =.99/.98 =1.01 and so treatment results in only a modest improvement in the probability of surviving.

Since 0 ≤ π1 ≤ 1, there are constraints placed on the values of RD andRR.

Specifically, for a given value ofπ2,RDandRRmust satisfy the inequalities 0 ≤ π2+RD ≤ 1 and 0 ≤ RRπ2 ≤ 1; or equivalently,−π2 ≤ RD ≤ (1−π2)and 0 ≤ RR ≤ (1/π2). In the case of a single 2×2 table, such as being considered here, these constraints do not pose a problem. However, when several tables are being analyzed and an overall measure of effect is being estimated, these constraints have greater implications. First, there is the added complexity of finding an overall measure that satisfies the constraints in each table. Second, and more importantly, the constraint imposed by one of the tables may severely limit the range of possible values for the measure of effect in other tables. The odds ratio has the attractive property of not being subject to this problem. Solving (2.1) forπ1gives

π1= ORπ2

ORπ2+(1−π2). (2.2)

Since 0≤π2≤1 andOR≥ 0, it follows that 0≤π1≤1 for any values ofORand π2for which the denominator of (2.2) is nonzero. Figures 2.1(a) and 2.1(b), which

FIGURE 2.1(a) π1as a function ofπ2, withOR=2

FIGURE 2.1(b) π1as a function ofπ2, withOR=5

are based on (2.2), show graphs ofπ1as a function ofπ2forOR=2 andOR=5.

As can be seen, the curves are concave downward in shape. By contrast, for given values ofRDandRR, the graphs ofπ1=π2+RDandπ1=RRπ2(not shown) are both linear; the former has a slope of 1 and an intercept ofRD, while the latter has a slope ofRRand an intercept of 0.

When choosing a measure of effect for a closed cohort study, it is useful to consider the properties discussed above—that is, whether the measure of effect is additive or multiplicative, intuitively appealing, exhibits reciprocal properties, and imposes restrictions on the range of parameter values. However, a more fundamental consideration is whether the measure of effect is consistent with the underlying mechanism of the disease process. For example, if it is known that a set of exposures exert their influence in an additive rather than a multiplicative fashion, it would be appropriate to select the risk difference as a measure of effect in preference to the risk ratio or odds ratio. Unfortunately, in most applications there is insufficient substantive knowledge to help decide such intricate questions. It might be hoped that epidemiologic data could be used to determine whether a set of exposures is oper- ating additively, multiplicatively, or in some other manner. However, the behavior of risk factors at the population level, which is the arena in which epidemiologic research operates, may not accurately reflect the underlying disease process (Siemi- atycki and Thomas, 1981; Thompson, 1991).

Walter (2000) has demonstrated that models based on the risk difference, risk ratio, and odds ratio tend to produce similar findings, a phenomenon that will be il- lustrated later in this book. Currently, in most epidemiologic studies, some form of multiplicative model is used. Perhaps the main reason for this emphasis is a practical consideration: In most epidemiologic research the outcome variable is categorical (discrete) and the majority of statistical methods, along with most of the statistical packages available to analyze such data, are based on the multiplicative approach

(Thomas, 2000). In particular, the majority of regression techniques that are widely used in epidemiology, such as logistic regression and Cox regression, are multiplica- tive in nature. For this reason the focus of this book will be on techniques that are defined in multiplicative terms.