Attributable cases and proportions

Suzuki et al. draw the distinction between the attributable cases and attributable proportions in their careful article in Am J Epidemiol. 2012;175(6):567–575.

Policy makers are often interested in the extent of reduction in incidence that would be achieved had the population been entirely unexposed, compared with its existing risk pattern. Note that this reasoning compares the observed risk $P(Y=1)$ with the counterfactual risk $P(Y_0=1)$ . In this paper, the article proposes to distinguish attributable “caseload” from attributable proportion on the basis of a simple rule that proportions have the numerator included in the denominator.

The first measure, attributable caseload, can be interpreted as a reduction/increment in observed cases had the population been entirely unexposed.

$\frac{P(Y=1) - P(Y_0=1)}{P(Y=1)}$

Notably, the numerator is not included in the denominator, and this measure ranges from $-\infty$ to 1. In most cases, the attributable caseload could be one of the most useful measures in public health should an intervention (e.g., vehicle emission control) be implemented to make everyone in the population unexposed.

The second measure, attributable proportion, is a proportion of observed cases in the total population that would not have occurred had the population been entirely unexposed

$\frac{P(Y=1) - P(Y=1,Y_0=1)}{P(Y=1)}$

$= P(Y_0=0\vert Y=1)$

The last probability is exactly what Pearl calls the probability of disablement.

The proportion of cases among those happened to be exposed that would not have occurred had the exposed been entirely unexposed is defined as follows:

$\frac{P(Y=1\vert E=1) - P(Y=1,Y_0=1\vert E=1)}{P(Y=1\vert E=1)}$