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B. **QUANTITATIVE EXTRAPOLATION WITHOUT CONSONANCE**

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B. **QUANTITATIVE EXTRAPOLATION WITHOUT CONSONANCE**

The extrapolation theorem presented in section 6.2.2 was limited insofar as it specified conditions only for extrapolating claims about positive or negative causal relevance and in presupposing consonance. Yet one might wish to extrapolate a quantitative claim about a causal effect, and one might also wish to extrapolate probabilistic causal claims in cases in which counteracting mechanisms may be present. Of course, quantitative extrapolation is easy when the base population is representative of the target, but it is often the case that this assumption is doubtful or known to be false. But even when the base population fails to be representative of the target, it may nevertheless be what I call *cell‐representative*. The base population is cell‐representative of the target when there is a partition of the base population into cells such that the strength of the causal effect within each cell in the base population is a good approximation of the strength of the effect in the corresponding cell in the target. A base population can be cell‐representative without being representative if the relative frequencies of the cells differ between the two populations.

Elaborating this idea requires a measure of strength of causal effect. One commonly used measure of causal efficacy is the *mean difference*,^{1} according to which the impact of *X* upon *Y* is given by E(Y|*do(x)*) − E(Y|*do(x* _{0})) =_{df} ΔE(Y|*do(x*)). Recall that x_{0} is the comparative value of *X*, usually zero. For simplicity, I restrict attention to the special case in which the cause and effect are binary. In this case, the mean difference is P(Y = 1 | *do*(X = 1)) − P(Y = 1 | *do*(X = 0)) =_{df} ΔP. A partition of a population consists of a mutually exclusive and collectively exhaustive collection of subsets of that population, in the simplest case, those who possess a particular property and those who do not. The cells of partitions will be numbered 1, 2, …, *n*. The probability function for the *i*th cell of the partition is represented by P_{i}. So, for example, ΔP_{2} is P_{2}(Y = 1 | *do*(X = 1)) −P_{2}(Y = 1 | *do*(X = 0)). In this context, the population P′ is *cell‐representative* of P with respect to *X* and *Y,* given the partition *i* = 1, 2, …, *n* just in case ΔP′_{i} ≈ ΔP_{i} for all *i*, where *P*′ and *P* are the probability functions for the populations P′ and P, respectively. Thus, the idea is that strengths of the causal effects are approximately equal within the cells in the two populations, and that differences in the overall effect between the two populations result only from differences in the proportions of these cells.

Clearly, whether it is reasonable to assume that the base population is cell‐representative of the target depends on the choice of partition. In section 6.2.1, it was presumed that the partition was by the particular set of undisrupted mechanisms possessed by the individual. If practically possible, this would be a promising way to partition, since differences in the strength of the causal effect presumably result from differences in mechanisms. However, it will generally be difficult, if not impossible, to accurately decide precisely which combination of mechanisms is present (p.206) in a given individual. The presence or absence of detectable factors that promote or interfere with the mechanisms in question would provide some guidance for such purposes. In the case of the effect of HIV exposure upon AIDS, this would include such things as availability of anti‐retroviral therapies or host resistance factors (such as the mutation affecting the R‐5 co‐receptor). But it is doubtful that the subgroups identified by such indicators will consist of individuals possessing the precisely same combinations of mechanisms.

This might seem like a serious problem, since equations (6.8) and (6.9) were derived on the assumption that one was partitioning by combinations of mechanisms. For instance, equation (6.9) told us that $\Delta P={\sum}_{i=1}^{n}{\varphi}_{i}\Delta {P}_{i}$, where each *i* indicates a specific combination of mechanisms from *X* to *Y*. Fortunately, however, these equations can be derived for different partitions, so long as the properties by which one partitions are independent of the cause when it is set by an intervention. That is, if the cells in the partition are *i* = 1, 2, …, *n*, then the key premise is that P(*i* | *do(x*)) = P(*i*), for each *i*. Given the definition of an ideal intervention, this premise is reasonable so long as the properties by which one partitions are not effects of *X*. For example, possession of the mutation inhibiting the expression of the R‐5 co‐receptor is presumably not an effect of HIV exposure.

Consider how quantitative extrapolation on the basis of a cell‐representative base population could work in the AFB_{1} example. Susceptibility to the carcinogenic effects of AFB_{1} is known to depend on exposure to the hepatitis B virus (HBV).^{2} There is also evidence that heightened sensitivity to mutagens is also a co‐factor (Wu et al. 1998), although the basis of these variations in AFB_{1} susceptibility remains somewhat unclear (cf. McGlynn et al. 2003). It is likely due in part to HBV exposure (Sohn et al. 2000), but the importance of other factors, such as congenital genetic variations, is still uncertain. At present, then, HBV exposure is the most firmly established factor for susceptibility to AFB_{1} carcinogenesis as well as something that can be measured reliably. Hence, given that it is likely that HBV exposure is not an effect of exposure to AFB_{1}, HBV would appear to be a good property by which to partition.

Imagine that one is interested in estimating the strength of the causal effect of AFB_{1} on liver cancer in North America from data from China, where exposure to AFB_{1} is more common and consequently where there are more extensive data sets. Thus, P′ and P in this case would be the populations of China and North America, respectively. Letting *X* represent exposure to AFB_{1} and *Y,* occurrence of liver cancer, ΔP is the strength of the causal effect in the North American population, and similarly for ΔP′. Since HBV is much more common in China than in North America, it is obvious that it would be unreasonable to regard ΔP′ as a good estimate of ΔP. Nevertheless, the Chinese population might serve as an approximate guide to the North American one if we partition by those who have been exposed to HBV and those who have not. Labeling these two groups
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1 and 2, respectively, ΔP_{1} = P_{1}(Y = 1 | *do*(X = 1)) − P_{1}(Y = 1 | *do*(X = 0)) is the strength of the causal effect among those in the North American population exposed to HBV, and similarly for ΔP_{2}. Let φ_{1} and φ_{2}, respectively, be the relative frequencies of those exposed and not exposed to HBV. Then if AFB_{1} is not a cause of HBV exposure, we can derive (as explained in section 6.2.1) the following equation.

Thus, if the Chinese population is approximately representative of the North American one with regard to the strength of the carcinogenic effect of AFB_{1} among those exposed to HBV and those not exposed (i.e., ΔP′_{1} and ΔP′_{2} provide reasonably good estimates of ΔP_{1} and ΔP_{2}, respectively), then the strength of the overall causal effect in the North American population, ΔP, can be computed, given the relative frequency of exposure to HBV in North America.^{3}

Notice that the above reasoning does not depend on assuming consonance: if it is possible to estimate the strength of the causal effect in each cell of a cell‐representative base population, then the overall effect in the target population can be estimated as explained above. For example, one could imagine a case like the above but in which ΔP_{1} is positive and ΔP_{2} is negative. However, consonance would be a useful assumption if it were possible to estimate the strength of the causal effect only in some cells and not others or if the base population were representative of the target for only some cells. For instance, suppose that in the aflatoxin example the strength of the causal effect could be estimated only for those who have been exposed to HBV. Then, given consonance, ΔP≥φ_{1}ΔP_{1}, which means that a lower bound can be placed on ΔP. This inference would not be valid, however, if consonance did not obtain, since in that case the second term on the right‐hand side of (6.10) could be negative.

But what if only some cells of P′ are representative of those in P and consonance is not plausible? Informative extrapolations may be possible even in this unfavorable situation. Since the maximum and minimum values of ΔP are 1 and −1, it is possible to compute extreme upper and lower bounds from φ_{1}ΔP_{1} when consonance is not assumed. That is, given that φ_{1}ΔP_{1} is estimated, φ_{1}ΔP_{1} − (1 − φ_{1})≤ΔP≤φ_{1}ΔP_{1} + (1 − φ_{1}). In other words, the lower bound is what results from the assumption that the strength of the effect is −1 in the remainder of the population (whose relative frequency is 1 − φ_{1}), while the upper bound results from the assumption that the strength of the effect in the remainder of the population is 1. The breadth of the range contained within these upper and lower bounds obviously varies inversely with the size of φ_{1}, that is, the greater the proportion of the population for which a causal strength is estimated, the narrower the interval of possible values of ΔP. If one is primarily concerned to know whether the overall effect is positive or negative, the value of ΔP_{1} also has a bearing on how informative the interval is, since for a given φ_{1}, the farther ΔP_{1} is from zero, the more
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the interval is skewed to the positive or negative side of the scale. Of course, some areas within the interval may reasonably be judged to be more probable than others. For instance, the strength of the causal effect is 1 when the cause is both necessary and sufficient for the effect, and one might have good reason to think it extremely improbable that this would be the case.

## Notes:

(1.) For example, the mean difference is often used as a measure of treatment impact in randomized controlled experiments. That is illustrated by the experimental evaluations of welfare‐to‐work programs discussed in section 8.2. However, other measures exist (for example, ratios) and may be preferable for some purposes.

(3.)
This procedure bears an obvious similarity to stratification in observational studies (cf. Rosenbaum 2002, 77–82). However, there is an important difference, since stratification is a method for estimating a causal effect in a population from *statistical data* concerning that *same* population. In contrast, the inference of concern here is an extrapolation: given the *causal effect* in one population, one wishes to draw conclusions about the effect in *another* population.