## 8.1 Need for individual causal effects

John had a stroke and was brought into the emergency department at the local hospital. After his initial rescue his cardiologist told him that his medical condition made him eligible for two types of surgery, a standard surgery and a new surgery. Both are known to prolong life, but the effect varies across patients. How do John and his doctor determine which of these two interventions would be best?

Table 8.1: Number of years lived for eight patients under a new surgery $$Y(1)$$ and under standard treatment $$Y(0)$$. Hypothetical data.
Patient Age $$Y(1)$$ $$Y(0)$$ $$\tau_i$$
John 68 14 13 +1
Caren 76 0 6 -6
Joyce 66 1 4 -3
Robert 81 2 5 -3
Ruth 70 3 6 -3
Nick 72 1 6 -5
Peter 81 10 8 +2
Torey 72 9 8 +1
Average -2

In order to answer this question, we would ideally like to know John’s survival under both options, and choose the option that gives him the longest life. Table 8.1 contains the hypothetical number of years lived for eight patients under a new surgery, labeled $$Y(1)$$, and the number for years under standard treatment, labeled $$Y(0)$$.

Let us define the individual causal effect $$\tau_i$$ for individual $$i$$ as the difference between the two outcomes

$\tau_i = Y_i(1) - Y_i(0) \tag{8.1}$

so John gains one year because of surgery. We see that the new surgery is beneficial for John, Peter and Torey, but harmful to the others.

In addition, let the average causal effect, or ACE, be the mean ICE over all units, i.e.,

$\tau = \frac{1}{n}\sum_{i=1}^n Y_i(1) - Y_i(0) \tag{8.2}$ s In Table 8.1 it is equal to $$\tau$$ = (1 - 6 - 3 - 3 - 3 - 5 + 2 + 1) / 8 = -2, so applying the new surgery will reduce average life expectancy in these patients by two years. Knowing this, we would be inclined to conclude that the new surgery is harmful, and should thus not be performed. However, that would also take away valuable life years from John, Peter and Torey.

What would the perfect doctor do instead? The perfect doctor would assign the best treatment to each patient, so that only John, Peter and Torey would get the new surgery. Under that assignment of treatments, these three persons live for another (14 + 10 + 9)/3 = 11 years, whereas the others live for another 5.4 years. Seeing these two numbers only, we might be tempted to conclude that surgery increases life expectancy by 11 - 5.4 = 5.6 years, but that conclusion would be far off. If, because of this apparent benefit, we were to provide surgery to everyone, we would actually be shortening their lives by an average of two years, a decision worse than withholding surgery for everybody. Evidently the best policy is to treat some, but not others. But how do we know whom to treat? The answer is that we need to know the ICE for every patient.

The ICE is of genuine interest in many practical settings. In clinical practice, we treat an individual, not a group, so we need an estimate of the effect for that individual. Distinguishing the ICE from the ACE allows for a more precise and clear understanding of causal inference. The ICE is more fundamental than the ACE. We can calculate ACE from a set of ICE estimates, but cannot go the other way around. Thus, knowing the ICE allows for easy estimation of every other causal estimand. It is true that estimates of the ICE might turn out to be more variable than group-wise causal estimates. But, paraphrasing Tukey, it might be better to have an approximate answer to the right question than a precise answer to the wrong one.

The case of the perfect doctor above is an example of a phenomenon known as heterogeneity in treatment effect (HTE). There is no consensus about the importance of HTE in practice. For example, Rothwell (2005) contends that genuine HTE is very rare, especially in those cases where treatment effects reverse in different groups. Brand and Xie (2010) however argued that HTE is “the norm, not an exception.” In an informal search of the scientific literature, I had no difficulty in locating examples of HTE in a wide variety of disciplines. Here are some:

• The effect of financial deterrents on giving birth to a third child depends on whether the first two children have the same sex (Angrist 2004);

• The effect of job training programs on earnings depends on age, sex, race and level of education (Imai and Ratkovic 2013);

• Coronary artery bypass grafting (CABG) reduces total mortality in medium- and high-risk patients, while low-risk patients showed a non-significant trend toward increased mortality (Yusuf et al. 1994);

• Estrogen replacement therapy increased HDL cholesterol, but the increase was twice as high in women with the ER-$$\alpha$$ IVS1-401 C/C genotype (Herrington et al. 2002);

• Social skills training programs had no effect in reconviction rates of criminal offenders, but did unintentionally increase reconviction rates of psychopaths (Hare et al. 2000);

• Individuals who are least likely to obtain a college education benefit most from college (Brand and Xie 2010).

In all of these cases treatment heterogeneity was partly explained by covariates. Of course, in practice heterogeneity may be present but we may be unable to explain it. Thus, here we see only a (possible tiny) subset of forms of HTE.