1.3 Measures of causal effect 7
Technical Point 1.1
Causal effects in the population. Let E[
] be the mean counterfactual outcome had all individuals in the population
received treatment level . For discrete outcomes, the mean or expected value E[
] is defined as the weighted sum
P
() over all possible values of the random variable
,where
(·) is the probability mass function of
,
i.e.,
()=Pr[
= ]. For dichotomous outcomes, E[
]=Pr[
=1]. For continuous outcomes, the expected
value E[
] is defined as the integral
R
() over all possible values of the random variable
,where
(·)
is the probability density function of
. A common representation of the expected value that applies to both discrete
and continuous outcomes is E[
]=
R
(),where
(·) is the cumulative distribution function (cdf)ofthe
random variable
. We say that there is a non-null average causal effect in the population if E[
] 6=E[
0
] for any
two values and
0
.
The average causal effect, defined b y a contrast of means of counterfactual outcomes, is the most commonly
used population causal effect. However, a population causal effect may also be defined as a contrast of functionals,
including medians, varian ces, hazards, or cdfs of counterfactual outcomes. In general, a population causal effect can be
defined as a contrast of any functional of the marginal distributions of counterfactual outcomes under different actions
or treatment values. For example the population causal effect on the variance is defined as (
=1
) − (
=0
),
which is zero for the population in Table 1.1 since the distribution of
=1
and
=0
are identical–both having 6
deaths out of 20. In fact, the equality of these distributions imply that for any functional (e.g., mean, variance, median,
hazard,etc.), the population causal effect on the functional is zero. However, in contrast to the mean, the difference
in population variances (
=1
) − (
=0
) does not in general equal the variance of the individual causal effects
(
=1
−
=0
). For example, in Table 1.1, since
=1
−
=0
isnotconstant(−1 for 6 individuals, 1 for 6
individuals and 0 for 8 individuals), (
=1
−
=0
) 0=(
=1
) − (
=0
). Wewillbeabletoidentify
(i.e., compute) (
=1
) − (
=0
) from the data collected in a randomized trial, but not (
=1
−
=0
)
because we can never simultaneously observe both
=1
and
=0
fo r any individual, and thus the covariance of
=1
and
=0
is not identified. The above discussion is true not only for the variance but for for any nonlinear functional
(e.g., median, hazard).
1.3 Measu res of causal effect
We have seen that the treatment ‘heart transplant’ does not have a causal
effect on the outcome ‘death’ in our population of 20 family members of
Zeus. The causal null hypothesis holds because the two counterfactual risks
Pr[
=1
=1]and Pr[
=0
=1]are equal to 05. There are equivalent ways
of representing the causal null. For example, we could say that the risk
Pr[
=1
=1]minus the risk Pr
£
=0
=1
¤
is zero (05 − 05=0)orthat
the risk Pr[
=1
=1]divided by the risk Pr
£
=0
=1
¤
is one (0505=1).
That is, we can represent the causal n u ll byThe causal risk difference in the
population is the average of the in-
dividual causal effects
=1
−
=0
on the difference scale, i.e., it is
a measure of the average individ-
ual causal effect. By contrast, the
causal risk ratio in the population
is not the average of the individual
causal effects
=1
=0
on the
ratio scale, i.e., it is a measure of
causal effect in the population but
is not the average of any individual
causal effects.
(i) Pr[
=1
=1]− Pr[
=0
=1]=0
(ii)
Pr[
=1
=1]
Pr[
=0
=1]
=1
(iii)
Pr[
=1
=1] Pr[
=1
=0]
Pr[
=0
=1] Pr[
=0
=0]
=1
where the left-hand side of the equalities (i), (ii), and (iii) is the causal risk
difference, risk ratio, and odds ratio, respectively.
Suppose now that another treatment , cigarette smoking, has a causal
effect on another outcome , lung cancer, in our population. The causal null
hypothesis does not hold: Pr[
=1
=1]and Pr[
=0
=1]are not equal. In
this setting, the causal risk difference, risk ratio, and odds ratio are not 0, 1,