2.3 CONFOUNDING
2.3.3 Some Hypothetical Examples of Closed Cohort Studies
As illustrated in the preceding section, stratification plays an important role in the analysis of epidemiologic data, especially in connection with confounding. In this section we examine a series of hypothetical closed cohort studies in order to develop a sense of how the risk difference, risk ratio, and odds ratio behave in crude and strat- ified 2×2 tables. This will motivate an analysis that will be useful in the discussion of confounding. In an actual cohort study, subjects are randomly sampled from a popu- lation, a process that introduces random error. For the remainder of this chapter it is convenient to avoid issues related to random error by assuming that the entire pop- ulation has been recruited into the cohort and that, for each individual, the outcome with respect to developing the disease is predetermined (although unknown to the investigator). In this way we replace the earlier probabilistic (stochastic) approach with one that is deterministic. Strictly speaking, we should now refer toπ1andπ2
in Table 2.1(b) as proportions rather than probabilities because there is no longer a stochastic context. However, for simplicity of exposition we will retain the earlier terminology. In what follows, we continue to make reference to the population, but will now equate it with the cohort at the start of follow-up.
Tables 2.2(a)–2.2(e) give examples of closed cohort studies in which there are three variables: exposure(E), disease(D), and a stratifying variable,(F). We use E = 1, D = 1, and F = 1 to denote the presence of an attribute and use E = 2, D = 2, and F = 2 to indicate its absence. Here, as elsewhere in the book, a dot•denotes summation over all values of an index. We refer to the tables with the
TABLE 2.2(a) Hypothetical Closed Cohort Study:FIs Not a Risk Factor for the Disease andFIs Not Associated with Exposure
F=1
D E
1 2
1 70 40
2 30 60
100 100
RD .30
RR 1.8
OR 3.5
F=2 E
1 2
140 80 60 120 200 200
.30 1.8 3.5
F= • E
1 2
210 120 90 180 300 300
.30 1.8 3.5
TABLE 2.2(b) Hypothetical Closed Cohort Study:FIs Not a Risk Factor for the Disease andFIs Not Associated with Exposure
F=1
D E
1 2
1 70 40
2 30 60
100 100
RD .30
RR 1.8
OR 3.5
F=2 E
1 2
160 80 40 120 200 200
.40 2.0 6.0
F= • E
1 2
230 120 70 180 300 300
.37 1.9 4.9
TABLE 2.2(c) Hypothetical Closed Cohort Study:FIs Not a Risk Factor for the Disease andFIs Associated with Exposure
F=1
D E
1 2
1 70 80
2 30 120
100 200
RD .30
RR 1.8
OR 3.5
F=2 E
1 2
160 40
40 60
200 100 .40 2.0 6.0
F= • E
1 2
230 120 70 180 300 300
.37 1.9 4.9
TABLE 2.2(d) Hypothetical Closed Cohort Study:FIs a Risk Factor for the Disease and FIs Not Associated with Exposure
F=1
D E
1 2
1 90 60
2 10 40
100 100
RD .30
RR 1.5
OR 6.0
F=2 E
1 2
80 20
120 180 200 200
.30 4.0 6.0
F= • E
1 2
170 80 130 220 300 300
.30 2.1 3.6
TABLE 2.2(e) Hypothetical Closed Cohort Study:FIs a Risk Factor for the Disease and FIs Associated with Exposure
F=1
D E
1 2
1 90 120
2 10 80
100 200
RD .30
RR 1.5
OR 6.0
F=2 E
1 2
30 10
170 90 200 100
.05 1.5 1.6
F= • E
1 2
120 130 180 170 300 300
−.03 .92 .87
TABLE 2.2(f) Hypothetical Closed Cohort Study:FIs a Risk Factor for the Disease andF Is Associated with Exposure
F=1
D E
1 2
1 140 50
2 60 50
200 100
RD .20
RR 1.4
OR 2.3
F=2 E
1 2
120 20 180 180 300 200
.30 4.06.0
F =3 E
1 2
70 90
30 210 100 300
.40 2.35.4
F= • E
1 2
330 160 270 440 600 600
.28 2.13.4
headings“F =1”and“F =2”as the stratum-specific tables and refer to the table with the heading“F = •”as the crude table. The crude table is obtained from the stratum-specific tables by collapsing overF—that is, summing over strata on a cell- by-cell basis. The interpretation of the subheadings of the tables will become clear shortly.
In Table 2.2(a), for each measure of effect, the stratum-specific values are equal to each other and to the crude value. In fact, the entries in stratum 2 are, cell by cell, double those in stratum 1. There would seem to be little reason to retain stratification when analyzing the data in Table 2.2(a). In Tables 2.2(b) and 2.2(c), for each measure of effect, the stratum-specific values increase from stratum 1 to stratum 2. Observe that each of the crude measures of effect falls between the corresponding stratum- specific values.
When some or all of the stratum-specific values of a measure of effect differ (across strata of F) we describe this phenomenon using any of the following syn- onymous expressions: The measure of effect is heterogeneous (across strata of F),
Fis an effect modifier (of the measure of effect), and there is an interaction between EandF. These expressions will be used interchangeably in subsequent discussions.
Note that the decision as to whether a measure of effect is heterogeneous is based exclusively on the stratum-specific values and does not involve the crude value. For each of the measures of effect under consideration, whenEandFare dichotomous, it can be shown thatF is an effect modifier of the E–Dassociation if and only if Eis an effect modifier of theF–Dassociation. This means that effect modification is a symmetric relationship between E andF. See Section 2.5.6 for a demonstra- tion of this result for the risk ratio. When heterogeneity is absent—that is, when all the stratum-specific values of the measure of effect are equal—we say there is ho- mogeneity. In Table 2.2(d) there is effect modification of the risk ratio, but not the risk difference or odds ratio. This illustrates that the decision as to whether effect modification is present depends on the measure of effect under consideration.
Surprisingly, it is possible for a crude measure of effect to be either greater or less than any of the stratum-specific values, a phenomenon referred to as Simpson’s para- dox (Simpson, 1951). In Table 2.2(e), all three measures of effect exhibit Simpson’s paradox. Here the crude values not only lie outside the range of the stratum-specific values but, in each instance, point to the opposite risk relationship. The odds ratio in Table 2.2(d) also exhibits Simpson’s paradox, afinding that is all the more striking given that there is no effect modification.