2.5 COUNTERFACTUAL APPROACH TO CONFOUNDING
2.5.4 Standardized Measures of Effect
The standardized measures of effect are defined to be sRD=π1−π1∗= O−sE
r1•
sRR= π1
π1∗ = O sE
and
sOR= π1(1−π1∗)
π1∗(1−π1)= O(r1•−sE) sE(r1•−O).
Note thatsRDwas denoted byRD∗in previous sections. WhenFis not a confounder, cE = sEand so the crude and standardized measures of effect are equal. When F is a confounder, the standardized measures of effect can be thought of as overall measures of effect for the cohort after controlling for confounding due toF.
It is readily verified that sRD=
J
j=1π1jr1j−J
j=1π2jr1j
r1• =
J j=1
p1jδj (2.21)
sRR= J
j=1π1jr1j
J
j=1π2jr1j
= J
j=1p1jπ2jρj
J
j=1p1jπ2j
(2.22) and
sOR= J
j=1p1jπ1j 1−J
j=1p1jπ2j
J
j=1p1jπ2j 1−J
j=1p1jπ1j
= J
j=1p1j(1−π1j)ω2jθj J
j=1p1j(1−π2j) J
j=1p1j(1−π2j)ω2j J
j=1p1j(1−π1j) . (2.23) The second equality in (2.23) follows from identities established in Section 2.4.4.
When the risk difference, risk ratio, and odds ratio are homogeneous, it follows from (2.21)–(2.23) thatsRD=δ,sRR=ρ, and
sOR=θ J
j=1p1j(1−π1j)ω2j J
j=1p1j(1−π1j) J
j=1p1j(1−π2j)ω2j J
j=1p1j(1−π2j). Condition (i) is sufficient to ensure thatsOR=θ; but in general,sOR=θ.
In an actual study the stratum-specific values of a measure of effect may be nu- merically close but are virtually never exactly equal. Once it is determined that (after
accounting for random error) there is no effect modification, the stratum-specific es- timates can be combined to create what is referred to as a summarized or summary measure of effect. Usually this takes the form of a weighted average of stratum- specific estimates where the weights are chosen in a manner that reflects the amount of information contributed by each stratum. Numerous examples of this approach to combining stratum-specific measures of effect will be encountered in later chapters.
A summarized measure of effect may be interpreted as an estimate of the common stratum-specific value of the measure of effect. Since we have used a determinis- tic approach here, the interpretation of Tables 2.2(a)–2.2(f) is that there is no ef- fect modification only if stratum-specific values are precisely equal. Accordingly, in Tables 2.2(a)–2.2(f), when there is no effect modification (in this sense) we take the summarized value to be the common stratum-specific value. For example, in Table 2.2(d) the summary odds ratio is 6.0.
When reporting the results of a study, a decision must be made as to which of the crude, standardized, summarized, and stratum-specific values should be presented.
Table 2.6(a) offers some guidelines in this regard with respect to the risk difference and risk ratio, and Table 2.6(b) does the same for the odds ratio. Here we assume that summarization is carried out by forming a weighted average of stratum-specific values. When there is no confounding, the crude value should be reported because it represents the overall measure of effect for the cohort. On the other hand, when con- founding is present, the crude value is, by definition, a biased estimate of the overall measure of effect and so the standardized value should be reported instead. When there is no effect modification, the summarized value should be reported because it represents the common stratum-specific value of the measure of effect. However, when effect modification is present, the stratum-specific values should be given in- dividually because the pattern across strata may be of epidemiologic interest.
Under certain conditions there will be equalities among the crude, standardized, summarized, and stratum-specific values of a measure of effect. When there is no effect modification, the summarized measure of effect equals the common stratum- TABLE 2.6(a) Guidelines for Reporting Risk Difference and Risk Ratio Results
Effect modification
Confounding No Yes
No crude=summarized crude and stratum-specific
Yes standardized=summarized standardized and stratum-specific
TABLE 2.6(b) Guidelines for Reporting Odds Ratio Results Effect modification
Confounding No Yes
No crude and summarized crude and stratum-specific
Yes standardized and summarized standardized and stratum-specific
specific value. When there is no confounding, the crude and standardized values of a measure of effect are equal (by definition). Also, when there is no confounding, identity (2.17) is satisfied and therefore so are (2.7) and (2.10). In this case the risk difference and risk ratio are averageable. If, in addition, there is no effect modifi- cation, these measures of effect are strictly collapsible. This justifies the equality in the upper left cell of Table 2.6(a). The equality in the lower left cell follows from re- marks made in connection with (2.21) and (2.22). In general, the preceding equalities do not hold for the odds ratio and so they have not been included in Table 2.6(b). This means that when there is no effect modification, two values of the odds ratio should be reported—the crude and summarized values when there is no confounding, and the standardized and summarized values when confounding is present.
We now illustrate some of the above considerations with specific examples. From Table 2.2(c) we haveO=230,cE=(120/300)300=120, and
sE= 80
200
100+ 40
100
200=120.
SincecE =sEthere is no confounding and so the crude value of each measure of effect should be reported. For each measure of effect there is effect modification and so the stratum-specific values should be given individually rather than summarized.
Now consider Table 2.2(d), whereO=170,cE=(80/300)300=80, and sE=
60 100
100+
20 200
200=80.
SincecE =sEthere is no confounding and so the crude value of each measure of effect should be reported. For the risk ratio, effect modification is present and so the stratum-specific values should be given separately. For the risk difference, there is no effect modification and, consistent with Table 2.6(a), the crude and summarized values are equal. For the odds ratio, effect modification is absent. Consistent with Table 2.6(b), the crude value, OR = 3.6, does not equal the summarized value, θ=6.0, and so both should be reported. We view the crude odds ratio as the overall odds ratio for the cohort, and we regard the summarized odds ratio as the common value of the stratum-specific odds ratios.
The fact that two odds ratios are needed to characterize the exposure–disease rela- tionship in Table 2.2(d) creates frustrating difficulties with respect to interpretation, as we now illustrate. Suppose that the strata have been formed by categorizing sub- jects according to sex. So for males and females considered separately the odds ratio isθ =6.0, whereas for the population as a whole it isOR=3.6. This means that, despite effect modification being absent, there is no single answer to the question
“What is the odds ratio for the exposure–disease relationship?”Intuitively it is diffi- cult to accept the idea that even though the odds ratios for males and females are the same, this common value is nevertheless different from the odds ratio for males and females combined. Furthermore, there is the frustration that the difference cannot be blamed on confounding.
As has been noted previously, the source of the difficulty is that condition (ii) is not sufficient to ensure that the odds ratio is averageable. This drawback of the odds ratio has led Greenland (1987) to argue that this measure of effect is epidemiologi- cally meaningful only insofar as it approximates the risk ratio or hazard ratio (defined in Chapter 8). As noted in Section 2.2.2, the odds ratio is approximately equal to the risk ratio when the disease is rare, and so using the odds ratio is justified when this condition is met. Alternatively, the failure of the odds ratio to be averageable can be acknowledged and the necessity of having to report two odds ratios accepted as an idiosyncrasy of this measure of effect.
It is instructive to apply the above methods to data from the University Group Diabetes Program (1970), a study that was quite controversial whenfirst published.
Rothman and Greenland (1998, Chapter 15) analyzed these data using an approach that is slightly different from what follows. The UGDP study was a randomized controlled trial comparing tolbutamide (a blood sugar-lowering drug) to placebo in patients with diabetes. Long-standing diabetes can cause cardiovascular complica- tions, and this increases the risk of such potentially fatal conditions as myocardial infarction (heart attack), stroke, and renal failure. Tolbutamide helps to normalize blood sugar and would therefore be expected to reduce mortality in diabetic patients.
Table 2.7 gives data from the UGDP study stratified by age at enrollment, with death from all causes as the study endpoint. The following analysis is based on the risk difference.
SincecRD=.045 it appears that, contrary to expectation, tolbutamide increases mortality. Note also that Simpson’s paradox is present. As will be discussed in Sec- tion 2.5.5, randomization is expected to produce treatment arms with similar patient characteristics. But this can only be guaranteed over the course of many replications of a study, not in any particular instance. From Table 2.7,p12 =98/204=.48 and p22 =85/205=.41. So the proportion of subjects in the 55+age group is greater in the tolbutamide arm than in the placebo arm. This raises the possibility that the ex- cess mortality observed in patients receiving tolbutamide might be a consequence of their being older. Since age is associated with exposure (type of treatment) and also increases mortality risk, age meets the two necessary conditions to be a confounder.
TABLE 2.7 UGDP Study Data Age<55 Survival Tolbutamide
yes no
dead 8 5
alive 98 115 106 120
RD .034
RR 1.81
OR 1.88
Age 55+ Tolbutamide
yes no
22 16
76 69
98 85
.036 1.19 1.25
All ages Tolbutamide
yes no
30 21
174 184 204 205
.045 1.44 1.51
Before employing the techniques developed above, we need to verify two as- sumptions, namely, that age is not in the causal pathway between tolbutamide and all-cause mortality, and there is no residual confounding in each of the age-specific strata. Since tolbutamide does not cause aging, the first assumption is clearly sat- isfied. There is evidence in the UGDP data (not shown) that variables other than age were distributed unequally in the two treatment arms. However, for the sake of illustration we assume that there is no residual confounding in each stratum. Then O=30,cE=(21/205)204=20.90, andsE=(5/120)106+(16/85)98=22.86.
The difference between cE andsE is not large, but there is enough of a dispar- ity to suggest that age is a confounder. On these grounds we takesRD = (30− 22.86)/204 = .035 to be the overall risk difference as opposed to the somewhat largercRD =(30−20.90)/204 =.045. So, even after accounting for age, tolbu- tamide still appears to increase mortality risk in diabetic patients.
At the beginning of this chapter we introduced the concept of confounding as a type of systematic error. The confounding in the UGDP data has its origins in the uneven manner in which randomization allocated subjects to the tolbutamide and placebo arms. The apparent conflict in terminology between confounding (system- atic error) and randomization (random error) is resolved once it is realized that con- founding is a property of allocation (Greenland, 1990). Therefore, given (conditional on) the observed allocation in the UGDP study, it is appropriate to consider age as a source of confounding.