4.1 ASYMPTOTIC UNCONDITIONAL METHODS
P(A1=a1,A2=a2|OR, π2)
= r1
a1
ORπ2
ORπ2+(1−π2) a1
1−π2
ORπ2+(1−π2) r1−a1
× r2
a2
π2a2(1−π2)r2−a2. (4.3)
Following Section 1.2.1, we view (4.3) as a likelihood that is a function of the pa- rametersORandπ2.
Point Estimate
The unconditional maximum likelihood equations are a1= ORuπˆ2r1
ORuπˆ2+(1− ˆπ2) (4.4)
and
m1= ORuπˆ2r1
ORuπˆ2+(1− ˆπ2)+ ˆπ2r2 (4.5) whereORudenotes the unconditional maximum likelihood estimate ofOR. This is a system of two equations in the two unknownsORuandπˆ2, which can be solved to give
ORu= ωˆ1
ˆ
ω2 = πˆ1(1− ˆπ2) ˆ
π2(1− ˆπ1)= a1b2
a2b1 (4.6)
and
ˆ π2= a2
r2.
The estimates ofπ1,ω1, andω2which appear in (4.6) are given by ˆ
π1= ORuπˆ2
ORuπˆ2+(1− ˆπ2) =a1 r1
ˆ
ω1= πˆ1
1− ˆπ1 =a1 b1
and
ˆ
ω2= πˆ2
1− ˆπ2 =a2 b2.
If any ofa1,a2,b1, orb2equals 0, we replace (4.6) with ORu= (a1+.5)(b2+.5)
(a2+.5)(b1+.5).
Other approaches to the problem of zero cells are available (Walter, 1987). It can be shown thatORuis less biased when .5 is added to all the interior cells, whether they are zero or not (Walter, 1985). However, as in Chapter 3, this practice will not be followed here.
Log-Odds Ratio Transformation
The log-odds ratio log(O R)plays an important role in the analysis of data from closed cohort studies. It can be shown that the unconditional maximum likelihood estimate of log(O R)is log(ORu). For convenience of notation we sometimes write logORuinstead of log(ORu). According to the observations made in Section 3.2.2,ωˆ can be rather skewed, while log(ω)ˆ is generally more or less symmetric. It is therefore not surprising thatORu = ˆω1/ωˆ2 can also be quite skewed and that log(ORu) = log(ωˆ1)−log(ωˆ2)is usually relatively symmetric. We illustrate this with examples.
Consider the binomial distributions with parameters (π1,r1) = (.4,10) and (π2,r2) = (.2,25). ThenORu = [a1(25−a2)]/[a2(10−a1)] and log(ORu) = log[a1(25 −a2)] − log[a2(10 −a1)]. The sample space of ORu extends from 9.34×10−4 to 1071, but the distribution is extremely skewed with odds ratios less than or equal to 12.25 accounting for 95.6% of the probability. Figure 4.1(a) shows the distribution ofORu after truncation on the right at 12.25. As in Figure 1, magnitudes are not shown on the axes because we are primarily concerned with the shapes of distributions. The data points for Figure 4.1(a) were constructed by dividing the truncated sample space into 10 equally spaced intervals and then sum- ming the probability elements within each interval. The distribution of log(ORu) is shown in Figure 4.1(b). The horizontal axis has been truncated on the left and
FIGURE 4.1(a) Distribution of odds ratio for binomial distributions with parameters (.4, 10) and (.2, 25)
FIGURE 4.1(b) Distribution of log-odds ratio for binomial distributions with parameters (.4, 10) and (.2, 25)
on the right, in both instances corresponding to a tail probability of 1%. As can be seen, log(ORu)is far more symmetric thanORuand so, with respect to normal approximations, it is preferable to base calculations on log(ORu)rather thanORu. Figures 4.2(a) and 4.2(b) show the distributions of ORu and log(ORu) based on binomial distributions with parameters(π1,r1) =(.4,25)and(π2,r2)= (.2,50).
Even though both binomial distributions have a mean of 10,ORu is quite skewed, while log(ORu)is relatively symmetric. Based on empirical evidence such as this, log(ORu) should be reasonably symmetric provided the means of the component binomial distributions are 5 or more, while much larger means are required to ensure thatORuis symmetric.
FIGURE 4.2(a) Distribution of odds ratio for binomial distributions with parameters (.4, 25) and (.2, 50)
FIGURE 4.2(b) Distribution of log-odds ratio for binomial distributions with parameters (.4, 25) and (.2, 50)
Confidence Interval
The maximum likelihood estimate of var(logORu)is
var(logORu)= 1 a1 + 1
a2 + 1 b1+ 1
b2. (4.7)
Note thatvar(logORu), likeORu, is expressed entirely in terms of the interior cell entries of the 2×2 table. A(1−α)×100% confidence interval for log(O R)is
logORu,logORu
=log(ORu)±zα/2
1 a1 + 1
a2 + 1 b1+ 1
b2
which can be exponentiated to give a confidence interval forOR, ORu,ORu
=ORuexp ±zα/2
1 a1 + 1
a2+ 1 b1+ 1
b2
.
If any ofa1,a2,b1, orb2equals 0, we replace (4.7) with
var(logORu)= 1
a1+.5 + 1
a2+.5 + 1
b1+.5+ 1 b2+.5.
The convention of adding .5 when there are zero cells applies to ORu and
var(logORu), but not to the other formulas discussed in this section.
Pearson’s Test of Association
Pearson’s test of association does not have any particular connection to the odds ratio, being equally applicable to analyses based on the risk ratio and risk difference. It is
introduced here as a matter of convenience. We say there is no association between exposure and disease when the probability of disease is the same in the exposed and unexposed cohorts, that is,π1 = π2. Under the hypothesis of no association H0:π1=π2, the expected counts are defined to be
ˆ
e1=r1m1
r eˆ2=r2m1 r fˆ1=r1m2
r fˆ2=r2m2
r
Using the term “expected” in this context is potentially confusing because these quantities are not expected values (constants). This is because m1, the number of cases, is unknown until the study has been completed, and hence is a random vari- able. It would be preferable to refer to the expected counts as “fitted counts under the hypothesis of no association”; however, the term “expected counts” is well es- tablished by convention. Note that the expected count for a given interior cell is calculated by multiplying together the corresponding marginal totals and then divid- ing byr. It is easily shown that the observed and expected marginal totals agree—for example,a1+a2=m1= ˆe1+ ˆe2—and so the expected counts can be displayed as in Table 4.2.
Large differences between observed and expected counts provide evidence that the hypothesis of no association may be false. This idea is embodied in Pearson’s test of association,
Xp2= (a1− ˆe1)2 ˆ
e1 +(a2− ˆe2)2 ˆ
e2 +(b1− ˆf1)2 fˆ1
+(b2− ˆf2)2 fˆ2
(df=1). (4.8)
Observe the similarity in form to (1.14) and (1.15). The normal approximation un- derlying Pearson’s test should be satisfactory provided all the expected counts are greater than or equal to 5. According to Yates (1984), this “rule of 5” originated with Fisher (1925). Froma1+a2 = ˆe1+ ˆe2 it follows that(a1− ˆe1)2 = (a2− ˆe2)2. There are similar identities for the other rows and columns, and this allows (4.8) to be expressed in any of the following equivalent forms:
TABLE 4.2 Expected Counts:
Closed Cohort Study Disease Exposure
yes no
yes eˆ1 eˆ2 m1 no fˆ1 fˆ2 m2 r1 r2 r
X2p=(a1− ˆe1)2 1
ˆ e1 + 1
ˆ e2 + 1
fˆ1+ 1 fˆ2
(4.9)
X2p=(a1b2−a2b1)2r
r1r2m1m2 (4.10)
and
X2p= r m2
(a1− ˆe1)2 ˆ
e1 +(a2− ˆe2)2 ˆ e2
. (4.11)
Wald and Likelihood Ratio Tests of Association
Sinceπ1=π2is equivalent toOR=1, which in turn is equivalent to log(O R)=0, the hypothesis of no association can be expressed asH0:log(O R)=0. UnderH0
an estimate of var(logORu)is
var0(logORu)= 1 ˆ e1 + 1
ˆ e2+ 1
fˆ1
+ 1 fˆ2
= r3 r1r2m1m2
which is obtained from (4.7) by replacing the observed with expected counts. The Wald test and likelihood ratio tests of association are
Xw2 =(logORu)2 1
ˆ e1+ 1
ˆ e2+ 1
fˆ1
+ 1 fˆ2
−1
= (logORu)2r1r2m1m2
r3 (df=1)
and X2lr=2
a1log
a1
ˆ e1
+a2log a2
ˆ e2
+b1log b1
fˆ1
+b2log b2
fˆ2
(df=1) (4.12) respectively. As x approaches 0, the limiting value of xlog(x)is 0. If any of the observed counts is 0, the corresponding term inX2lris assigned a value of 0.
Provided the sample size is large, and sometimes even when it is not so large, Wald, score, and likelihood ratio tests (which can be shown to be asymptotically equivalent) tend to produce similar findings. When there is a meaningful difference among test results the question arises as to which of the tests is to be preferred.
Based on asymptotic properties, likelihood ratio tests are generally the first choice, followed by score tests and then Wald tests (Kalbfleisch and Prentice, 1980, p. 48;
Lachin, 2000, p. 482). Problems can arise with Wald tests when the variance is not estimated under the null hypothesis (Mantel, 1987). A major disparity among test
TABLE 4.3 Observed Counts:
Antibody–Diarrhea Diarrhea Antibody
low high
yes 12 7 19
no 2 9 11
14 16 30
results may be an indication that the sample size is too small for the asymptotic approach and that exact methods should be considered.
Example 4.1 (Antibody–Diarrhea) Table 4.3 gives a portion of the data from a cohort study conducted in Bangladesh which investigated whether antibodies present in breast milk protect infants from diarrhea due to cholera (Glass et al., 1983). These data have been analyzed by Rothman (1986, p. 169).
We first analyze the exposed and unexposed cohorts separately using the methods of Chapter 3. The estimatesπˆ1 =12/14 =.86 andπˆ2 =7/16=.44 suggest that low antibody level increases the risk of diarrhea. Exact 95% confidence intervals for π1 andπ2 are [.57, .98] and [.20, .70], respectively. The degree of overlap in the confidence intervals suggests thatπ1andπ2may be equal, but this impression needs to be formally evaluated using a test of association.
The odds ratio estimate isORu =(12×9)/(7×2)=7.71, and so once again it appears that low antibody level increases the risk of diarrhea. To be technically correct we should express this observation by saying that low antibody level seems to increase the odds of developing diarrhea. From
var(logORu)= 1 12+1
7 +1 2 +1
9 =.84 the 95% confidence interval for log(O R)is log(7.71)±1.96√
.84 = [.25,3.84].
Exponentiating, the 95% confidence interval forORis[1.28,46.37]. With a sample size as small as the one in this study, it is not surprising that the confidence interval is extremely wide. Our impression is thatORmay be larger than 1, but how much larger is difficult to say. The expected counts, shown in Table 4.4, are all greater
TABLE 4.4 Expected Counts:
Antibody–Diarrhea Diarrhea Antibody
low high yes 8.87 10.13 19
no 5.13 5.87 11
14 16 30
than 5. The Pearson, Wald, and likelihood ratio tests are similar in value and provide considerable evidence that low antibody level is associated with the development of diarrhea,
X2p= (12−8.87)2
8.87 +(7−10.13)2
10.13 +(2−5.13)2
5.13 +(9−5.87)2 5.87
=5.66(p=.02) X2w=(log 7.71)2
1
8.87+ 1
10.13+ 1 5.13+ 1
5.87 −1
=7.24(p=.01)
and Xlr2=2
12 log
12 8.87
+7 log
7 10.13
+2 log
2 5.13
+9 log
9 5.87
=6.02(p=.01).
Example 4.2 (Receptor Level–Breast Cancer) The data for this example were kindly provided by the Northern Alberta Breast Cancer Registry. This is a population- based registry that collects information on all cases of breast cancer treated in the northern half of the province of Alberta, Canada. After initial treatment, patients are reviewed on an annual basis, or more frequently if necessary. When an annual follow-up appointment is missed, an attempt is made to obtain current informa- tion on the patient by corresponding with the patient and the treating physicians.
When this fails, a search is made of provincial and national vital statistics records to determine if the patient has died and, if so, of what cause. Due to the intensive methods that are used to ensure follow-up of registrants, it is reasonable to assume that patients who are not known to have died are still alive.
The cohort for this example was assembled by selecting a random sample of 199 female breast cancer patients who registered during 1985. Entry into the cohort was restricted to women with either stage I, II, or III disease, thereby excluding cases of disseminated cancer (stage IV). It has been well documented that breast cancer mortality increases as stage of disease becomes more advanced. Another predictor of survival from breast cancer is the amount of estrogen receptor that is present in breast tissue. Published reports show that patients with higher levels of estrogen receptor generally have a better prognosis. Receptor level is measured on a continuous scale, but for the present analysis this variable has been dichotomized into low and high levels using a conventional cutoff value.
For this example the maximum length of follow-up was taken to be 5 years and the endpoint was defined to be death from breast cancer. Of the 199 subjects in the cohort, seven died of a cause other than breast cancer. These individuals were dropped from the analysis, leaving a cohort of 192 subjects. Summarily dropping subjects in this manner is methodologically incorrect, but for purposes of illustration this issue will be ignored. Methods for analyzing cohort data when there are losses to follow-up are presented in later chapters.
TABLE 4.5(a) Observed Counts:
Receptor Level–Breast Cancer Survival Receptor Level
low high
dead 23 31 54
alive 25 113 138
48 144 192
Table 4.5(a) gives the breast cancer data with receptor level as the exposure vari- able. The estimatesπˆ1=23/48=.479 andπˆ2 =31/144 =.215 suggest that low receptor level increases the mortality risk from breast cancer. Based on the explicit method, the 95% confidence intervals forπ1andπ2are [.338, .620] and [.148, .282], respectively. The confidence intervals are far from overlapping which suggests that π1andπ2are likely unequal. The odds ratio estimate isORu=(23×113)/(31× 25) = 3.35. From var(log ORu) = .125, the 95% confidence interval for ORis [1.68, 6.70]. The confidence interval is not especially narrow but does suggest that receptor level is meaningfully associated with breast cancer mortality.
At this point it is appropriate to consider the potential impact of misclassification on the odds ratio estimate. Leta1,a2,b1, andb2denote what would have been the observed counts in the absence of misclassification. From Tables 2.11 and 4.5(a), the following linear equations must be satisfied:
α1a1 +(1−β1)b1=23 α2a2 +(1−β2)b2=31 (1−α1)a1 +β1b1=25 (1−α2)a2 +β2b2=113
whereα1andα2are the sensitivities, andβ1andβ2are the specificities (Section 2.6).
One potential source of misclassification is that Registry staff may have failed to identify all the deaths in the cohort. For purposes of illustration we setα1 =α2 = .90; that is, we assume that only 90% of deaths were ascertained. It seems unlikely that someone who survived would have been recorded as having died, and so we set β1=β2=.99. The above equations become
(.90a1)+(.01b1)=23 (.90a2)+(.01b2)=31 (.10a1)+(.99b1)=25 (.10a2)+(.99b2)=113
TABLE 4.5(b) Observed Counts after Adjusting for Misclassification: Receptor Level–Breast Cancer Survival Receptor Level
low high
dead 25.30 33.21 58.51
alive 22.70 110.79 133.49
48 144 192
which have the solutions given in Table 4.5(b). After accounting for misclassifica- tion, the estimated odds ratio isORu=(25.30×110.79)/(33.21×22.70)=3.72, which is only slightly larger than the estimate based on the (possibly) misclassified data. This shows that misclassification is unlikely to be a major source of bias in the present study.
Returning to an analysis of the data in Table 4.5(a), the expected counts, given in Table 4.6, are all much greater than 5. The Pearson, Wald, and likelihood ratio tests are similar in value and provide considerable evidence that low receptor level is associated with an increased risk of dying of breast cancer,
X2p= (23−13.5)2
13.5 +(31−40.5)2
40.5 +(25−34.5)2
34.5 +(113−103.5)2 103.5
=12.40 (p< .001) X2w=(log 3.35)2
1 13.5+ 1
40.5 + 1
34.5+ 1 103.5
−1
=10.66 (p=.001).
Xlr2=2
23 log 23
13.5
+31 log 31
40.5
+25 log 25
34.5
+113 log 113
103.5
=11.68 (p=.001).
TABLE 4.6 Expected Counts: Receptor Level–Breast Cancer
Survival Receptor Level
low high
dead 13.5 40.5 54
alive 34.5 103.5 138
48 144 192
4.2 EXACT CONDITIONAL METHODS FOR A SINGLE 2×2 TABLE