This example is taken from Canner (1987). It concerns the overview of six major clinical trials of aspirin compared with placebo in coronary heart disease. The all- cause mortality figures are given in Table 6.8. The meta-analysis is based on the unconditional maximum likelihood estimation of the log-odds ratio for mortality on aspirin relative to placebo (formulae (3.1) and (3.2)). A CI plot (Figure 6.3) shows that the first five trials are in remarkably good agreement. The test of heterogeneity is not significant (p=0.96), and the overall test of a treatment difference is highly significant (p=0.001). However, when study 6 is added the picture is changed dramatically. In study 6 there is higher mortality on aspirin than on placebo. Because this study is much larger than the other studies, its inclusion reduces the positive effect to a level which is not statistically significant (p=0.11).
In addition, the test for heterogeneity is pushed towards borderline significance (p=0.08). Canner presents his investigations of this apparent heterogeneity of the findings, focusing on the large difference between study 6 and the others.
The first potential source of heterogeneity explored was that to do with the design and operational features of the six trials. Table 6.9 shows some of the design features of the trials. Study 6 had the smallest mean age, but the range across all trials was very small. Two of the trials included males only, but study 6 was one of the four that included both sexes. The total daily dose of aspirin varied from 300 mg to 1500 mg, but study 6 with a dose of 1000 mg was close to three other studies (2, 4 and 5). The dosage schedule ranged from once to
Study 1 Study 2 Study 3 Study 4 Study 5 Study 6 Fixed (1−6) Random (1−6)
Fixed (1−5) Random (1−5)
−2.0 −1.5 −1.0 −0.5 0.0 Log-odds ratio
0.5 1.0 1.5 2.0
Figure 6.3 Aspirin in coronary heart disease: the log-odds ratio of mortality on aspirin relative to placebo. Individual study estimates and overall fixed and random effects estimates are presented, with 95% confidence intervals.
An investigation 171 Table 6.8 Aspirin in coronary heart disease: log-odds ratio of mortality on aspirin relative to placebo, using formulae (3.1) and (3.2)
Study Aspirin Placebo ˆθi wi θˆiwi ˆθ2iwi
Number of Total Number of Total deaths number deaths number
patients patients
1 49 615 67 624 −0.329 25.7 −8.46 2.78
2 44 758 64 771 −0.385 24.3 −9.34 3.59
3 27 317 32 309 −0.216 13.3 −2.86 0.62
4 102 832 126 850 −0.220 48.8 −10.71 2.35
5 85 810 52 406 −0.225 28.4 −6.41 1.44
Total (1–5) 140.5 −37.78 10.78
6 246 2267 219 2257 0.125 104.0 12.96 1.62
Total (1–6) 244.5 −24.82 12.40
Studies 1–5
U=(−37.78)2/140.5=10.16; (1 df)p=0.001 Q=10.78−10.16=0.63; (4 df)p=0.96 θˆ= −37.78/140.5= −0.269; se(ˆθ)=1/√
140.5=0.084 95% CI=(−0.269±1.96/√
140.5)=(−0.434,−0.104) Studies 1–6
U=(−24.82)2/244.5=2.52; (1 df)p=0.11 Q=12.40−2.52=9.88; (5 df)p=0.08 θˆ= −24.82/244.5= −0.102; se(ˆθ)=1/√
244.5=0.064 95% CI=(−0.102±1.96/√
244.5)=(−0.227,0.024)
three times daily, although study 6 had a twice daily dosing regimen. The mean time from the qualifying myocardial infarction to entry into the trial ranged from 8 days to 85 months, with study 6 having a mean of 25 months. The mean duration of follow-up varied from study to study from 11.9 to 41.0 months, with studies 5 and 6 having the longest follow-up times. Canner concluded that there was nothing obvious in the design features of the studies that might explain any possible differences in the mortality results.
The next line of investigation undertaken by Canner was to consider adjustment of the individual study estimates for prognostic factors. For each of seven risk factors (history of congestive heart failure, history of angina pectoris, history of ECG-documented arrhythmia, use of digitalis, use of nitroglycerin, use of propranolol or other beta-blockers, and use of other drugs), it was found that the occurrence was significantly higher in the aspirin group than the placebo group in study 6. This might explain the more negative findings of the study.
For three of the studies (2, 5 and 6) it was possible to adjust the log-odds ratio estimate for a variety of baseline characteristics. As different baseline variables were collected in each study, the adjustment was undertaken separately for each
Table 6.9 Aspirin in coronary disease: design features of the studies
Study1 Study 2 Study 3 Study 4 Study 5 Study 6 Time period 1971–73 1972–75 1970–77 1975–79 1975–79 1975–79 Number of
patients
1126 1529 626 1682 1216 4524
Mean age 55.0 56.5 58.9 56.2 56.3 54.8
Gender M M M, F M, F M, F M, F
Total daily dose (mg)
300 972 1500 900 972 1000
Dosage schedule
o.d. t.i.d. t.i.d. t.i.d. t.i.d. b.i.d.
Time from qualifying MI to entry
mean 70 days 85 mo. 40 days 8 days 20 mo. 25 mo.
range 0.5–6 mo. 21 days– 28–42 days– 2–60 mo. 2–60 mo.
22 yr days weeks
Duration of patient follow-up (months)
mean 11.9 22.0 24.0 12.0 41.0 39.6
range 2–30 10–28 24–24 12–12 35–48 35–48
Notes: MI=myocardial infarction; o.d.=once daily; b.i.d.=twice daily; t.i.d.=three times daily.
study. The estimates of the log-odds ratios presented in the rest of this section are calculated from summary statistics from the Canner paper and so will be approximate. Adjustment in study 6 resulted in a reduction of the log-odds ratio to 0.054, but there was only a minor effect on studies 2 and 5. No adjustment was possible for the other three studies. A repeated fixed effects meta-analysis of the six studies using the three adjusted estimates in place of the unadjusted estimates was undertaken. TheQstatistic changed from 9.88 to 7.30, resulting in ap-value of 0.20. The fixed effects estimate of the log-odds ratio changed from−0.102 to
−0.128, a statistically significant effect (p=0.04). Thus the baseline imbalance in study 6 may have contributed to the heterogeneity.
Although the data so far have been treated as binary responses, it may be more appropriate to treat them as survival times as this would allow for the differing follow-up times of the patients. In addition, mortality rates over specific time periods, such as one-year mortality rates, could be investigated using survival analysis techniques. In the paper, mortality within each year of follow-up was analysed, using the log-odds ratio approach based on binary data. The results are
An investigation 173 Table 6.10 Aspirin in coronary heart disease: log-odds ratio of mortality on aspirin relative to placebo by year of follow-up
Study 1st year 2nd year 3rd and 4th years
1 −0.312 −0.676 ∞
2 −0.200 −0.842 0.938
3 −0.245 −0.146 –
4 −0.214 – –
5 −0.063 −0.673 −0.059
6 −0.178 0.214 0.260
θˆ −0.211 −0.174 0.231
U (1 df) 6.00;p=0.01 1.72;p=0.19 3.06;p=0.08 Q 0.48; (5 df)p=0.99 11.52; (4 df)p=0.02 2.53; (3 df)p=0.47
presented in Table 6.10. For mortality during the first year of follow-up, there is a consistent beneficial effect of aspirin amongst all six trials. The fixed effects estimate of the log-odds ratio was−0.211, a statistically significant effect (p=0.01). The test for heterogeneity was not significant (Q=0.48 (5 df),p=0.99). For the second year of follow-up study 4 is excluded because it only had a 1-year follow- up period. There is evidence of heterogeneity amongst the other five studies (Q=11.52 (4 df),p=0.02). In study 6 there is higher mortality in the aspirin group than in the placebo group, whereas the opposite is true for the other studies.
During the third and fourth years of follow-up the four studies contributing data show no effect or an adverse effect of aspirin over placebo. As there were no deaths during this period in the placebo group in study 1, the estimated log-odds ratio is
∞, although this study appears to have been included in the analysis presented in the paper. Heterogeneity is not significant (Q=2.53 (3 df),p=0.47). It appears that after a consistently positive effect of aspirin in the first year, the benefit disappears by the third year. In study 6 the reversal of the effect occurs earlier than in the other studies, causing the apparent heterogeneity.
To see whether or not heterogeneity was confined just to mortality, fixed effects meta-analyses were undertaken on a number of non-fatal outcomes. Data on the occurrence of non-fatal myocardial infarction reported in studies 2 – 6 provided evidence of a strong beneficial aspirin effect but no significant evidence of heterogeneity. On a number of other cardiovascular and gastrointestinal outcomes reported in studies 2, 5 and 6 there was good agreement between the studies.
Summarizing the results of the investigation into the apparent heterogeneity of the mortality results amongst the studies, the following conclusions were drawn. There was no obvious difference in the design of study 6 to offer an explanation. The heterogeneity was confined to the second year of follow-up, during which a reversal of the beneficial effect of aspirin began for study 6 but not the other studies. This reversal did not begin in the other longer-term studies
until later. With respect to a number of other outcomes recorded, there was good agreement between study 6 and the other studies. Adjustment for imbalance in the distribution of risk factors between the two treatment groups in study 6 helped to reduce the amount of heterogeneity. The overall conclusion was that it seemed as if there was no real heterogeneity in mortality findings amongst the six studies, and that the results were consistent with a true aspirin effect that was beneficial in the short term of 1 – 2 years.