Confidence Intervals for Pooled Estimates

To obtain confidence intervals for the pooled estimates of effect we need variance formulas to combine with the point estimates. Table 10-4 lists variance formulas for the various pooled estimates that we consider in this chapter.

Although the formulas may look complicated, they are easy to apply. Each variance formula corresponds to a particular type of stratified data. First consider the pooled risk difference. For the data in Table 10-3, we calculated an RDMH of 0.035. We can derive the variance for this estimate and a confidence interval by applying the first formula from Table 10-4 to the data in Table 10-3.

This gives a standard error of (0.001028)% = 0.0321 and a 90% confidence interval of 0.035 ± 1.645 • 0.0321 = 0.035 ± 0.053 = -0.018 to 0.088. The

Table 10-4 Variance Formulas for Pooled Analyses

confidence interval is broad enough to indicate a fair amount of statistical uncertainty in the finding that tolbutamide is worse than placebo. It is notable, however, that the data are not very compatible with any compelling benefit for tolbutamide.

A confidence interval can be constructed for the risk ratio estimated from the same stratified data. In that case, an investigator would use the second formula in Table 10-4, setting limits on the log scale, as we did in the previous chapter for crude data. The variance for the logarithm of the RRMH can be calculated as

This result gives a standard error for the logarithm of the RR of (0.0671)1/2 = 0.259 and a 90% confidence interval of 0.87 to 2.0.

The interpretation for this result is similar to the interpretation for the confidence interval of the risk difference, which is as expected because the two measures of effect and their respective confidence intervals are alternative ways of expressing the same finding from the same set of data.

As another example, consider again the data in Table 1-2. We can calculate the risk ratio for 20-year risk of death among smokers compared with nonsmokers across the seven age strata using Equation 10-2. This calculation gives an overall Mantel-Haenszel risk ratio of 1.21, with a 90% confidence interval of 1.06 to 1.38. The Mantel-Haenszel risk ratio is different from the crude risk ratio of 0.76, and as discussed in Chapter 1, it points in the opposite direction.

 
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