Impact of Haplotype-Frequency Estimation Error on Test Statistics in Association Studies

To the Editor: Curtis and Sham (2006 [in this issue]) discuss the impact of haplotype-frequency estimation error on several statistics in case-control association studies. Of course, it is worth investigating how the error of haplotype-frequency estimation affects the results of association studies. However, to assess systematically the impact of haplotype-frequency estimation errors on association studies is not a trivial task. Here, we only consider some simple cases, to respond to the letter from Curtis and Sham (2006). Consider the simple case of two loci, each with two alleles, yielding four haplotypes with frequencies f1, f2, f3, and f4. Consider an example in which sample size n=500, f1=f, f2=2f, f3=3f, and f4=1-f1-f2-f3. Under the null hypothesis of equal haplotype frequencies in both cases and controls, figure 1 shows the asymptotic average values for the χ2, entropy-based, and likelihood-ratio–based heterogeneity test statistics as a function of the parameter f. In theory, under the null hypothesis, the values of these three statistics should be equal to zero. However, because of sampling error and haplotype-frequency estimation error, these three statistics are not actually equal to zero. Figure 1 shows that, on average, the χ2 statistic and the entropy-based statistic are always smaller than the likelihood-ratio–based heterogeneity statistic. This implies that, for these haplotype frequencies, the χ2 statistic and the entropy-based statistic should have smaller type I error rates than that of the likelihood-ratio–based heterogeneity test statistic. To analyze the example given by Curtis and Sham (2006), we created the graph in figure 2. Figure 2 shows the values of these three statistics when n=500, f1=4f, f2=1.5f, f3=1.5f, and f4=1-f1-f2-f3. We can see that, in most ranges of the parameter f, the average values of the χ2 statistic and the entropy-based statistic are smaller than that of the likelihood-ratio–based heterogeneity test statistic. However, when the frequencies of the four haplotypes are close to 0.5, 0.2, 0.2, and 0.1—which are the haplotype frequencies used to generate table 1 in the letter from Curtis and Sham (2006)—the average values of the χ2 statistic and the entropy-based statistic are much larger than that of the likelihood-ratio–based heterogeneity test statistic. The range of the haplotype frequencies leading to this result is not large. Both figures ​figures11 and ​and22 demonstrate that the average values of the χ2 statistic and the entropy-based statistic are similar and that, in most cases, the average values of both the χ2 statistic and the entropy-based statistic are smaller than that of the likelihood-ratio–based heterogeneity test statistic. Figure 1 Asymptotic average values for the χ2, entropy-based, and likelihood-ratio–based heterogeneity test statistics as a function of the frequency parameter f, under the assumptions n=500, f1=f, f2=2f, f3=3f, and f4=1-f1-f2-f3. Figure 2 Asymptotic average values for the χ2, entropy-based, and likelihood-ratio–based heterogeneity test statistics as a function of the frequency parameter f, under the assumptions n=500, f1=4f, f2=1.5f, f3=1.5f, and f4=1-f1-f2-f3. If the covariance matrices of the moment estimates of haplotype frequencies in the standard χ2 statistic and the entropy-based statistic are replaced with the covariance matrix of maximum-likelihood estimates of the haplotype frequencies, the average values of the three statistics will be asymptotically the same. Therefore, if we were to use the covariance matrix from the maximum-likelihood estimates, any differences among the three test statistics would be, on average, small. In summary, (1) we formulate the entropy-based statistic in terms of the estimated frequencies, not counts as implemented in the simulation performed by Curtis and Sham (2006); (2) the estimation error for haplotype frequencies will have an impact on the test statistic under the null hypothesis, and the magnitude of this effect will depend on the haplotype frequencies; and (3) asymptotically, the impact of the haplotypefrequency estimation error on the standard χ2 statistic and the entropy-based statistic is smaller than the impact on the likelihood-ratio–based heterogeneity test statistic. Therefore, the claim that type I error rates of the heterogeneity test are always much smaller than those of the standard χ2 statistic and the entropy-based statistic is incorrect. The effects of the haplotype-frequency estimation errors on type I error rates of a test are complex and should be investigated by both theoretical analysis and intensive simulation studies over large parameter spaces, not just over a small range of haplotype frequencies.