Simple approximations for the maximal transmission/disequilibrium test with a multi-allelic marker

Spielman et al. (1993) popularized the transmission/disequilibrium test (TD T) to test for linkage between disease and marker loci that show a populati on association. Several authors have proposed extensions to the TDT for mul ti-allelic markers. Many of these approaches exhibit a 'swamping' effect in which a marker with a strong effect is not detected by a global test that includes many markers with no effect. To avoid this effect, Schaid (1996) p roposed using the maximum of the Isi-allelic TDT statistics computed for ea ch allele versus all others combined. The maximal TDT statistic, however. n o longer follows a chi-square distribution. Here, a refinement to Bonferron i's correction for multiple testing provided by Worsley (1982) based on max imal spanning trees is applied to calculate accurate upper bounds for the t ype I error and p-values for the maximal TDT. In addition, an accurate lowe r Bonferroni bound, is applied to calculate power. This approach does not r equire any simulation-based analysis and is less conservative than the stan dard Bonferroni correction. The bounds are given for both the exact probabi lity calculations and for those based on the normal approximation. The resu lts are assessed through simulations.