Transmission disequilibrium test

The transmission disequilibrium test (TDT) was proposed by Spielman, McGinnis and Ewens (1993) as a family-based association test for the presence of genetic linkage between a genetic marker and a trait. It is an application of McNemar's test. The transmission disequilibrium test (TDT) was proposed by Spielman, McGinnis and Ewens (1993) as a family-based association test for the presence of genetic linkage between a genetic marker and a trait. It is an application of McNemar's test. A specificity of the TDT is that it will detect genetic linkage only in the presence of genetic association.While genetic association can be caused by population structure, genetic linkage will not be affected, which makes the TDT robust to the presence of population structure. We first describe the TDT in the case where families consist of trios (two parents and one affected child). Our description follows the notations used in Spielman, McGinnis & Ewens (1993). The TDT measures the over-transmission of an allele from heterozygous parents to affected offsprings.The n affected offsprings have 2n parents. These can be represented by the transmitted and the non-transmitted alleles M 1 {displaystyle M_{1}} and M 2 {displaystyle M_{2}} at some genetic locus. Summarizing the data in a 2 by 2 table gives: The derivation of the TDT shows that one should only use the heterozygous parents (total number b+c).The TDT tests whether the proportions b/(b+c) and c/(b+c) are compatible with probabilities (0.5, 0.5).This hypothesis can be tested using a binomial (asymptotically chi-square) test with one degree of freedom: χ 2 = [ b − ( b + c ) / 2 ] 2 ( b + c ) / 2 + [ c − ( b + c ) / 2 ] 2 ( b + c ) / 2 = ( b − c ) 2 b + c {displaystyle chi ^{2}={frac {^{2}}{(b+c)/2}}+{frac {^{2}}{(b+c)/2}}={frac {(b-c)^{2}}{b+c}}} A derivation of the test consists of using a population genetics model to obtain the expected proportions for the quantities a , b , c {displaystyle a,b,c} and d {displaystyle d} in the table above. In particular, one can show that under nearly all disease models the expected proportion of b {displaystyle b} and c {displaystyle c} are identical. This result motivates the use of a binomial (asymptotically χ 2 {displaystyle chi ^{2}} ) test to test whether these proportions are equal. On the other hand, one can also show that under such models the proportions a , b , c {displaystyle a,b,c} and d {displaystyle d} are not equal to the product of the marginals probabilities ( a + b ) / 2 n {displaystyle (a+b)/2n} , ( c + d ) / 2 n {displaystyle (c+d)/2n} and ( a + c ) / 2 n {displaystyle (a+c)/2n} , ( b + d ) / 2 n {displaystyle (b+d)/2n} . A rewording of this statement would be that the type of the transmitted allele is not, in general, independent of the type of the non-transmitted allele. A consequence is that a χ 2 {displaystyle chi ^{2}} test for homogeneity/independence does not test the appropriate hypothesis, and thus, only heterozygous parents are included. The TDT can be readily extended beyond the case of trios. We keep following the notations of Spielman, McGinnis & Ewens (1993). Consider a total of h {displaystyle h} heterozygous parents. We use the fact that the transmission to different children are independent. The information can be then summarized in three categories: