Transmission disequilibrium test

Description of the test 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 affected offsprings have parents. These can be represented by the transmitted and the non-transmitted alleles and at some analogical 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 ). The TDT tests whether the proportions and are compatible with probabilities . This hypothesis can be tested using a binomial (asymptotically chi-square) test with one degree of freedom: \chi^2 = \frac{ [b - (b+c)/2]^2}{(b+c)/2} + \frac{ [c - (b+c)/2]^2}{(b+c)/2} = \frac{(b-c)^2}{b+c} Outline of the test derivation A derivation of the test consists of using a population genetics model to obtain the expected proportions for the quantities in the table above. In particular, one can show that under nearly all disease models the expected proportion of and are identical. This result motivates the use of a binomial (asymptotically ) test to test whether these proportions are equal. On the other hand, one can also show that under such models the proportions are not equal to the product of the marginals probabilities \tfrac{a+b}{2n}, \tfrac{c+d}{2n} and \tfrac{a+c}{2n}, \tfrac{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 test for homogeneity/independence does not test the appropriate hypothesis, and thus, only heterozygous parents are included. == Extension to two affected child per family ==

Extension of the test The TDT can be readily extended beyond the case of trios. We keep following the notations of Spielman, McGinnis & Ewens (1993). is informative. The Blackwelder and Elston approach uses the total number of haplotypes identical by descent (mean haplotype sharing). This measure ignores the allelic state of a marker and simply compares the number of times a parent transmits the same allele to both affected children with the number of times a different allele is transmitted. The test statistic is: \chi^2_{hs} = \frac{(2i+2j-h)^2}{h}. Under the null hypothesis of no linkage the expected proportions of are . One can derive a simple chi-square statistic with 2 degrees of freedom: \chi^2_{total} = \frac{(i - h/4)^2}{h/4} + \frac{(h-i-j-h/2)^2}{h/2} + \frac{(j-h/4)^2}{h/4} = \chi^2_{tdt} + \chi^2_{hs}. It clearly appears that the total statistic (with two degree of freedom) is the sum of two independent components: one is the traditional linkage measure and the other is the TDT statistic. == Modified version ==

More recently, Wittkowski KM, Liu X. (2002/2004) proposed a modification to the TDT that can be more powerful under some alternatives, although the asymptotic properties under the null hypothesis are equivalent. The motivating idea for this modification is the fact that, while the transmissions of both allele from parents to a child are independent, the effects of other filial genetic or environmental covariates on penetrance are the same for both alleles transmitted to the same child. This situation can be important if, for example, the genetic marker is linked to a disease locus with a strong selection against heterozygous individuals. This observation suggests to shift the statistical model from a set of independent transmissions to a set of independent children (see Sasieni (1997) for the corresponding problem in case-control association tests). While this observation does not affect the distribution under the null hypothesis of no linkage, it allows, for some disease models, to design a more powerful test. In this modified TDT test the children are stratified by parental type and the modified test statistic becomes: \chi^2 = \frac{ \bigl( [n_{\rm PQ} - n_{\rm QQ}]_{\rm PQ \sim QQ} + 2\times[n_{\rm PP} - n_{\rm QQ}]_{\rm PQ \sim PQ} + [n_{\rm PP} - n_{\rm PQ}]_{\rm PP \sim PQ} \bigr)^2}{[n_{\rm PQ} + n_{\rm QQ}]_{\rm PQ \sim QQ} + 4\times[n_{\rm PP} + n_{\rm QQ}]_{\rm PQ \sim PQ} + [n_{\rm PQ} + n_{\rm PP}]_{\rm PP \sim PQ}} where [n_{\rm PQ}]_{\rm PQ \sim QQ} is the number of PQ children from parents with the PQ and QQ types. ==References==