of heterozygous parents. We use the fact that the transmission to different children are independent. The information can be then summarized in three categories: = number of parents who transmit to both children. = number of parents who transmit to one child and to another. = number of parents who transmit to both children. Using the notations of the previous paragraph we have: leading to the chi-squared test statistic: Relation with another linkage statistic The comparison with the more traditional (at least at the time when the TDT was proposed) linkage test proposed by Blackwelder and Elston 1985 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: Under the null hypothesis of no linkage the expected proportions of (i, h − i − j, j) are (0.25, 0.5, 0.25). One can derive a simple chi-square statistic with 2 degrees of freedom: 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