The nonparametric linkage test of de Vries et al. [10] uses affected sibling data. Consider a nuclear family with s affected sibs and a heterozygous parent with genotype a/b at some marker locus. Let na and nb count the number of affected sibs receiving the a and b alleles, respectively, from the parent. If the other parent is typed, then this determination is always possible unless both parents and the child are simultaneously of genotype a/b. de Vries et al. [10] suggest the statistic T = |na − nb|. Under the null hypothesis of independent transmission of the disease and marker genes, Badner et al. [3] show that T has mean and variance E(T) = & s( 1 2 )s  s s 2  s even s( 1 2 )s−1s−1 s−1 2  s odd Var(T) = s − E(T)
2.
Prove these formulas. If there are n such parents (usually two per family), and the ith parent has statistic Ti, then the overall statistic n i=1[Ti − E(Ti)]
n i=1 Var(Ti)
should be approximately standard normal. A one-sided test is appropriate because the Ti tend to increase in the presence of linkage between the marker locus and a disease predisposing locus. (Hint:
The identities s 2 −1 i=0 s i 
= 2s−1 −
 s s 2 
2 s 2 −1 i=0 i s i 
= s 
2s−2 −
s − 1 s 2 
for s even and similar identities for s odd are helpful.)