Page 124 - Applied Probability

6. Applications of Identity Coefficients
108
Let Υ r denote the probability of the condensed identity state S r under
these conventions. Then

) | S r )Υ r .
E(Z ij Z kl )=
E(1 {G i =G j } f(p G i
)1 {G k =G l } f(p G k
r
Table 6.3 lists the necessary conditional expectations. The entries in
the table are straightforward to verify. For instance, consider the entry
for state S 8 . In this condensed identity state, one of the two genes on
the top row is i.b.d. with one of the two genes on the bottom row. Thus,
1 {G i=G j } 1 {G k =G l } = 1 only when G i , G j , G k , and G l all agree in state. By
independence, all four genes coincide with the mth allele with probability
3
p .
m
To compute the probabilities Υ r , we reason as we did in Chapter 5 in
passing between generalized kinship coefficients and condensed identity co-
efficients. Consider the 15 detailed identity states possible for 4 genes as
depicted in Figure 5.2. Now imagine in all states that the sampled genes
G i and G j occupy the top row in some particular order and that G k and
G l occupy the bottom row in some particular order. The probability of any
detailed identity state is just a generalized kinship coefficient involving the
four sampled genes G i , G j , G k , and G l . Under the usual correspondence
between detailed and condensed states, adding the appropriate generalized
kinship coefficients yields each Υ r .
TABLE 6.3. Conditional Expectations for Marker Sharing
State r E(1 {G i =G j } f(p G i )1 {G k =G l } f(p G k ) | S r )
1 p m f(p m) 2
m
2 { p m f(p m)} 2
m
2
3, 5, 7 p f(p m) 2
m m
2
4, 6 { p f(p m)}{ p m f(p m )}
m m m
3
8 p f(p m) 2
m m
2
9 { p f(p m)} 2
m m
Given a collection of pedigrees, it is helpful to combine the marker-
sharing statistics from the individual pedigrees into one grand statistic.
The grand statistic should reflect the information content available in the
individual pedigrees and should lead to easily approximated p-values. If Z m
is the statistic corresponding to the mth pedigree, then these goals can be
achieved by defining
w m [Z m − E(Z m )]

m
T = ,
2
w Var(Z m )
m m