Page 47 - Applied Probability

2. Counting Methods and the EM Algorithm
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These considerations lead us to rewrite the updates (2.4) and (2.5) as

s k p m (1−π m )(1−p m π m ) k −1
r k +

k
s
1−(1−p m π m ) k
=
p m+1
s
(1−p m π m ) k

s ]
s k [1 +
k
1−(1−p m π m ) k s
= k a k ,
π m+1
s
r k +
s k p m (1−π m )(1−p m π m ) k −1
s
k 1−(1−p m π m ) k
where all sums extend over the ascertained families alone.
TABLE 2.3. Cystic Fibrosis Data
Siblings s Affecteds r Ascertaineds a Families n
10 3 1 1
9 3 1 1
8 4 1 1
7 3 2 1
7 3 1 1
7 2 1 1
7 1 1 1
6 2 1 1
6 1 1 1
5 3 3 1
5 3 2 1
5 2 1 5
5 1 1 2
4 3 2 1
4 3 1 2
4 2 1 4
4 1 1 6
3 2 2 3
3 2 1 3
3 1 1 10
2 2 2 2
2 2 1 4
2 1 1 18
1 1 1 9
Example 2.6.1 Segregation Analysis of Cystic Fibrosis
The cystic fibrosis data of Crow [3] displayed in Table 2.3 offer an op-
portunity to apply the EM algorithm. In this table the column labeled
“Families n” refers to the number of families showing a particular config-
uration of affected and ascertained siblings. For these data the maximum