Page 266 - Applied Probability

11. Radiation Hybrid Mapping
253
9. Under the polyploid model for two loci, show that the expected in-
formation for θ is
2 2
c−2
c
c
(1 − r) [1 − 2(1 − r) +(1 − θr) ]
c r (1 − θr)
=
J θθ
c
c
c
[1 − (1 − θr) ][1 − 2(1 − r) +(1 − r) (1 − θr) ]
Argue that J θθ has a maximum as a function of r near r = 1 c c . when
c+1
θ is near 0. (Hint: Be careful because lim θ→0 J θθ = ∞. The singularity
at θ = 0 is removable in the function ∂ ln J θθ .)
∂r
This result suggests that the value r = 1 is nearly optimal for
c+1
small θ in the sense of providing the smallest standard error of the
ˆ
maximum likelihood estimate θ of θ. In this regard note that J θr and
J rr have finite limits as θ → 0. Thus for small θ, the approximate
ˆ 1
standard error of θ is proportional to √ even when r is jointly
J θθ
estimated with θ.
10. In computing the distribution of the number of obligate breaks per
clone in the polyploid model, how must the initial conditions and
recurrences for the probabilities p k (i, j) be modified when some loci
are untyped?
11. The construction of radiation hybrids always involves a selectable
enzyme such as HPRT or TK. On the chromosome containing the
selectable locus, at least one fragment containing the locus is neces-
sary to form a viable clone. This requirement invalidates our model
of fragment retention for the chromosome in question. The obvious
amendment of the model is to condition on the event of retention of
at least one fragment bearing the selectable locus. Discuss how this
change of the model affects likelihood calculation via Baum’s forward
and backward algorithms [16].
11.12 References
[1] Barrett JH (1992) Genetic mapping based on radiation hybrid data.
Genomics 13:95–103
[2] Baum LE (1972) An inequality and associated maximization tech-
nique in statistical estimation for probabilistic functions of Markov
processes. Inequalities 3:1–8

[3] Bishop DT, Crockford GP (1991) Comparisons of radiation hybrid
mapping and linkage mapping. Genetic Analysis Workshop 7: Is-
sues in Gene Mapping and Detection of Major Genes. MacCluer JW,
Chakravarti A, Cox D, Bishop DT, Bale SJ, Skolnick MH, editors,
Cytogenetics and Cell Genetics, S Karger, Basel