Maximum Likelihood Estimates and the EM Algorithms II. Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University email@example.com http://tigpbp.iis.sinica.edu.tw/courses.htm. Part 1 Computation Tools. Include Functions in R. source( “ file path ” ) Example
Henry Horng-Shing Lu
Institute of Statistics
National Chiao Tung University
Two linked loci with alleles A and a, and B and b
A, B: dominant
a, b: recessive
A double heterozygote AaBb will produce gametes of four types: AB, Ab, aB, ab
F ( Female) 1- r’ r’ (female recombination fraction)
M (Male) 1-r r (male recombination fraction)
r and r’ are the recombination rates for male and female
Suppose the parental origin of these heterozygote is from the mating of . The problem is to estimate r and r’ from the offspring of selfed heterozygotes.
Fisher, R. A. and Balmukand, B. (1928). The estimation of linkage from the offspring of selfed heterozygotes. Journal of Genetics, 20, 79–92.
Four distinct phenotypes: A*B*, A*b*, a*B* and a*b*.
A*: the dominant phenotype from (Aa, AA, aA).
a*: the recessive phenotype from aa.
B*: the dominant phenotype from (Bb, BB, bB).
b* : the recessive phenotype from bb.
A*B*: 9 gametic combinations.
A*b*: 3 gametic combinations.
a*B*: 3 gametic combinations.
a*b*: 1 gametic combination.
Total: 16 combinations.
Hence, the random sample of n from the offspring of selfed heterozygotes will follow a multinomial distribution:
Suppose that we observe the data of
y = (y1, y2, y3, y4) = (125, 18, 20, 24),
which is a random sample from
Then the probability mass function is
Maximize likelihood: Solve the score equations, which are setting the first derivates of likelihood to be zeros.
Under regular conditions, the MLE is consistent, asymptotic efficient and normal!
A B C
We will define some functions and variables for finding the MLE in Example 1 by R
and c<1 for p=1.
Hence, we can use regression to estimate p.
R=Newton(y1, y2, y3, y4, initial)
#Newton method can be substitute for different method
Write your own programs for those examples presented in this talk.
Write programs for those examples mentioned at the following web page:
Write programs for the other examples that you know.
Example 3 in genetics: The observed data are (nO, nA, nB, nAB) = (176, 182, 60, 17) ~ Multinomial(r^2, p^2+2pr, q^2+2qr, 2pq), where p, q, and r fall in [0,1] such that p+q+r = 1. Find the MLEs for p, q, and r.
Example 4 in the positron emission tomography (PET):The observed data are n*(d) ~Poisson(λ*(d)), d = 1, 2, …, D, and
The values of p(b,d) are known and the unknown parameters are λ(b), b = 1, 2, …, B.
Find the MLEs for λ(b), b = 1, 2, …, B.
Example 5 in the normal mixture:The observed data xi, i = 1, 2, …, n, are random samples from the following probability density function:
Find the MLEs for the following parameters: