# Likelihood in Lander and Botstein 1989

Dr. Gerard Tromp tromp
Wed Apr 3 12:43:27 EST 1996

```Greetings,

I've been trying to grasp the Lander and Botstein article "Mapping
Mendelian Factors Underlying Quantitative Traits Using RFLP Linkage Maps"
Genetics 121:185-199.
L&B start with a quantitative trait in two strains A and B where A
is the low strain and B is the high strain. The trait phenotype is normally
distributed, or transformed to be normal. They will examine B1, the
offspring of an A X F1 backcross. L&B derive a likelihood function on page
189, eqn (4), as follows.
L(a,b,var) = PI(i) z((psi(i) - (a + bg(i)), var)
where:
z      is the normal distribution function
psi(i) is the variance of the ith individual
b      is the phenotypic effect of a single high phenotype allele
(= slope in the linear regression model)
g(i)   is the genotype status --
number of high phenotype alleles
var    is the variance
PI     is the product operator

I can follow the logic of the derived likelihood equation. BUT ...

The QUESTION I have is about the unlinked likelihood for the likelihood ratio
test.
L&B state: "These constrained MLEs are easily seen to be
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
(mean of A, 0, var(B1))."
^^^^^^^^^^^^^^^^^^^^^^^^
where A is the low phenotype strain and B1 is the
F1 X A backcross. What I don't understand is the estimate of the "mean of
A", the other two are easily seen. I expected the estimate of the mean to
revert to that of the population under study, i.e. B1. Since there is no
information to discriminate the genotype effect (b is equivalent to the
estimate of 1/2 delta, i.e. the effect of a single allele -- it is also the
slope under the linear regression model), and the phenotype data is that of the
B1 progeny, how do we estimate the low trait mean?

I must be missing something elementary, but, I can't recognize what I'm
missing.

Gerard
--
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Gerard Tromp, Ph.D.
CMMG, Wayne State University    vox:	313-577-8773
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```