Page 78 - Applied Probability

4. Hypothesis Testing and Categorical Data
of allele frequencies among patients, controls, and the combined sample,
respectively. To test the hypothesis p = q, we compute separate maximum
likelihoods L u (ˆ p), L n(ˆ q), and L c(ˆ r) for the ulcer patients, normal controls,
and combined sample under the assumption of Hardy-Weinberg equilib-
rium. The appropriate likelihood ratio statistic is
L u (ˆ p)L n (ˆ q) 61
2
χ 2 =2 ln
L c(ˆ r)
=2 ln L u (ˆ p)+ 2 ln L n (ˆ q) − 2ln L c(ˆ r).
2
The degrees of freedom of the χ are the difference 4 − 2 = 2 between
the number of independent parameters for the two populations treated
separately versus in combination.
Gene counting for the normal controls yields the maximum likelihood
estimates ˆ q A = .2492, ˆ q B = .0655, and ˆ q O = .6853 and for the combined
sample ˆ r A = .2335, ˆ r B = .0588, and ˆ r O = .7077. Straightforward compu-
tations yield
ln L u (ˆ p)= −173.903 − 189.955 − 97.750 − 49.963
= −511.571
ln L n (ˆ q)= −238.134 − 253.114 − 163.050 − 58.161
= −712.459
ln L c(ˆ r)= −414.198 − 443.848 − 261.644 − 107.846
= −1227.536.

2
Hence, the homogeneity χ =2 (−511.571 − 712.459 + 1227.536) = 7.012.
2
This statistic is significant at the .05 level but not at the .01 level. Sub-
sequent studies have substantiated the association between duodenal ulcer
and blood type O.
Example 4.2.2 Color Blindness
The data for this color-blindness example were mentioned in Problem
2 of Chapter 2. If Hardy-Weinberg equilibrium does not hold, then we
postulate a probability q B for normal females, q b for color-blind females,
r B for normal males, and r b for color-blind males. The only functional
relationship tying these frequencies together are the constraints q B +q b =1
and r B + r b = 1. If in a random sample there are f B normal females, f b
color-blind females, m B normal males, and m b color-blind males, then the
likelihood of the sample is

f B + f b m B + m b
q f B f b r m B m b .
r
q
B b B b
f B m B
q
q
Maximizing this likelihood leads to the estimates ˆ B = f B ,ˆ b = f b ,
f B +f b f B +f b
r
ˆ r B = m B , and ˆ b = m b .
m B +m b m B +m b