Part III
Back to our Maximum Likelihood estimate of r. The general
formula is:
log(L) = C + a1 log m1 + a2
log m2 +...+ ai log mi
We take the first derivative of L with respect to r
and set this equal to zero to maximize r.
dL |
= 0 +a1 |
d log m1 |
+ a2 |
d log m2 |
+...+ai |
d log mi |
dr |
dr |
dr |
dr |
| |
|
|
|
|
|
|
| |
0 = a1 |
d log mi |
+ a2 |
d log m2 |
+...+ai |
d log mi |
dr |
dr |
dr |
| |
Numbers |
| Classes |
Observed |
Expected |
| PpTt |
a1 |
m1
= (1- r)/2 |
| pptt |
a4 |
m4
= (1- r)/2 |
| Pptt |
a2 |
m2
= r/2 |
| ppTt |
a3 |
m3
= r/2 |
We have shown that m = 1/2(1-r) where we multiply 1/2(1-r)
for a single event by n to get the total number expected
for n progeny.
0 = a1 |
d log m1 |
+ a2 |
d log m2 |
+...+ ai |
d log mi |
dr |
dr |
dr |
|
|
|
|
|
|
0 = a1 |
d log [1 - r] |
+ a2 |
d log [r] |
|
|
|
dr |
dr |
|
|
|
|
|
|
|
|
|
+ a3 |
d log [r] |
+ a4 |
d log [1 - r] |
|
|
|
dr |
dr |
|
|
|
Because we substituted m = (1 - r)/2, etc.
n/2 is a constant so when q =
a1
log (1-r)
dq |
= |
-a1 |
because |
d log (1-r) |
= |
-1 |
dr |
1-r |
d lodr(1-r) |
1-r |
and a1 |
d log (1-r) |
= a1 x |
[ |
-1 |
] |
= |
-a1 |
dr |
1 - r |
1 - r |
When q = a2 |
log [r] |
, then dq = |
a2 |
dr |
r |
