Mar. is, 1924 
Measurement of Linkage Values 
887 
X 2 AS A MEASURE OF LINKAGE 
One of the most fundamental determinations in connection with 
linkage problems is to discriminate between a low linkage value and 
independent inheritance. This is done by calculating the x 2 of the ob¬ 
served population from a theoretical population of the same size with 
no linkage. 
If the theoretical population is constructed in accordance with the 
expected Mendelian ratio the result will be erroneous unless the observed 
population conforms exactly to the expected Mendelian ratios. Any 
irregularity in the behavior of the individual characters will increase x 2 
and may indicate a linkage when none exists. 
It would seem that the correct procedure should be to construct the 
theoretical population from the observed instead of the expected Mende¬ 
lian ratios. 
A theoretical population, A'B', A'b ', a'B' and a'b' of the same size 
and having the same Mendelian ratios may be constructed as follows: 
a '6' — (a# + ab)X(Ab + ab) 
n 
tT>f (aB-\- ab) X (AB-\- aB) 
a Jt> - - 
n 
(ABAb) X (Ab-\-ab) 
n 
VB , (AB-\-Ab) X (AB+aB) 
n 
The observed distribution is compared with the theoretical distribution 
and x 2 is determined as follows: 
2 (AB — A'B') 2 , (Ab-A'b') 2 , (aB-a'B') 2 , (ab-a'b') 2 
: - A'B' + A'b' + a'B’ + a'b’ 
The degree of confidence to be placed in the observed departure is 
indicated in the following table: 
Odds against 
the deviation 
being due to 
X* chance 8 . 
1 . .2 to I 
2 . .7toi 
3 . I. 6 to 1 
4 . 2. 8 to 1 
5 . 4- 8 to 1 
6 . 8. o to 1 
7 . 12. 9 to 1 
8 . 20. 7 to 1 
9 . 33. 1 to 1 
10 . 52. 9 to 1 
11 . 84. 3 to 1 
12 .. 134. 4 to 1 
13 . 214. 7 to I 
14 .. 343. 2 to I 
15 . ’••••• 549-4 to i 
16 . 880. 8 to 1 
17 . 1, 413. 4 to 1 
18 . 2, 271. 7 to 1 
19 . 3, 662. o to I 
20 . 5, 88l. 4 tO I 
8 Calculated-from the values of P given in Pearson's “Tables for Statisticians and Biometricians,” 
Table XII (6). 
