LOD score analysis is
used to estimate whether the observed degree of concordance (co-occurrence,
co-segregation) of a genetic marker with a trait of interest
indicates signification genetic linkage between the two. LOD analysis is a basic
technique of Genome-Wide
Association Studies (GWAS) used to map traits of interest to
particular chromosomal regions. A full-scale LOD score analysis
requires a fully-mapped genome with many thousands of genetic
markers closely space on all chromosomes, and is complicated
by many factors built into the mathematical model. Even a
simple example, involving a single
chromosome with complete
knowledge of the genetics of the participants
Appreciation of a LOD score analysis can be developed with respect to poker. At its heart, a LOD ratio is the ratio of two unlikely events, so as to determine whether one event is significantly less unlikely (i.e., more likely) than the other.
Consider a deck of 52 cards (four suits, 13 cards in each suit ranked 2-10, J, Q, K, A). There are then 525 = 380,205,032 = 3.80 x 108 possible five-card hands, the large majority of which are 'junk' with no particular value [many games require a pair of "jacks or better" to open]. Thus the a priori probability of getting one particular random hand is 1/525 = 2.6 x 10-9.
Suppose we wish to calculate the probability of a Flush (five cards of the same suit):
The a priori probability that the first card dealt will be the same suit as itself is clearly 1, and the probability that the next card will be the same suit [e.g., a heart] as the first is (12 / 51), since there are twelve hearts remaining among the 51 cards left in the deck..
The probabilities that the next three cards will be same suit as the first two are then calculated in the same manner, so
(1) (12/51) (11/50) (10/49) (9/48) = 1.98 x 10-3
and log10 ratio of the odds of a flush versus any random hand is log10 (1.98 x 10-3) / ( 2.6 x 10-9) = 5.88
Appreciation of a LOD score analysis can be developed with respect to poker. At its heart, a LOD ratio is the ratio of two unlikely events, so as to determine whether one event is significantly less unlikely (i.e., more likely) than the other.
Consider a deck of 52 cards (four suits, 13 cards in each suit ranked 2-10, J, Q, K, A). There are then 525 = 380,205,032 = 3.80 x 108 possible five-card hands, the large majority of which are 'junk' with no particular value [many games require a pair of "jacks or better" to open]. Thus the a priori probability of getting one particular random hand is 1/525 = 2.6 x 10-9.
Suppose we wish to calculate the probability of a Flush (five cards of the same suit):
The a priori probability that the first card dealt will be the same suit as itself is clearly 1, and the probability that the next card will be the same suit [e.g., a heart] as the first is (12 / 51), since there are twelve hearts remaining among the 51 cards left in the deck..
The probabilities that the next three cards will be same suit as the first two are then calculated in the same manner, so
(1) (12/51) (11/50) (10/49) (9/48) = 1.98 x 10-3
and log10 ratio of the odds of a flush versus any random hand is log10 (1.98 x 10-3) / ( 2.6 x 10-9) = 5.88