LOD Score

LOD is an abbreviation for "log of the odds."

The lod score z = log10 .

Example, Binomial distribution lod score:

Let Q = probability(success); 1-Q = probability(failure);
Let x = number of successes; n-x = number of failures.

The probability of X successes out of n trials is:

     n!          Q^X(1-Q)^n-X
x! (n-x)!

H_o: Q = 0.5
H_A: Q is unequal to 0.5

z = Log₁₀

z = xLog₁₀ Q + (n-x)Log₁₀ (1-Q) - nLog₁₀(1/2)

This is because:

z = L(Q_A|x) =       n!       Q^x(1-Q)^n-x
                          x!(n-x)!

and

z Log[L(Q_A|x)]
z = Log + xLog(Q) + (n-x)Log(1-Q)

and

z Log[L(Q_N|x)]

z = Log[ (0.5)^x (0.5)^n-x ]

z = Log [ (0.5)ⁿ ]

z = Log + nLog(0.5)

Then

z = Log₁₀

z = Log + xLog(Q) + (n-x)Log(1-Q)
z = - Log - nLog(0.5)

z = xLog₁₀(Q) + (n-x)Log₁₀(1-Q) - nLog₁₀(0.5)

** Stated another way, the probability of observing a sample of size n with x successes and n-x failures when the probability of success in any one event equals Q.

L(Q) = Probability(x successes out of n trials with Q = the probability of one successful event)

= Q^x (1-Q)^x

Then when Q = 1/2, then

L(Q) = (1/2)ⁿ(1/2)^n-x
L(Q) = (1/2)ⁿ

* L(Q_N) is not the maximum likelihood estimate, it is the probability of observing X successes out of n trials for a given value of Q. L(Q_N)is the maximum likelihood estimate of observing X successes out of n trials when Q = 1/2.