The log-odds ratio, lor sub s and a, is defined as the natural logarithm of the ratio of two quantities. The numerator of the ratio is pi 2 sub s and a divided by 1 minus pi 2 sub s and a. The denominator of the ratio is pi 1 sub s and a divided by 1 minus pi 1 sub s and a.
The estimate of the log-odds ratio, lor hat sub s and a, is defined as the natural logarithm of the ratio of two quantities. The numerator of the ratio is p 2 sub s and a divided by 1 minus p 2 sub s and a. The denominator of the ratio is p 1 sub s and a divided by 1 minus p 1 sub s and a, where p 1 sub s and a are the 2009-2010 State estimates and p 2 sub s and a are the 2010-2011 State estimates.
Variance v of the estimate of the log-odds ratio, lor hat sub s and a, is a function of three quantities: q1, q2, and q3. It is expressed as the sum of q1 and q2 minus q3. Quantity q1 is the variance v of the natural logarithm of Theta 1 hat, quantity q2 is the variance v of the natural logarithm of Theta 2 hat, and quantity q3 is 2 times the covariance between the natural logarithm of Theta 1 hat and the natural logarithm of Theta 2 hat.
The covariance between the natural logarithm of Theta 1 hat and the natural logarithm of Theta 2 hat is equal to the correlation between the natural logarithm of Theta 1 hat and the natural logarithm of Theta 2 hat multiplied by the square root of the product of the variance v of the natural logarithm of Theta 1 hat and variance v of the natural logarithm of Theta 2 hat.