The logodds ratio, lor sub s, is defined as the natural logarithm of the ratio of two quantities. The numerator of the ratio is pi 2 sub s divided by 1 minus pi 2 sub s. The denominator of the ratio is pi 1 sub s divided by 1 minus pi 1 sub s, where pi 1 sub s is the 20062008 prevalence rate and pi 2 sub s is the 20082010 prevalence rate for substate area s.
The estimate of the logodds ratio, lor hat sub s, is defined as the natural logarithm of the ratio of two quantities. The numerator of the ratio is p 2 sub s divided by 1 minus p 2 sub s. The denominator of the ratio is p 1 sub s divided by 1 minus p 1 sub s, where p 1 sub s is the 20062008 small area estimate and p 2 sub s is the 20082010 small area estimate for substate area s.
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 variances of the natural logarithm of Theta 1 hat and the natural logarithm of Theta 2 hat.
