Capital S R R is equal to the ratio of two quantities. The numerator is the summation of the product of w sub h h and complete sub h h. The denominator is the summation of the product of w sub h h and eligible sub h h.

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Capital I R R is equal to the ratio of two quantities. The numerator is the summation of the product of w sub i and complete sub i. The denominator is the summation of the product of w sub i and selected sub i.

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Capital O R R is equal to the product of capital S R R and capital I R R.

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The model is given by the following equation: log of pi sub a, i, j, k divided by 1 minus pi sub a, i, j, k is equal to the sum of three terms. The first term is given by x transpose sub a, i, j, k times beta sub a. The second term is eta sub a, i. And the third term is nu sub a, i, j.

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Lower sub s and a is defined as the exponent of capital L sub s and a divided by the sum of 1 and the exponent of capital L sub s and a. And upper sub s and a is defined as the exponent of capital U sub s and a divided by the sum of 1 and the exponent of capital U sub s and a.

Click here to return to Equation 5.


Capital L sub s and a is defined as the difference of two quantities. The first quantity is the natural logarithm of the ratio of Theta sub s and a and 1 minus Theta sub s and a. The second quantity is the product of 1.96 and the square root of MSE sub s and a, which is the mean square error for State-s and age group-a.

Click here to return to Equation 6.


Capital U sub s and a is defined as the sum of two quantities. The first quantity is the natural logarithm of the ratio of Theta sub s and a and 1 minus Theta sub s and a. The second quantity is the product of 1.96 and the square root of MSE sub s and a, which is the mean square error for State-s and age group-a.

Click here to return to Equation 7.


The mean square error, MSE sub s and a, is defined as the sum of two quantities. The first quantity is the square of the difference of two parts. Part 1 is defined as the natural logarithm of the ratio of P sub s and a and 1 minus P sub s and a. Part 2 is defined as the natural logarithm of the ratio of Theta sub s and a and 1 minus Theta sub s and a. The second quantity is the posterior variance of the natural logarithm of the ratio of P sub s and a and 1 minus P sub s and a.

Click here to return to Equation 8.


The average annual rate is defined as 100 times quantity q divided by 2. Quantity q is defined as capital X sub 1 divided by the sum of 0.5 times capital X sub 1 plus capital X sub 2.

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The logit of pi hat sub w is equivalent to the logarithm of pi hat sub w divided by the quantity 1 minus pi hat sub w, which is equal to the sum of the following three quantities: negative 4.7500, the product of 0.2098 and capital X sub k, and the product of 0.3839 and capital X sub w.

Click here to return to Equation 10.


The logit of pi hat sub s is equal to the sum of the following three quantities: negative 4.4924, the product of 0.2960 and capital X sub k, and the product of 0.2242 and capital X sub s.

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Pi hat sub w is equal to the ratio of two quantities. The numerator is 1. The denominator is 1 plus e raised to the negative value of the sum of the following three quantities: negative 4.7500, the product of 0.2098 and capital X sub k, and the product of 0.3839 and capital X sub w.

Click here to return to Equation 12.


Pi hat sub s is equal to the ratio of two quantities. The numerator is 1. The denominator is 1 plus e raised to the negative value of the sum of the following three quantities: negative 4.4924, the product of 0.2960 and capital X sub k, and the product of 0.2242 and capital X sub s.

Click here to return to Equation 13.


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 the variance v of the natural logarithm of Theta 2 hat.

Click here to return to Equation 14.


Variance v of the natural logarithm of Theta sub i is equal to the square of quantity q. Quantity q is calculated as the difference between capital U sub i and capital L sub i divided by 2 times 1.96, where i takes values 1 and 2.

Click here to return to Equation 15.


Capital U sub 1 is defined as the natural logarithm of the ratio of 0.1562 and 1 minus 0.1562, which is negative 1.6868. Capital L sub 1 is defined as the natural logarithm of the ratio of 0.1100 and 1 minus 0.1100, which is negative 2.0907. Capital U sub 2 is defined as the natural logarithm of the ratio of 0.1964 and 1 minus 0.1964, which is negative 1.4089. Capital L sub 2 is defined as the natural logarithm of the ratio of 0.1436 and 1 minus 0.1436, which is negative 1.7857.

Click here to return to Equation 16.


The estimate of the log-odds ratio, lor hat sub a, is defined as the natural logarithm of the ratio of two quantities. The numerator of the ratio is p 2 sub a divided by 1 minus p 2 sub a. The denominator of the ratio is p 1 sub a divided by 1 minus p 1 sub a, where p1 sub a is 0.1314 and p 2 sub a is 0.1684. The estimate lor hat sub a is calculated to be 0.2916.

Click here to return to Equation 17.


The variance v of the natural logarithm of Theta 1 hat is equal to the square of quantity q. Quantity q is calculated as the difference between capital U sub 1 and capital L sub 1 divided by the product of 2 and 1.96. Here, capital U sub 1 is negative 1.6868, and capital L sub 1 is negative 2.0907. Hence, the variance v of the natural logarithm of Theta 1 hat is calculated to be 0.01062. The variance v of the natural logarithm of Theta 2 hat is equal to the square of quantity q. Quantity q is calculated as the difference between capital U sub 2 and capital L sub 2 divided by the product of 2 and 1.96. Here, capital U sub 2 is negative 1.4089, and capital L sub 2 is negative 1.7857. Hence, the variance v of the natural logarithm of Theta 2 hat is calculated to be 0.00924.

Click here to return to Equation 18.


Quantity z is the estimate of the log-odds ratio, lor hat sub a, divided by the square root of the sum of the variance v of the natural logarithm of Theta 1 hat and the variance v of the natural logarithm of Theta 2 hat, where lor hat sub a is 0.2916, the variance v of the natural logarithm of Theta 1 hat is 0.01062, and the variance v of the natural logarithm of Theta 2 hat is 0.00924. The statistic z is calculated to be 2.0695.

Click here to return to Equation 19.

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