Calculating APC with Weighting

Let d₁,¼,d_n be the directly standardized rates for the n years for which we wish to calculate an APC. Let n_ij be the population size for the ith year for the jth age group, let z_ij be the count of the number of cancer cases for the ith year for the jth age group, and let c_j be the standard population for the jth age group. Then

Let

y_i = log(d_i)

and let x_i be the ith year. Then we estimate the variance of Y_i with

and the weight for the ith year is . See:

Kim HJ, Fay MP, Feuer EJ, Midthune DN. Permutation test for joinpoint regression with applications to cancer rates. Statistics in Medicine 2000;19;335-51 (correction: 2001;20:655).

Then the estimate of the slope of the line (on the log scale) is

Then the APC is just 100 ×( e^m - 1 ). To estimate the standard error of m, s, we use the square root of

where

and

See:

Neter J, Wasserman W, Kutner M. Applied Linear Statistical Models. R.D. Irwin, 2nd edition, 1985; pages 167, 220.

Then the 100×(1 - a)% confidence intervals for the APCs are

CI_low = 100 ×( e^{[ m - s · t_1-a/2,n-2 ]} - 1 )

CI_high = 100 ×( e^{[m + s · t_1-a/2,n-2 ]} - 1 )

where t_1-a/2,n-2 is the inverse of the t distribution evaluated at 1-a/2 with n-2 degrees of freedom.