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

\( {\Huge d_i = \frac{\sum_{j=1}^{A}\frac{c_j z_{ij}}{n_{ij}}}{\sum_{j=1}^{A}c_j} } \)

Let

y_i = log(d_i)

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

\( {\Huge \hat{v_i} = \frac{\sum_{j=1}^{A}\frac{c_j^{2} z_{ij}}{n_{ij}^{2}}}{ (\sum_{j=1}^{A}\frac{c_j z_{ij}}{n_{ij}})^{2}} } \)

and the weight for the i^th year is \( {\Large w_i = \frac{1}{\hat{v_i}} } \) .  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

\( {\Huge m = \frac{\left(\frac{\sum_{i=1}^{n}w_ix_iy_i}{\sum_{i=1}^{n}w_i} \right) - \left( \frac{\sum_{i=1}^{n}w_ix_i}{\sum_{i=1}^{n}w_i}\right)\left( \frac{\sum_{i=1}^{n}w_iy_i}{\sum_{i=1}^{n}w_i}\right)}{\left( \frac{\sum_{i=1}^{n}w_ix^{2}_i}{\sum_{i=1}^{n}w_i}\right) - \left( \frac{\sum_{i=1}^{n}w_ix_i}{\sum_{i=1}^{n}w_i}\right)^{2}} } \)

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

\( {\Huge s^{2} = \frac{\sum_{i=1}^{n} w_i \left( y_i-\hat{y}_i\right)^{2}}{n - 2} \left( \frac{\sum_{i=1}^{n}w_i}{\left(\sum_{i=1}^{n}w_i\right) \left(\sum_{i=1}^{n}w_ix^{2}_i \right) - \left(\sum_{i=1}^{n}w_ix_i \right)^{2}} \right) }\)

where

     \( {\Large \hat{y}_i = b + mx_i } \)

and

     \( {\Huge b = \frac{\sum_{i=1}^{n}w_iy_i - m\sum_{i=1}^{n}w_ix_i}{\sum_{i=1}^{n}w_i} } \)

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:

     \( {\Large CI_{low} = 100 \times \left( e^{[m - s \cdot t_{1-\frac{\alpha}{2},n-2}]} - 1 \right)} \)

     \( {\Large CI_{high} = 100 \times \left( e^{[m + s \cdot t_{1-\frac{\alpha}{2},n-2}]} - 1 \right)} \)