# Annual Percent Change (APC) and Confidence Interval

A description of APC and Confidence Interval.

Annual Percent Change (APC) is one way to characterize trends in cancer rates over time. When the Log Transformation option on the Input File tab is ln(y)=xb, then the output calculates the estimated annual percentage rate change (APC).  With this approach, the cancer rates are assumed to change at a constant percentage of the rate of the previous year. For example, if the APC is 1%, and the rate is 50 per 100,000 in 1990, the rate is 50 x 1.01 = 50.5 in 1991 and 50.5 x 1.01 = 51.005 in 1992. Rates that change at a constant percentage every year change linearly on a log scale. For this reason, to estimate the APC for a series of data, the following regression model is used: $\log \left( R_y \right) =\ b_0 +\ b_1 y where \log \left( R_y \right) is the natural log of the rate in year y. The APC from year y to year y + 1 = \left[ \frac{R_{y+1} - R_y}{R_y} \right] \times\ 100 =\ \frac{\left\{ e^{b_0 + b_1 \left( y + 1 \right)} - e^{b_0 + b_1 \left( y \right)} \right\}}{e^{b_0 + b_1 \left( y \right)}} \times\ 100 =\ \left( e^{b_1} - 1 \right) \times\ 100.$

One advantage of characterizing trends this way is that it is a measure that is comparable across scales, for both rare and common cancers. For example, it is reasonable to think that rates for a rare cancer and a common cancer could both change at 1% per year, but it is not reasonable to think that a rare cancer and a common cancer would change in the same increments on an absolute (or arithmetic) scale. That is, a cancer with a rate of 100 per 100,000 could be changing by 2 per 100,000 every year, but a cancer with a rate of 1 per 100,000 would probably not change in the same increments.

It is not always reasonable to expect that a single APC can accurately characterize the trend over an entire series of data. The joinpoint model uses statistical criteria to determine when and how often the APC changes. For cancer rates, it is fit using joined log-linear segments, so each segment can be characterized using an APC. For example, cancer rates may rise gradually for a period of several years, rise sharply for several years after that, and then drop gradually for the next several years. Finding the joinpoint model that best fits the data allows us to determine how long the APC remained constant, and when it changed.

## Confidence Intervals - Parametric Method

For any segment with slope β, the APC is 100{ exp(β) -1 }. The 100(1-α)% confidence limits are: $Lower = 100 \left\{ exp \left( \beta - s^*t_d^{-1} \left( 1 - \alpha / 2 \right) \right) - 1 \right\} Upper = 100 \left\{ exp \left( \beta + s^*t_d^{-1} \left( 1 - \alpha / 2 \right) \right) - 1 \right\}$

where d is the degrees of freedom and s is the standard error for the slope listed in the output (i.e., from the unconstrained linear models), and td-1(q) is the qth quantile of a t distribution with d degrees of freedom.

The p-value for a two-sided test that the true APC is zero is calculated based on a t distribution.

## Confidence Intervals - Empirical Quantile Method

For a more detailed description of the Empirical Quantile methodology, please go here.

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