# Average Annual Percent Change (AAPC)

As of version 3.3, an important new feature was added to Joinpoint, the Average Annual Percent Change (AAPC). While Joinpoint computes the trend in segments whose start and end are determined to best fit the data, sometimes it is useful to summarize the trend over a fixed predetermined interval. The AAPC is a method which uses the underlying Joinpoint model to compute a summary measure over a fixed pre-specified interval.

Annual Percent Change (APC) is one way to characterize trends in cancer rates over time. This means that 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 50.000 per 100,000 in 1990, the rate is 50 × 1.01 = 50.500 in 1991 and 50.5 × 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:

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, 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.

Average Annual Percent Change (AAPC) is a summary measure of the trend over a pre-specified fixed interval. It allows us to use a single number to describe the average APCs over a period of multiple years. It is valid even if the joinpoint model indicates that there were changes in trends during those years. It is computed as a weighted average of the APC's from the joinpoint model, with the weights equal to the length of the APC interval.

## How is the AAPC computed?

AAPC is derived by first estimating the underlying joinpoint model that best fits the data. The accompanying figure shows the joinpoint model for prostate cancer incidence from 1975-2003 (from the 2005 data submission), which found joinpoints in 1988, 1992, and 1995. (This model is fit under the default joinpoint parameters). The AAPC over any fixed interval is a weighted average of the slope coefficients of the underlying joinpoint regression line with the weights equal to the length of each segment over the interval. The final step of the calculation transforms the weighted average of slope coefficients to an annual percent change. If we denote bis as the slope coefficients for each segment in the desired range of years, and the wis as the length of each segment runs in the range of years, then:

APCi = { (Exp(bi) − 1) } × 100

and

In the prostate cancer example, to compute the AAPC from 1994 to 2003, we first note that an APC of −10.7 runs for 1 year (with a slope coefficient of −0.113), while an APC of 0.7 runs for 8 years (with a slope coefficient of 0.007). Thus, the AAPC is computed as:

{ Exp( ( 1 x −0.113 + 8 × 0.007 ) / 9 ) − 1 } × 100 = −0.6

The AAPC for a year range which is entirely within a single joinpoint segment is equal to the APC for that segment, e.g., the AAPC for 1999 to 2003 is computed as:

{ Exp( ( 4 × 0.007 ) / 4 ) − 1 } × 100 = 0.7

The AAPC for the entire range of data is computed as:

{ Exp( (13 × 0.026 + 4 × 0.151 + 3 × −0.113 + 8 × 0.007 ) / 28 ) − 1 } × 100 = 2.4

Confidence Interval for AAPC:

Denote the normalized weight as . We can rewrite the AAPC as

An approximate 100(1-α)% confidence interval for the true average annual percent change is

, where

are the lower and upper confidence limits of the intervals, is the α th quantile of the standard normal distribution, and denotes the estimate of the variance of obtained from the fit of the joinpoint model.

In the Joinpoint software, the AAPC confidence interval is based on the normal distribution, and the APC confidence interval is based on a t distribution. If an AAPC lies entirely within a single joinpoint segment, the AAPC is equal to the APC for that segment. To obtain consistency between the APC and AAPC confidence intervals in this situation, the confidence interval for the AAPC has been modified to be identical to that used for the APC using the t distribution instead of the normal distribution. For details on the confidence intervals for the APC, see: Estimated AAPC.

If the confidence interval contains zero, then there is no evidence to reject the null hypothesis that the true AAPC is zero at the significance level of α; otherwise, we reject the null hypothesis in the favor of the alternative hypothesis that the true AAPC is different from zero.

## Can we compare AAPCs for two independent groups?

The true AAPCs for two independent groups, for example, males and females, can be compared by using the following approximate 100(1-α)% confidence interval with the estimated difference of AAPC(1) - AAPC(2), where

and

.

Using that, an approximate 100(1-α)% confidence interval for φ(1) - φ(2) is (dL(α),dU(α)), where

and

,

and using a Taylor series expansion, an approximate 100(1-α)% confidence interval for the difference between the two true average annual percent change rates is obtained as (100xdL(α), 100xdU(α)).

## What are the relative advantages and disadvantages of reporting an AAPC over APCs?

Reporting APCs for each joinpoint segment provides a complete characterization of the trend over time. However, sometimes a summary measure over a fixed interval may be desirable. The statistical power to determine if an APC is different from 0 is a function of the length of the interval. Thus, a short segment rising at a steep rate may not be statistically significant. Comparing the last segment of two series (e.g. males and females) sometimes yields seemingly contradictory results when the segments are of very different lengths. Comparing AAPC's of equal lengths from both series is usually a more meaningful comparison. For example, the delay-adjusted thyroid cancer incidence trend (using data from 1975-2002) was rising for males at 2.2% per year from 1980-2000 (and is characterized as rising since it is statistically significant) and at 11.4% per year from 2000 to 2002 (characterized as a non-significant change because the APC is not statistically significant). However, for females it was rising at 5.3% per year from 1993-2002 (characterized as rising since it is statistically significant). Because the last segment for males is relatively short, it introduces uncertainty, and we arrive at the conclusion that recent rates for females are rising, while for males we are uncertain if the rates are rising. To make the comparison between males and females more comparable, it is useful to compute the AAPC over the same fixed interval for both series. The AAPC for 1993-2002 is 4.2% for males and 5.3% for females (each characterized as rising since they are both statistically significant). The AAPC for 1998-2002 is 6.8% for males and 5.3% for females (each characterized as rising since they are both statistically significant).

Rather than reporting the APC for the final segment for a long list of cancer sites, there may be advantages to reporting the AAPCs over specified fixed intervals. If space permits, reporting both the AAPC and the final segment APC gives an even more complete picture, since each give a somewhat different perspective.