# Average Annual Percent Change (AAPC) and Confidence Interval

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.

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 APCs from the joinpoint model, with the weights equal to the length of the APC interval.

What is an AAPC?

How is the AAPC Computed?

How are the AAPC Confidence Intervals Computed?

Can We Compare AAPCs for Two Independent Groups?

What are the Relative Advantages and Disadvantages of Reporting an AAPC over APCs?

What is the Advantage of Reporting an AAPC over an APC Computed by Fitting a Single Line (on a log Scale) to the Data?

Effect of Independent Variable Interval Type on the Labeling of APC and AAPCs

## 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 calculated using 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 in the range of years, then:

APCi = { Exp(bi) - 1 } x 100

and $AAPC = \left\{ exp \left( \frac{\sum w_i b_i}{\sum w_i} \right) - 1 \right\} \times\ 100.$

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 x 0.007 ) / 9 ) - 1 } x 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 x 0.007 ) / 4 ) - 1 } x 100 = 0.7.

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

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

## How are the AAPC Confidence Intervals Computed?

### Parametric Confidence Interval for True AAPC

Denote the normalized weight as $\~w_i = w_i / \sum w_j$

and rewrite the AAPC as $AAPC = \left\{ exp \left( \sum \~w_i b_i \right) - 1 \right\} \times\ 100.$

An approximate 100(1- α)% confidence interval for the true average annual percent change is $\left( AAPC_{L \left( \alpha \right)}, AAPC_{U \left( \alpha \right)} \right)$, where $AAPC_{L \left( \alpha \right)} = \left\{ exp \left[ \log \left( \left( AAPC / 100 \right) + 1 \right) - z_{1 - \alpha / 2} \sqrt{\sum \~w_i^2 \^\sigma_i^2} \right] - 1 \right\} \times\ 100 AAPC_{U \left( \alpha \right)} = \left\{ exp \left[ \log \left( \left( AAPC / 100 \right) + 1 \right) + z_{1 - \alpha / 2} \sqrt{\sum \~w_i^2 \^\sigma_i^2} \right] - 1 \right\} \times\ 100$

are the lower and upper confidence limits of the interval, Ζα is the αth quantile of the standard normal distribution, and $\^\sigma_i^2$ denotes the estimate variance of bi 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: Annual Percent Change (APC) and Confidence Interval.

### Empirical Quantile Confidence Interval for True AAPC

Motivated by a conservative tendency of the asymptotic confidence interval for the true AAPC, a new method called the empirical quantile method is implemented in V 4.2. as an improved confidence interval. The idea of the empirical quantile method is to generate resampled data by (i) generating resampled residuals as the inverse function values of the uniform random numbers over (0,1) where the function is the empirical distribution function of the original residuals and then (ii) adding resampled residuals to the original fit. For each resampled data, the model fit was made and the AAPCs are estimated. Then, the 100(α/2)th and 100(1-α/2)th percentiles of the resampled AAPC values are obtained as the lower and upper limits of the 100(1-α)% empirical quantile confidence interval for the true AAPC. For more information on the empirical quantile confidence interval, please go here.

### Confidence Interval and Hypothesis Test

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 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 $AAPC^{\left( 1 \right)} = \left\{ exp \left( \sum \~w_i^{\left( 1 \right)} b_i^{\left( 1 \right)} \right) - 1 \right\} \times\ 100 = \left\{ exp \left( \^\varphi^{\left( 1 \right)} \right) - 1 \right\} \times\ 100 and AAPC^{\left( 2 \right)} = \left\{ exp \left( \sum \~w_i^{\left( 2 \right)} b_i^{\left( 2 \right)} \right) - 1 \right\} \times\ 100 = \left\{ exp \left( \^\varphi^{\left( 2 \right)} \right) - 1 \right\} \times\ 100.$

Note that an approximate 100(1-α)% confidence interval for φ(1) - φ(2) is (dL(α),dU(α)), where $d_{L \left( \alpha \right)} = \^\varphi^{\left( 1 \right)} -\ \^\varphi^{\left( 2 \right)} -\ z_{1 - \alpha / 2} \sqrt{\widehat{Var} \left( \^\varphi^{\left( 1 \right)} \right) + \widehat{Var} \left( \^\varphi^{\left( 2 \right)} \right) } and d_{U \left( \alpha \right)} = \^\varphi^{\left( 1 \right)} -\ \^\varphi^{\left( 2 \right)} +\ z_{1 - \alpha / 2} \sqrt{\widehat{Var} \left( \^\varphi^{\left( 1 \right)} \right) + \widehat{Var} \left( \^\varphi^{\left( 2 \right)} \right) },$

Then, using a Taylor series expansion, an approximate 100(1-α)% confidence interval for the difference between the two true average annual percent change rates can be obtained as $\left( \^d_{L \left( \alpha \right)}, \^d_{U \left( \alpha \right)} \right)$, where $\~d_{L \left( \alpha \right)} = AAPC^{\left( 1 \right)} -\ AAPC^{\left( 2 \right)} -\ z_{1 - \alpha / 2} 100 \sqrt{e^{2 \^\varphi^{\left( 1 \right)}} \widehat{Var} \left( \^\varphi^{\left( 1 \right)} \right) +\ e^{2 \^\varphi^{\left( 2 \right)}} \widehat{Var} \left( \^\varphi^{\left( 2 \right)} \right)} \~d_{U \left( \alpha \right)} = AAPC^{\left( 1 \right)} -\ AAPC^{\left( 2 \right)} +\ z_{1 - \alpha / 2} 100 \sqrt{e^{2 \^\varphi^{\left( 1 \right)}} \widehat{Var} \left( \^\varphi^{\left( 1 \right)} \right) +\ e^{2 \^\varphi^{\left( 2 \right)}} \widehat{Var} \left( \^\varphi^{\left( 2 \right)} \right)}$

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

Reporting an APC 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 AAPCs 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.

## What is the advantage of reporting an AAPC over an APC computed by fitting a single line (on a log scale) to the data?

Prior to the development of the joinpoint and AAPC methodology, to characterize a trend over a fixed interval, a single regression line (on a log scale) over the fixed interval was fit, and the slope coefficient was then transformed to an APC. This older methodology has two disadvantages over the AAPC. First, the older methodology assumes linearity of the trend (on a log scale) over the interval, while the AAPC does not. Secondly, the AAPC can be used to characterize a short segment based on a joinpoint model fit over a much longer series. This is especially advantageous for situations when the data are sparse (e.g. a rare cancer or data from a small geographic area). In these cases, because of the variability of the underlying data, the older method might estimate an APC for the period 1996-2005 with a very wide confidence interval. The AAPC from 1996-2005 might be based on an underlying joinpoint model using data from 1975-2005, and thus the resulting AAPC would usually be more stable.