## Percent Change (PC)

The percent change (PC) in rates over a particular time period is calculated by taking the difference between the initial rate and the end rate. The rates can either be a single year rate or a two year average. The difference is then divided by the initial rate and multiplied by 100 to convert it to a percent.

\( PC_{x-y} = \left( {\Large \frac{\left( End \textbf{ } Rate \textbf{ } - \textbf{ } Initial \textbf{ } Rate\right)}{Initial \textbf{ } Rate} }\right) \times {\Large10 }\)

\( Initial \textbf{ }Rate = Rate_x \textbf{ } or \textbf{ } {\Large \left( \frac{Rate_x \textbf{ } + \textbf{ } Rate_{x+1}}{2} \right) } \)

\( End \textbf{ }Rate = Rate_y \textbf{ } or \textbf{ } {\Large \left( \frac{Rate_y \textbf{ } + \textbf{ } Rate_{y+1}}{2} \right) } \)

## Annual Percent Change (APC)

The Annual Percent Change (APC) is calculated by fitting a least squares regression line to the natural logarithm of the rates, using the calendar year as a regressor variable.

*n* = number of years

*r* = rates

*y* = *Ln(r)*

*x = calendar year*

*y* = *mx* + *b ('b' is the point, where the line intersects the 'y axis' and 'm' denotes the slope of the line)*

\( APC = 100 \times (e^m - 1) \)

Because the methods used in the calculation of PC and APC are not directly related, it is possible that the signs of the PC and the APC may disagree.

See Calculating APC with Weighting for the formulas used in calculating Weighted APCs.

## Significance Test: APC to 0

Testing the hypothesis that the APC is equal to zero is equivalent to testing the hypothesis that the regression parameter m is equal to zero. The hypothesis is rejected at a significance level *p* if \( ProbT(abs(\frac{m}{SEm}), n - 2) >= 1 - \frac{p}{2}\), where *ProbT*(*x*,*n*) is the t distribution function evaluated at *x* and with *n* degrees of freedom, and where *SEm* is the standard error of m from the regression.

## Standard Error for APC

The standard error, i.e., *SEm*, is obtained from the fit of the regression. This calculation assumes that the rates increased or decreased at a constant rate over the entire calendar year interval. The validity of this assumption is not assessed. In those few instances where at least one of the rates is equal to zero, the linear regression is not calculated.

See:

Kleinbaum, Kupper, and Muller. Applied Regression Analysis and Other Multivariable Methods. PWS-Kent, Boston, Mass., 2nd edition, 1988.

## Confidence Intervals for APC

The endpoints of a (1 - p) × 100% confidence interval are calculated as:

\( {\Large CI_{low} = } \left( {\Large e^{\left( m - \left( T_{val} \times SEm\right)\right)} - 1 }\right) \times {\Large100 }\)

\( {\Large CI_{high} = } \left( {\Large e^{\left( m + \left( T_{val} \times SEm\right)\right)} - 1 }\right) \times {\Large100 }\)

Where \( T_{val} = T_{inv}(1 - \frac{p}{2}, n -2)\) is the inverse of the* t* distribution function evaluated at \(1 - \frac{p}{2}\) and with \( n - 2\) degrees of freedom.

**Note:** In SEER*Stat, you specify the p-value as opposed to the confidence interval. The confidence interval = (1 – *p*) x 100. Therefore, a 95% confidence interval is specified by *p *= .05.

## Significance Test: APC to APC

The differences between trends for two time periods are tested for statistical significance by comparing the difference in regression coefficients divided by the standard error of that difference with a T distribution with degrees of freedom defined as the sum of the years in both time periods minus 4 (Kleinbaum, 1988).