Estimation procedures: Interannual Trends and Power Analysis 



Linear regression analyses were conducted to determine whether a trend was 

 present in the indices or estimates of abundance (i.e., the slope of the regression line 

 of abundance vs. year was significantly different from zero). 



We used a power analysis to calculate the number of surveys or the CVs of the 

 estimates required to detect a trend (Gerrodette 1987). The power analysis relates five 

 parameters: alpha (the probability of making a Type-1 error, i.e. concluding that a 

 trend exists when in fact it does not), the power, or 1 - beta (beta is the probability of 

 making a Type-2 error, i.e. concluding that a trend does not exist when in fact it 

 does), n (the number of surveys), r (the rate of change in population size), and the 

 CV of the abundance estimate. Additionally, one must choose whether a t- or z- 

 distribution and a one- or two-tailed test is appropriate, and whether r changes 

 exponentially or linearly. It is also necessary to determine whether the CV is 

 constant with abundance, the square root of abundance, or to the inverse of the 

 square root of abundance. Notice that the actual estimate is not used, only the 

 coefficient of variation of the estimate. This estimate can be the actual abundance 

 (population size as determined from mark-resight methods or censuses) or indices of 

 abundance (such as total number of marked animals in the photo-ID catalog for a 

 particular year, or total number of dolphins sighted per survey or time period). 



One of the objectives of this research was to determine whether the photo-ID 

 method could detect a doubling or halving of population size with 80% certainty. 

 Thus, alpha = 0.05, beta = 0.20, power = 0.80, r = 1.00 or -0.50, n = 2 annual surveys, and 

 it is only necessary to calculate the CV required to detect a trend and compare it with 

 the CV of the abundance estimate calculated from the data. Alternatively, one can 

 use the CV of the estimate to solve for n, the number of surveys necessary to detect 

 the trend. In general, the lower the CV, the fewer the number of surveys required to 

 detect a trend (Gerrodette 1987). For mark-resight estimates, the CV decreases as the 

 proportion of marked animals in the population increases (Wells and Scott 1990). 



Traditionally in research, one is concerned mainly with alpha and Type-1 

 errors. This is conservative when considering whether to accept an alternate 

 hypothesis as truth or not, but may not be conservative from a management point of 

 view. Such a case might occur when the null hypothesis that a population is stable is 

 accepted when, in fact, it is declining (Type-2 error). Gerrodette (1987) applied 

 power analysis to linear regressions of abundance. Because the question posed is 

 whether a large change can be detected from one year to the next, and because we 

 used an annual survey period as the sampling unit, the sample size (n), equals two. A 

 linear regression is not feasible with only two data points, so it is necessary to 

 compare two distributions presumed to have known variances rather than use a 

 linear regression (TRENDS2 does this automatically). 



Given the initial parameters specified by the NMFS (alpha = 0.05, power = 0.80, 

 r = 1.00 or -0.50, and n = 2), one can calculate the CV necessary to detect trends in 

 abundance. We used a 1-tailed t-distribution for the TRENDS2 program, and specified 

 that rates of increase or decrease be exponential. We made this choice because an 

 exponential function is more typical of biological processes and because detecting a 

 50% linear decline is a moot exercise given that the population would be reduced to 

 zero at the end of the second year. TRENDS2 also requires that the model of the 

 relationship between CV and abundance be specified. As suggested by Gerrodette 

 (1987) and a graph of our data, the "CV proportional to the square root of abundance" 



