1 1 



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- 

 distnbution 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" option was selected. Given these parameters, a maximum CV of 0.05 is 



