372 



Fishery Bulletin 89(3), 1991 



ANCOVA model with a null hypothesis of no trend in 

 abundance. The a-error (Type I error) is the fraction 

 of simulations which falsely detected a trend. 



For all four simulations, the resulting a-errors were 

 close to the theoretical ones (Table 5). The average root 

 mean-square-error term obtained for these data sets 

 (16.63) was also close to the error for the actual data 

 (17.01). The estimates of the covariate for year (d) in 

 the simulated data were approximately normally 

 distributed with a zero mean, as expected (Fig. 4). This 

 confirms that the simulation procedures do not intro- 

 duce substantial bias into the data or error structure. 



Simulations with trends (d ^O) To analyze the power 

 of this procedure to detect given trends, random data 

 sets spanning five, six, eight, and ten years were 

 created with artificial changes in abundance of ± 5% 

 and ± 10% per year. All other parameters were taken 

 from Table 4, set (B), as above. For each combination 

 of survey years and trend, 500 data sets were created 

 and analyzed with the ANCOVA procedure. 



In each simulation, a fraction of the analyses did not 

 detect a trend: this represents the /?-error (Type II 

 error). A much smaller fraction detected a trend in the 

 opposite direction of the true trend. The latter presents 

 a special case (dilemma), and we have termed this type 

 of error y-error (Type III, cf. Carmer 1976). Figure 5 

 graphically illustrates a, p, and y for a situation where 

 an increasing trend is occurring and being tested 

 against the null hypothesis in a two-tailed test (in a one- 

 tailed test, y is zero). The three types of errors are inter- 

 dependent: as a increases (i.e., the bars in Figure 5 

 move closer to zero), p decreases, and y increases. 



Power has been defined as the probability of correctly 

 rejecting the null hypothesis when it is false, which 

 numerically is 1 - p (Rotenberry and Wiens 1985, 

 Peterman 1990ab). However, this definition does not 

 address the error associated with accepting a false 

 alternate hypothesis (y). In the case of trend analysis, 



120 



80 



O 60 

 O 



40 



20 



-0.3 



-0.2 



-0.1 0.1 



COVARIATE 



0.2 



0.3 



Figure 4 



Distribution of covariate estimates (d) representing yearly 

 change in abundance of harbor porpoise (from ANCOVA) for 

 500 simulations of five survey years with no annual trend in 

 abundance. 



this is the probability of rejecting the null hypothesis 

 (no trend) in favor of a trend in the wrong direction. 

 We therefore suggest that power be defined more 

 precisely to include only the probability of detecting 

 the correct alternate hypothesis, which numerically is 

 1-flJ + y). 



Using this definition, the power to correctly detect 

 trends in harbor porpoise abundance is displayed in 

 Table 6 for six different levels of a. The values listed 

 under a = 1 .0 correspond to the fraction of the time that 

 the sign of the covariate is correct, regardless of 

 significance level. At this a-level, the /J-error is zero, 

 because the null hypothesis of no trend is always re- 

 jected in favor of either an increasing or a decreasing 

 trend. Both power and y-errors are maximized when 

 a = 1.0 (see Discussion below). 



At a = 0.05, the ability to detect trends in abundance 

 of harbor porpoise is poor (0.07-0.79) for all tested 

 trends and numbers of survey years. This is below the 

 level of power = 0.80 which has been suggested as a 

 minimum standard (Skalski and McKenzie 1982, Peter- 

 man and Bradford 1987). Raising o-levels improves the 

 ability to detect trends, but also increases the chance 

 of detecting a trend in the opposite direction of the true 

 trend (y-error). When a = 0.05, y-errors are less than 

 0.01 for the levels of change tested. In contrast, at 

 a = 0.20, y-errors range from to 0.05, and with 

 a = 1.0, y-errors are between and 0.33. Both p and 

 y-errors are reduced with larger trends and more 

 survey years. 



