Forney et al.: Aerial surveys of Phocoena phocoena abundance trends 



373 



-Cutoffs lor a = .05 



H A] istrue 



Distribution of values 



if H is true 



Distribution of values 



if Ha, is true 



fj™. Region of correctly 

 iiS&l rejecting H in favor 

 of H A ,(=Power) 



 Region of falsely 

 rejecting H in favor 

 of Ha 2 (=y error) 



r— i Region of falsely not 

 Y/A rejecting H when Ha, 

 is true (=p error) 



Figure 5 



Graphic illustration of the errors associated with a two-tailed 

 test, such as trend analysis. Solid line represents the distribu- 

 tion of coefficients for a hypothetical increasing trend. Dashed 

 line represents the null distribution of coefficients (when there 

 is no trend). Shaded areas represent the three error types, 

 a, p, and y (see text). H A1 represents an increasing trend, 

 H A2 represents a decreasing trend, and H represents no 

 trend. 



Discussion 



The number of years necessary to detect trends in har- 

 bor porpoise abundance with the techniques described 

 above will depend on two things: the rate of change 

 to be detected, and the degree of certainty desired. A 

 5% annual change will be more difficult to detect than 

 a 10% change over the same time period. If a large 

 change must occur before the trend is detected, such 

 methods may be of limited use in the management of 

 populations, and more powerful techniques may be 

 required. 



If one does not need the ability to determine both in- 

 creases and decreases, but merely wishes to determine 

 whether a population is declining (objective 3, Peter- 

 man and Bradford 1987), one-tailed statistical tests can 

 be used and will increase statistical power. Alternative- 



ly, if one is willing to accept a larger probability of in- 

 ferring a trend when none is actually present, the 

 power to detect trends can be improved by raising the 

 level of a used to determine statistical significance. 



It has been suggested that appropriate levels for a 

 and ft should be determined based on the relative costs 

 of committing each type of error (Toft and Shea 1983, 

 Rotenberry and Wiens 1985, Hayes 1987, Peterman 

 1990b). If the cost of failing to detect a change in abun- 

 dance is high relative to the cost of falsely detecting 

 a trend for a stable population, then the traditional 

 a-level of 0.05 may be inappropriate. In such cases it 

 may be preferable to minimize /?-errors by increasing 

 a. For example, in the context of ecological monitor- 

 ing, Hinds (1984) suggests that a should be made equal 

 to p. However, it is important to remember that in- 

 creasing a when power is low also raises y from vir- 

 tually zero to potentially large levels. Rather than 

 equalizing a and /?, a tradeoff must be made between 

 all three types of error. The magnitude of these errors 

 can be estimated using simulations. 



When a is raised to 0.10, ten years of data are suffi- 

 cient to yield power greater than 0.80 and a y-error of 

 virtually zero when a 10% annual change is occurring. 

 However, this corresponds to a very large total change 

 in abundance (236% increase or 61% decrease). A 



