370 



Fishery Bulletin 89(3), 1991 



20 

 18 

 16 

 14 

 12 

 10 



6 — 

 4— 

 2 







20 



18 



16 



14 



12 



10 



8 



6 



4 



2 







n,l in n, 



n. 



0.04 



0.08 0.12 0.16 0.2 

 PORPOISE/KM 



0.24 0.28 



-7 -6 -5 -4 -3 -2 -1 

 LN(PORPOISE/KM + 0.001) 



Figure 3 



Distribution of observed porpoise/km values (A), and log- 

 transformed porpoise/km values (B) for five years of aerial 

 survey data. The transformation was ln(x + 0.001). where x 

 is the observed number of porpoise/km. 



Simulation methods 



Once the best model had been selected (see Results), 

 Monte Carlo simulations were performed to determine 

 the power of the ANCOVA to correctly detect a given 

 trend in porpoise abundance. The analysis of power was 

 divided into two main steps: (1) Simulations without 

 a trend, to determine whether the procedure can create 

 and correctly analyze simulated data; and (2) simula- 

 tions with trends, to estimate how often the procedure 

 correctly identifies a known trend in harbor porpoise 

 abundance. Annual changes of +5% and ±10% were 

 tested over periods of five, six, eight, and ten years. 

 The random data sets were generated using the 

 parameters and error structure obtained for the actual 

 data from the best model (see Results). First, the ex- 

 pected logarithmic value of porpoise per kilometer for 



each combination of conditions was calculated from the 

 fitted parameters. A random error term for each ex- 

 pected value was then drawn from a normal distribu- 

 tion with a mean of zero and standard error from the 

 ANCOVA results of the best model. To allow weighted 

 analysis of the simulated data, this error ter m was 

 weighted inversely, i.e., multiplied times ^(1/w), 

 where w is the number of kilometers flown under the 

 given conditions. A set of 60 values for w, one for each 

 of the 60 simulated porpoise-per-kilometer values, was 

 obtained for each year by randomly selecting the ac- 

 tual numbers of kilometers flown from one of the five 

 survey years. Complete yearly sets were chosen rather 

 than individual values to avoid unlikely combinations 

 of kilometers flown. 



A yearly trend was incorporated into the simulation 

 data by multiplying the calculated value of porpoise per 

 kilometer times a factor representing the desired ex- 

 ponential change in porpoise abundance. To make the 

 simulated data more like potential real data, all values 

 were rounded to yield only integer values of porpoise 

 over the given number of kilometers flown. In addition, 

 to prevent unfeasible values of porpoise per kilometer, 

 a new error term was drawn if the original one resulted 

 in a value which was negative or greater than 0.4 por- 

 poise per kilometer. The highest value observed in 

 1986-90 was 0.24 porpoise per kilometer; multiplying 

 this value times the maximum simulated increasing 

 trend yields an upper limit of approximately 0.4 por- 

 poise per kilometer. Less than 5% of all error terms 

 were redrawn in the simulations. 



