74 
Fishery Bulletin 99(1 ) 
Now consider the variation in loge ;; . We can write 
where d l ■ = an effect applying to all estimates made 
during dive i; and 
= an effect applying solely to the estimate for 
fish j during that dive. 
The value of d l will be influenced by the conditions at loca- 
tion i, as well as by the diver used at that location. The 
value of £jj will be influenced by the activity of the fish and 
its orientation to the diver. 
We can now write 
Loge y = logrf, + log £jj, 
with Var(logd-) = a 2 d and, VarOoge^) = ct 2 , analogous to the 
equation for log* . Again, a 2 d and ct 2 are the between-dive 
and within-dive variance components. 
The power analysis that follows involves predicting the 
variation we would expect in logy, for a given number of 
dive locations in) and a fixed number of fish at each loca- 
tion (m). The equations above can be combined to show 
that this variation has four components. Because 
logy y = a, + b tJ + log d, + log Ejj, 
we have 
Vflo gy u ) = ct 2 + a\ + a 2 d + ct 2 . 
Going one step further, the predicted variance of the mean 
of logy, over all fish (j) and all dives (i) at that site can be 
written as 
U a T U d T 
y = — + EL. 
n n 
Assuming the number of dives and fish recorded per dive 
is the same at the second site, the predicted power to 
detect a difference D in the mean log-lengths at the two 
sites is given by 
D 
SED 0,2 
where F r [.] = cumulative distribution function; 
t uh = the upper a /2% point for the ^-distribution 
with 2(n-l) degrees of freedom, and SED = 
V2V, which is the standard error of the dif- 
ference in the two mean log-lengths (see 
Sokal and Rohlf, 1995, p. 263). Note that if 
the analysis involved the comparison of s>2 
sites, the degrees of freedom for the f-distri- 
bution would be s(/?-l). Because the anal- 
ysis is on a log-scale, the difference D is 
calculated as D=log(R+l), where R is the 
percentage change of interest. 
In order to evaluate the power, we needed estimates of the 
four variance components. The first two, ct 2 and o\, were 
estimated by using catch data on true length for populations 
of red cod, blue cod, and snapper from around New Zealand 
from trawl and trap surveys by the Fisheries Division of the 
National Institute of Water and Atmospheric Research. The 
length data for red cod and snapper came from a number of 
locations around New Zealand (Fig. 1). The depths at which 
these fish were collected ranged between 20 and 40 m. The 
locations at which they were collected were grouped into 
sites, such that two locations that were within approximate- 
ly 30 km of each other were considered to be at the same 
site. For red cod, there were 12 sites, each with between 
two and four locations: at each location, lengths were record- 
ed for between 12 and 20 fish. For snapper, there were six 
sites, two of which contained 14 locations, and the other four 
had just two locations each. At each location, lengths were 
recorded for between 10 and 91 fish. The log-lengths were 
then analyzed by using nested analysis of variance. Site and 
location were specified to be random factors, and location 
was nested within the site. The location and residual vari- 
ance components were used as estimates of ct 2 and a re- 
spectively. The data for blue cod came from one site, at five 
locations off Stewart Island. There were between 47 and 51 
fish lengths recorded per location. The log-lengths were ana- 
lyzed by using analysis of variance, with location being spec- 
ified as a random factor. The location and residual variance 
components were used as estimates of a 2 and o 2 h , respec- 
tively. The remaining two variance components, o 2 d and ct 2 
were estimated by using data on the measurement error of 
novice scientific divers, experienced scientific divers, and a 
stereo-video system. These errors were determined by using 
a simple testing procedure for calibrating diver estimates of 
the lengths of reef fish. Silhouettes of fish were placed in the 
water and their lengths estimated by following the meth- 
ods of the GBRMPA (1979), Bell et al. (1985), and English 
et al. (1994). There were eight novice divers and six expe- 
rienced divers, each of who swam five repeat transects. On 
each transect they estimated the length of sixteen silhou- 
ettes. The same procedure was used for the stereo-video es- 
timates: for each image the estimate used was the mean 
of ten measurements. The novice divers and three of the 
experienced divers made their length estimates in a salt- 
water pool. For the stereo-video system and the other three 
experienced divers, measurements were made in a swim- 
ming pool. Definitions of novice and experienced scientific 
divers are the same as those given in Harvey et al. (2000), 
which contains a full presentation of the data and detailed 
description of the method used. A full description of the de- 
sign and calibration of the system can be found in Harvey 
and Shortis (1996, 1998). 
The novice and experienced diver estimates were both 
analyzed by using nested analysis of variance of the loga- 
rithm of the relative accuracy (estimated length divided 
by true length). Diver and silhouette were specified to be 
random factors, and silhouette was nested within diver. 
The diver and silhouette variance components were used 
as estimates of o 2 d and a 2 , respectively. The stereo-video 
estimates were analyzed by using analysis of variance 
of the logarithm of the relative accuracy, and silhouette 
