Harvey et a!.: Improving the statistical power of length estimates of reef fish 
73 
a number of authors (Green, 1979; Andrew and Mapstone, 
1987; Gerrodette, 1987; Hayes, 1987; Peterman, 1990a, 
1990b; Fairweather, 1991). Statistical power is defined as 
the probability of correctly rejecting a null hypothesis and 
is 1-/3, where [i is the probability of a type-II error (An- 
drew and Mapstone, 1987; Gerrodette, 1987; Fairweather, 
1991). An example of a type-II error in environmental 
monitoring would be to conclude that no impact has oc- 
curred when one has. Therefore, low statistical power can 
be disastrous for environmental monitoring because ad- 
verse environmental impacts go undetected (Fairweather, 
1991). Despite this problem, few marine ecologists and bi- 
ologists make use of power analysis (Fairweather, 1991). 
Power analysis has been used to determine the optimum 
size of sample units and levels of replication needed to 
detect an effect of a particular size with a desired level 
of probability (Andrew and Mapstone, 1987; Gerrodette, 
1987; Fairweather, 1991). Power is a function of sample 
size, the probability of a type-I error (a) and the effect size 
(Gerrodette, 1987). Fairweather (1991) discussed the is- 
sues associated with deciding upon an appropriate level of 
power. Low power can be attributed not only to the sample 
design, but also to biases and errors inherent in the sam- 
pling method (Andrew and Mapstone, 1987) and power 
analysis must account for the uncertainty of measurement 
error (Gerrodette, 1987). 
Historically, reef-fish ecologists have failed to calculate 
and publish the power of their sampling programs. Fur- 
thermore, it is frequently assumed by many researchers 
that their visual estimates of reef-fish length are both 
accurate and precise. In the published literature on reef 
fish studies containing data on visual length estimates, 
we found only three examples out of forty-three papers in 
which the authors stated the accuracy of their in situ vi- 
sual length estimates (Sweatman, 1985; Polunin and Rob- 
erts, 1993; Green, 1996). 
The aims of our study were 1) to examine the accuracy 
and precision of length estimates made by a number of ex- 
perienced and novice scientific SCUBA divers, and so de- 
termine their power to detect changes in the mean length 
of populations of three common species of reef fish from 
around New Zealand coastal waters and 2) to demonstrate 
that the power to detect changes in mean length can be 
greatly improved for two of the three species by using an 
underwater stereo-video system instead of divers’ visual 
estimates. 
The three fish species that we consider are blue cod ( Pa - 
rapercis colias), red cod ( Pseudophycis backus), and snap- 
per ( Pagrus auratus ). All three species support commer- 
cial trawl (red cod and snapper), long line (snapper), and 
trap (blue cod) fisheries. Blue cod and snapper are also the 
focus of popular recreational fisheries and thus are impor- 
tant species in New Zealand. 
Methods and materials 
To assess the extent to which measurement error will 
affect the power of visual estimates to detect changes in 
mean length of a population of fish, we considered the fol- 
lowing simple scenario. Suppose we are interested in com- 
paring the mean lengths of two fish populations and we 
collect length estimates by randomly selecting dive loca- 
tions within each site. At each location, the dive involves 
the visual collection of data from a strip-transect or point- 
count method to measure the length of each of a number 
of fish of the species concerned. Later, we will assume that 
the same numbers of fish are encountered on each dive. 
This is clearly unrealistic because the numbers encoun- 
tered will obviously differ: it merely helps to simplify the 
discussion of power analysis. The analysis we consider 
here involves first transforming the estimated lengths by 
using natural logarithms, calculating the mean log-length 
at each location in each site, and then comparing sites by 
a standard Ltest, with the locations acting as replicates. 
The reason for considering log-length rather than length 
is twofold. First, it may be more prudent to perform such 
an analysis on the log-scale, for the usual reason of want- 
ing to satisfy the assumptions of the t-test. Second, the 
power analysis can then be framed in terms of our ability 
to detect a percentage change in mean length. As a conse- 
quence, the standard allometric relationship between log- 
length and log-weight (Kulbicki, 1989) implies that the 
results given here for the power to detect a percentage 
change in mean length will also apply to an equivalent 
proportional change in mean weight. 
The estimated length of fish j at dive location i can be 
written as 
where x if = the true length of the fish; and 
e = the relative accuracy of the estimate (see St 
John et ah, 1990). 
This equation shows that variation in estimated length 
will arise from two sources: first, from the natural varia- 
tion, both between and within dive locations, of the true 
lengths of the fish; second, from the variation, between 
and within dives, in the relative accuracy of the estimate. 
It is this second component of variation that will be influ- 
enced by using stereo-video system as opposed to experi- 
enced or novice scientific divers. The relative magnitudes 
of the two sources of variation will determine the benefits 
to be expected from improving the measurement of length. 
Thus, if the natural variation in true length is large in 
relation to the measurement error, there will be little sta- 
tistical benefit in reducing the latter. 
On a log-scale this equation can be written as 
logy,, = logx y + logCy 
The variation in logjt (/ between and within dive locations 
can be expressed in a one-way random effects model as 
logv (/ = a, + bjj, 
with Var(a ( ) = o 2 a and Var(6 i; ) = o\ (Sokal and Rolf, 1995). 
Thus o 2 and a 2 , are the between-dive and within-dive 
a o 
variance components, respectively. 
