Optimum Sample Size to Detect Perturbation Effects: The ...

Power analysis for sample size estimations 3
size (n), sampling design, variability of sampling and effect size. Effect size is the magnitude
of the true effect. The larger it is, the more likely a given design with a given
sample size will correctly reject H 0 at a stated a level. Additionally, it is possible to
evaluate the quality of sampling design and the size of sampling units, especially when
it is necessary to satisfy the assumptions of normality of a data set for parametric tests
and the homogeneity of variances for parametric and non-parametric tests (Underwood,
1981, 1997).
There are two types of statistical power analysis: the first is before the start of data
collection programs, experiments or management manipulations (a priori power analysis),
the second afterwards when the work has been done (a posteriori power analysis).
A priori analysis is commonly used in fisheries sciences before starting an experiment
or management program in order to estimate the sample size necessary to generate acceptably
high power. Another application is to plan the magnitude of treatment perturbations
(effect sizes) necessary to produce high power. It is also used to determine
beforehand how large an effect size would need to be in order to give an acceptable
power, given a planned sample size. On the other hand, a posteriori analysis is relevant
only when interpreting the results of a statistical test that has already failed to reject the
null hypothesis (for more details see the excellent work of Peterman, 1990a).
2. Data analysis
Based on the results obtained by Mouillot et al. (1999) (from their Tables 5 and 6), estimations
of power curves were calculated for N° of samples having variability coefficients
of 10 and 25 %. This procedure was carried out utilising the power tables for
ANOVA design (from 2 to 25 N° of treatments) described by Cohen (1988) at three
different levels of effect size: small, medium and large (0.1, 0.25 and 0.4, respectively).
Note that these levels of effect size were taken from the social sciences context and
`small', `medium' and `large' may not be suitable for biological and ecological studies.
In the present analysis, these are only used as extreme possible situations. The a rate
was fixed at 0.05 and the acceptable power at 0.80 (b = 0.20). These levels of a and b
are usually defined as the minimum acceptable (Peterman, 1990a; Peterman & M'Gonigle,
1992; Schlese & Nelson, 1996; Ribic & Ganio, 1996; Underwood, 1981, 1997),
although, under a conservative test hypothesis, they must be set a = b, or at a desired
power = 0.95 for a = 0.05 (Peterman, 1990a). Additionally, Mapstone (1995) proposed
a procedure for determining the a and b rate that is based on the magnitude of impacts
(effect sizes). Using power analysis it is possible to increase the robustness of any statistical
test, which means that in testing a working hypothesis (e.g., perturbation) the
probability of Type I and Type II statistical errors is simultaneously decreased (Tiku et
al., 1986).
Finally, the results of Mouillot et al.'s (1999) work (see their Tables 5 and 6) were
separated by habitat and time: (1) Seagrass habitat west for June, July and August, (2)
Rocky bottoms south for October, November and December and (3) Rocky bottoms
east for November and December (Table 1). The sample size (n) used to calculate the
power curves was the average of those obtained by Mouillot et al. (1999) per habitat for
the two coefficients of variation (10 % and 25 %).