L.2.1.1.2 Analysis of example data

n(d)

t 1-β ,v =

- t 1-α ,v

(L-12)

2(s)

We then enter a t table at v df and find the column closest to the value of t1-;

power . 1 - P, where P is the probability for that column. SAS can calculate

power more exactly using the PROBT function for t1- and v df. Note that t-

values can be used because both n and v are known. One can also calculate the

difference that can be detected for any given power and sample size:

d = ( t 1-α ,v + t 1-β ,v ) 2s 2/n

(L-13)

The simplest power to use is 0.50, because then t1- = 0. Many computer

programs will provide this difference, usually referred to as the "minimum sig-

nificant difference," "least significant difference," or some similar term. The

term "average detectable difference" would also be applicable, as this is the

difference we expect to be able to detect 50 percent of the time. In this Appen-

dix, we recommend reporting the minimum significant difference or some other

indication of power along with the results of statistical analyses. If power is

consistently and regularly reported, investigators will gain an appreciation of the

strengths and limitations of various toxicity tests and analyses.

If values are transformed prior to analyses, all power calculations should be

done on the transformed scale. In the case of arcsine-transformed survival, a

constant effect size d on the percentage or proportion scale will not be constant

on the arcsine scale, because the latter scale spreads out high and low values.

Therefore, a reference survival must be specified and arcsine-transformed, and

the effect size also transformed to a difference on the arcsine scale. For

example, suppose we wanted to calculate the power of a t-test to detect a 25

percent reduction in survival from the reference. A reasonable reference

survival (e.g., 90 percent) would be specified and arcsine-transformed (=1.249).

We would also arcsine-transform a 25 percent reduction (=65 percent survival or

0.938 after transformation). The difference d would then be 1.249 - 0.938 or

0.311, and that value would be used in power calculations. Experimentation

with arcsine-transformed data will rapidly reveal that toxicity tests are more

powerful, in terms of the size of differences that can be detected on the original

(untransformed) scale, when reference survival is higher. In other words, we are

more likely to detect a 25 percent reduction in survival if reference survival is 90

percent than if reference survival is 75 percent. This is precisely what happens

in real toxicity tests, which is why the arcsine transformation is used for survival

data.

Simple formulae for calculation of sample size or power are not available for

the tests of assumptions recommended in this Appendix.

L.2.1.1.2 Analysis of example data.

Table L-3 contains example data from a 96-hr water column toxicity test

using a dilution water and a dredged-sediment elutriate at four serial dilutions.

In this example, control (laboratory) water was also used for dilutions, and no

L15

Appendix L

Statistical Methods