Statistical Power

In comparisons involving only two treatments, there is no real need to test

assumptions on the rankits or ranks; simply proceed with a one-tailed t-test for

unequal variances using the rankits or ranks.

Statistical Power. For a t-test, the basic formula for calculating the sample

size (number of replicate experimental units, n) per treatment necessary to

provide a specified power (1 - ) to detect a given effect size (d) is:

n = 2 ( t 1-α ,v + t 1-β ,v )2 ( s 2/ d 2 )

(L-10)

where

v = degrees of freedom (df) or (n1 + n2 - 2)

t1-a,v = Student t-value for probability 1 - a and v df

t1-,v = Student t-value for probability 1 - and v df

d = the effect size or difference to be detected.

Recall that is the probability of committing a Type II error. This formula

for n must be solved iteratively, because an initial value of n must be used to

determine v. A new n is then calculated using the initial value, and the process

is repeated until n and v are consistent. The iterative process can be tedious if

computer programs are not used. It is easier to use the following approximate

formula (from Alldredge 1987):

n = 2 ( z 1-α + z 1-β )2 ( s 2/ d 2 ) + 0.25( z 1-α )

(L-11)

where

z1-a = normal deviate for 1 - a

z1- = normal deviate for 1 -

0.25(z21-a) = correction term to increase sample size when n is small

Calculated n derived from this formula should be regarded as approximate

for n < 5. Regardless of which formula is used, a fractional n is always rounded

up to the next integer.

A useful exercise when sample sizes are fixed because of budget or logistic

constraints is to calculate the power of the test to detect a specific effect size (d).

In a test comparing 100 percent elutriate survival with dilution water survival, d

is some selected reduction in mean 100 percent elutriate survival from mean

dilution water survival. Equation L-8 can be rearranged and solved for t1- to

determine the power:

L14

Appendix L

Statistical Methods