Two-sample t-tests

Bartlett's Test (from SYSTAT) and FN both indicated that the variances of

arcsine-transformed data were not significantly different for the two treatments,

with P > 0.10 (a level from Table L-2, n = 5, balanced design). Thus, on the

basis of these tests, we would proceed with a t-test for equal variances (7).

Two-sample t-tests. Table L-4 provides the results of t-tests for equal (7)

and unequal variances (8). The t-test for equal variances indicated that survival

in the 100 percent elutriate was significantly (P < 0.05) less than in the dilution

water (9). If the data had been normally distributed with unequal variances, the

t-test for unequal variances would have been used. With the example data, both

test results are the same, but this will not always be the case.

Nonparametric Test. Nonparametric tests would generally not be

performed on these data because the sample data did not depart significantly

from a normal distribution. However, the data were converted to rankits (10),

and a t-test for unequal variances (11) was conducted on the rankits (SAS Pro-

gram WATTOX) for illustrative purposes. The t-test indicated that median

survival in the 100 percent elutriate was significantly lower than in the dilution

water (Table L-4).

Statistical Power. The difference in survival between the 100 percent

elutriate and the dilution water was so large (63 percent) that it was easily

detected (declared significant), even though there were only five replicates per

treatment. The power of a t-test to detect such a large decrease in survival (d =

0.848 on the arcsine scale) when n = 5 and s = 0.1055 (also on the arcsine scale)

is >0.99. However, it is reasonable to ask if n = 5 is adequate for detecting

smaller differences. For example, what sample size would be required to pro-

vide a 0.95 chance (1 - = 0.95; z1- = 1.645) of detecting a reduction of

survival to 80 percent, with a = 0.05 (z1-a = 1.645)? In the example data, mean

arcsine-transformed dilution water survival was 1.4806 (99 percent survival;

back-transformation of means of transformed values will not be the same as

means based on original data, although the difference is trivial in this case); the

arcsine-transformed value for 80 percent survival is 1.1071, giving a reduction

(d) of 0.3736 on the arcsine scale; and the pooled s was 0.1055. Using

Equation L-14:

n = 2(1.645 + 1.645 )2 (0.1055 2/0.3736 2 ) + 0.25(1.645 2 ) = 2.40 (L-14)

Rounding up gives n = 3. A more exact iterative computer program

(SYSTAT DESIGN) based on t-values (Equation L-13) also yields n = 3. The

sample size required for a 0.95 probability of detecting a reduction in survival to

90 percent is n = 6, again calculated with the iterative program. The minimum

significant difference (i.e., the difference we have a 0.50 probability of

detecting) when n = 5 is t0.95,8(2s2/n)2 or 1.86[2(0.10552/5)]2 = 0.1241. Sub-

tracting that from the mean transformed dilution water survival, and back-

transforming gives 95.5 percent survival. In other words, given the example

data, the test can be expected to detect a reduction in survival from .99 percent

to .95-96 percent approximately half the time.

L17

Appendix L

Statistical Methods