Table L-2. Suggested a Levels to Use for Tests of Assumptions

nonparametric tests, but careful laboratory practices can reduce the frequency of

outliers.

Tests for Normality. The most commonly used test for normality for small

sample sizes (<50 observations total) is the Shapiro-Wilk's Test. This test

determines if residuals are normally distributed. Residuals are the differences

between individual observations and the treatment mean. Residuals, rather than

raw observations, are tested because subtracting the treatment mean removes any

differences among treatments. This scales the observations so that the mean of

residuals for each treatment and over all treatments is zero. The Shapiro-Wilk's

Test provides a test statistic W, which is compared to values of W expected from

a normal distribution. W will generally vary between 0.3 and 1.0, with lower

values indicating greater departure from normality. Because normality is

desired, one looks for a high value of W with an associated probability greater

than the prespecified a level.

Table L-2 provides a levels to determine whether departures from normality

are significant. Normality should be rejected when the probability associated

with W (or other normality test statistic) is less than a for the appropriate total

number of replicates (N) and design. A balanced design means that all

treatments have an equal (or nearly equal) number of replicate experimental

units. For applications in this Appendix, a design may be considered unbalanced

when the treatment with the largest number of replicates (nmax) has at least twice

as many replicates as the treatment with the fewest replicates (nmin). Note that

higher a levels are used when number of observations is small, or when the

design is unbalanced, because these are the cases in which departures from

normality have the greatest effects on t-tests and other parametric comparisons.

If data fail the test for normality, even after transformation, nonparametric tests

should be used (see Nonparametric Tests below).

Table L-2

Suggested a Levels to Use for Tests of Assumptions

a When Design Is

Test

Number of Observations

Unbalanced2

Balanced

0.10

0.25

N = 3 to 9

Normality

0.05

0.10

N = 10 to 19

0.01

0.05

N = 20 or more

0.10

0.25

n = 2 to 9

Equality of Variances

0.05

0.10

n = 10 or more

N = total number of observations (replicates) in all treatments combined; n = number of

observations (replicates) in an individual treatment.

nmax =? nmin.

Tables of quantiles of W can be found in Shapiro and Wilk (1965), Gill

(1978), Conover (1980), USEPA (1989) and other statistical texts. These

references also provide methods of calculating W, although the calculations can

be tedious. For that reason, computer programs are preferred for the calculation

L11

Appendix L

Statistical Methods