**ENVS 541**

Sampling and Analysis of Environmental Contaminants

**ANOVA Post Hoc Tests**

**This is
some example output from Systat showing how to carry out tests after an
ANOVA to see which of the factor level means is different from each other.
Remember, doing all pairwise comparisons using simple t tests corrupts the
significance levels.
**

**Here’s what the Systat manual has to
say:**

The results in an ANOVA
table serve only to indicate whether means differ significantly or not.
They do not indicate which means differ from another. To report
which pairs of means differ significantly, you might think of computing a
two-sample t test for each pair; however, do not do this. The
probability associated with the two-sample t test assumes that only one
test is performed. When several means are tested pairwise, the
probability of finding one significant difference by chance alone
increases rapidly with the number of pairs. If you use a 0.05
significance level to test that means A and B are equal and to test that
means C and D are equal, the overall acceptance region is now 0.95*0.95 or
0.9025. Thus, the acceptance region for two independent comparisons
carried out simultaneously is about 90%, and the critical region is 10%
(instead of the desired 5%). For six pairs of means tested at the
0.05 significance level, the probability of a difference falling in the
critical region is not 0.05 but 1-(0.95)6 = 0.265. For 10 pairs,
this probability increases to 0.40. The result of following such a
strategy is to declare differences as significant when they are not.

**Here’s some data from 4 levels of a
factor with 10 data points per level:**

10
1

14
1

15
1

13
1

17
1

12
1

19
1

15
1

10
1

11
1

15
2

18
2

17
2

16
2

18
2

16
2

19
2

22
2

19
2

14
2

10
3

08
3

12
3

11
3

14
3

13
3

07
3

10
3

11
3

09
3

13
4

15
4

14
4

17
4

16
4

13
4

15
4

19
4

11
4

16
4

**Here’s the output from Systat for the
ANOVA:
**

Effects
coding used for categorical variables in model.

Categorical
values encountered during processing are:

LEVELS (4
levels)

1, 2,
3, 4

Dep Var: A
N: 40 Multiple R: 0.729 Squared multiple R: 0.531

Analysis of Variance