statement and confidence limits for the differences in means for each pairwise
comparison can be output using the CLDIFF option. When specific hypothesis tests are
suggested prior to data analysis preplanned comparisons can be performed in SAS using
the contrast and estimate statements. The contrast statement generates a sums of
squares for the contrast and an F-value for testing the null hypothesis of no difference
(linear combination of parameters=0). The estimate statement, used in the same way as
the contrast statement generates an estimate of the difference in means, the standard error
of the difference, a t-test to show whether the estimate of the difference is significantly
different from zero and an associated p-value. Confidence intervals can then be
constructed for the estimate of the difference in the means.
In the remainder of this chapter, four ANOVA procedures are described: One-way
ANOV As (both unrelated and related); a Two-way ANOVA (2×2) Factorial design
(unrelated); and a Two-way ANOVA Split Plot design (mixed, a related and an unrelated
factor). Worked examples of a One-way related and unrelated analysis are presented and
compared with computer output so that the reader can grasp the general principles of the
ANOVA approach in the context of the general linear model. These principles can be
extended to Two-way and more complex factorial designs. Calculations by hand for the
Two-way analyses are tedious (and prone to error) and are therefore omitted. The
researcher is likely to use a proprietary computer package for analyses of more complex
designs and emphasis is therefore given to interpretation of computer output for a Two-
way factorial and a split plot design. All of these ANOVA procedures and associated
hypothesis tests are based on assumptions underpinning, use of the F-test statistic, and
underlying assumptions of the general linear model. These assumptions, and ways to
verify them, rather than listing them under each ANOVA procedure are presented here as
a unified set.
Assumptions for ANOVA1 The response variable should be a continuous metric, at least at the interval level of
measurement (equal intervals).
2 The distribution of the response variable should be approximately normal in the
population, but not necessarily normal in the sample.
3 The variance of the response variable should be equal in all population subgroups
(treatment groups) represented in the design. This is the homogeneity of variance
assumption. (Verify by plotting residual against predicted values. A random scatter of
points about the mean of zero indicates constant variance and satisfies this
assumption. A funnel shaped pattern indicates nonconstant variance. Outlier
observations are easily spotted on this plot.)
4 Errors should be independent. This is the most important assumption for use of the F-
statistic in ANOVA. To prevent correlated errors subjects should be sampled at
random (independent of each other) and subjects’ responses should be independent.
Assumptions specific to the general linear model include:
5 Effects should be additive, that is the relationship among the independent variables and
the response variable is assumed to be additive. Each independent variable contributes
an effect to the response variable independent of all other factors in the model. (Check
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