For a Two-way ANOVA, the effect of any interaction can be evaluated by comparing
the full model (Equation 8.21) with a reduced model where the interaction term is
deleted.
yijk=μ+αi+βj+εijk
Reduced model for Two-way
ANOVA—8.22
The interaction sums of squares is evaluated as the difference between the error sums of
squares for the full model and the error sums of squares for the reduced model. There are
more direct ways of estimating the interaction sums of square but this approach works
with both balanced and unbalanced designs (unequal numbers in the cells of the design).
In a One-way ANOVA, for example, the null hypothesis tested is H 0 : μ 1 =μ 2 =μ 3 =μn,
the means of the treatment groups are equal or in model terms, all αi,s are equal. The
alternative hypothesis is that the means are not equal. Two measures of variation are used
in the test of significance of the overall model: 1) Sums of squares describing the
variation between treatment groups in terms of how different the treatment group means
are, and 2) Sums of squares describing variation attributable to individuals within the
treatment groups (chance variation among individuals). The ratio of the between to
within sources of variance (sums of squares), each sums of squares divided by their
appropriate degrees of freedom, forms the F-statistic and is an overall test of the model
fit. The sums of squares divided by degrees of freedom is called the Mean Square.
Degrees of freedom (df) are values associated with sums of squares, the total df are
partitioned into df associated with each source of variance. If a model fits the data well,
then differences among treatment group means will be large in comparison to the
differences among individuals. That is the Mean Square Between groups MSb a
summary of treatment mean differences, will be larger than the measure of differences
among individuals within all groups called the Mean Square Within groups. In this
situation the effects of the treatment groups will be distinguished from random
differences among individuals, and the null hypothesis of equal treatment group means is
likely to be rejected. The F-test statistic will be larger than 1 (equivalent to the null
hypothesis of equal variation among treatments and individuals—treatment group effects
are no more than random fluctuations) and a small p-value will indicate a significant
statistical model has been fitted to the data. We would conclude there are differences
between treatment group means.
Once a model has been fitted to empirical data and an F-test statistic is found to be
significant, then the investigator will need to determine the nature of the differences
among treatment means. Even if the overall null hypothesis of equal group means is
rejected there may be some means that do not differ from one another. Comparisons
among means may be suggested by the data itself and these are called post hoc
comparisons. Alternatively planned comparisons may have been determined before the
analysis and in this case contrasts of differences between means are performed following
the F-test.
In SAS, One-way ANOV As can be performed by several procedures two of which are
PROC ANOVA, which can only handle balanced designs, and PROC GLM (meaning
General Linear Models). Since the latter is the most flexible it is illustrated in this
chapter. In PROC GLM post hoc comparisons can be performed using the means
Inferences involving continuous data 313