Page 162 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

142 4 Parametric Tests of Hypotheses

the alternative hypothesis being that there is at least one pair with unequal means,
µ i ≠ µ j.
c
We now assume that H 0 is assessed using two-means tests for all ( ) pairs of
2
the c means. Moreover, we assume that every two-means test is performed at a
95% confidence level, i.e., the probability of not rejecting the null hypothesis when
true, for every two-means comparison, is 95%:

( P µ i = µ j | H 0ij ) = . 0 95 , 4.18

where H 0ij is the null hypothesis for the two-means test referring to the i and j
samples.
The probability of rejecting the null hypothesis 4.17 for the c means, when it is
true, is expressed as follows in terms of the two-means tests:

α = ( P reject H 0 | H 0 ) . 4.19
= ( P µ ≠ µ 2 | H 0 or µ ≠ µ 3 | H 0 orK or µ c− 1 ≠ µ c | H 0 )
1
1

Assuming the two-means tests are independent, we rewrite 4.19 as:

α = 1− ( P µ = µ 2 | H 0 )P (µ = µ 3 | H 0 )K ( P µ c− 1 = µ c | H 0 ) . 4.20
1
1

Since H 0 is more restrictive than any H 0ij, as it implies conditions on more than
two means, we have P (µ ≠ µ j | H 0ij ) ≥ P (µ ≠ µ j | H 0 ) , or, equivalently,
i
i
( P µ = µ j | H 0ij ) ≤ (µ = µ j | H 0 ) .
P
i
i
Thus:

α ≥ 1− ( P µ 1 = µ 2 | H 012 )P (µ 1 = µ 3 | H 013 )K ( P µ c− 1 = µ c | H 0c− , 1 c ) . 4.21

For instance, for c = 3, using 4.18 and 4.21, we obtain a Type I Error
3
α ≥ 1−0.95 = 0.14. For higher values of c the Type I Error degrades rapidly.
Therefore, we need an approach that assesses the null hypothesis 4.17 in a “global”
way, instead of assessing it using individual two-means tests.
In the following sections we describe the analysis of variance (ANOVA)
approach, which provides a suitable methodology to test the “global” null
hypothesis 4.17. We only describe the ANOVA approach for one or two grouping
variables (effects or factors). Moreover, we only consider the so-called “fixed
factors” model, i.e., we only consider making inferences on several fixed
categories of a factor, observed in the dataset, and do not approach the problem of
having to infer to more categories than the observed ones (the so called “random
factors” model).