One-Way Analysis of Variance (ANOVA)

Logic of ANOVA

The null hypothesis in analysis of variance is that all samples (e.g., the samples of students with secure, fearful, preoccupied, and dismissing attachment from Dr. Bliwise's data set) came from populations that have the same mean (in this case, the same mean score on the interpersonal anxiety measure). The test is called analysis of variance and not analysis of means because it actually uses variance calculations. You can think about it as a test that examines the variation among the means: Are all the means very similar to each other or do they differ significantly.

In class, Dr. Bliwise used the Lifesavers example to demonstrate the null hypothesis. Lifesavers are a kind of candy that comes in several different colors, each of which has its own flavor (e.g., orange, cherry, pine-apple, and so on.) In this example, there were four colors/flavors. The company that makes Lifesavers uses the same solution for all of them and then adds flavorless scents that combine with the generic flavor to give consumers the illusion that they experience different tastes. Based on this knowledge, you would predict that people would have a difficulty trying to guess the color of a Lifesaver with their eyes and noses closed tight, since the only remaining information would be the taste, which is identical in the absence of scents. Dr. Bliwise asked each student in the class to try and guess the color of a Life Saver with eyes and nose closed tight, and she predicted that they would essentially be guessing at random without the ability to smell the identifying essence. Thus, assuming that there were equal numbers of the four different kinds of Lifesavers, 25% of the students would guess right simply by chance. We can divide the students into 4 groups, depending on the color they named (e.g., red, green, blue, and yellow). Under the null hypothesis, each group would have gotten equal numbers of the four kinds of Lifesavers. In other words, the four groups did not differ in the composition of Lifesavers.

Click here for a Flash demonstration of the null.
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If we apply the above example to the question about the four attachment styles and mean interpersonal anxiety, then we could think of the four kinds of Lifesavers as different levels of anxiety. Under the null hypothesis, the four samples of students (four different attachment styles) do not differ in their composition of anxiety levels (their mean anxiety is the same).

The alternate hypothesis is that there is a difference somewhere among the means of the samples. For example, if all students cheated on the Lifesavers exercise by not closing their eyes or noses, or if the Lifesavers did indeed have different tastes, then the composition of Lifesavers across the four groups of students would differ. Students who names their Lifesaver red, green, blue, or yellow would have actually had a red, green, blue, or yellow piece of candy. Notice, however, that all four groups do not have to differ for the null hypothesis to be false. For example, in the case of attachment style and interpersonal anxiety, we might predict that students with secure and dismissing attachment have similar (low) levels of anxiety and that they differ significantly from students with fearful and preoccupied attachment (high anxiety). The omnibus ANOVA F-test tells you that there is a difference somewhere (that the null hypothesis is false) but it does not tell you the nature of the difference.

Click here for a Flash demonstration of the alternate hypothesis
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Notice that in the first example, where the null hypothesis is true, the variability of Lifesavers (or anxiety levels) within each group was the same as the variability of Lifesavers between the groups. The extent to which there were different kinds of Lifesavers within each group roughly corresponds to the within-group estimate of the population variance. The extent to which the composition of Lifesavers in each group differed from the composition of the other groups corresponds to the between-group variance. (Technically, it is variance among several groups and should be called among-groups variance, but traditionally the word "between" has been used.) When the null hypothesis is true, the variance within groups equals the variance between groups. This is so, because the means of the groups differ by chance (due to sampling error) much like the individual scores within the groups differ by chance (due to the same sources of sampling error). Notice that in the second example, where the null hypothesis was false, the groups differed from each other more so than did the Lifesavers within each group. Since students who said "red" tended to have mostly red Lifesavers, the variability of Lifesavers within the "red" group was very small. Since students who said "blue" tended to have mostly blue Lifesavers and did not have blue, green, or yellow ones, the variability within the "blue" group was small. The variability among the groups (between-group variance) is much larger, because "mostly red" is very different from "mostly blue." If you think of the different Lifesaver colors as different levels of interpersonal anxiety, then anxiety scores were similar within each attachment group, and then were different across the groups. Thus, a group with mostly low anxiety scores would have a low mean score, whereas a group with mostly high anxiety scores would have a high mean score, and the difference (variance) among or between group means would be larger than the variance among individual scores within the groups.

Thus, F = (between-group variance estimate)/(within-group variance estimate)
When F<1 or F=1, between-group variance is smaller than within group variance, suggesting that the groups differ as much as one would expect due to sampling error. When F >> 1, then we gain confidence to say that there is something going on, because the variability due to sampling error (within-group variance) is much smaller than the variability among the groups. Something else is going on: peeking or sniffing in the Lifesavers example, or real differences in anxiety across attachment styles. The between-group and within-group estimates of variance are calculated based on different information from the sample. This tutorial does not cover the calculations. The F-statistic has 2 kinds of degrees of freedom - one for the between-group estimate and one for the within-group estimate. Dfbetween = k-1, where k=number of groups. Dfwithin = n-k, where n = total sample size. The sampling distribution of F is called and F-distribution. Once you look-up the F critical value using the alpha-level and the degrees of freedom, you can compare it to the F-observed. Variances are always positive, and therefore F is always > 0. Hypotheses for F are always non-directional.

Assumptions of the F-test:
Just like the independent t-test:

  1. Random and independent sampling;

  2. The dependent variable is interval or ratio;

  3. The dependent variable is distributed normally within each group;

  4. All groups have equal variances.

[Introduction][Logic of ANOVA][Structural Model]
[Logic of Planned Comparisons] [Logic of Post-hoc Tests]
[Annotated SPSS Example][References]

Pavel Blagov & Joanne Peart