One-Way Analysis of Variance

One-way analysis of variance is part of the family of tests known as analysis of variance (ANOVA). Typically, it is used to analyze experimental designs in which only one independent variable has been manipulated. Usually, one-way ANOVA is used to test whether differences exist between three or more means; however, it can be applied to situations in which there are only two means to be compared. Although the t test is preferred by many in such situations, the F test produced in one-way ANOVA is a direct function of t, and so it also is a legitimate way to compare two means.

The two types of one-way ANOVA differ in terms of the experimental design to which they are applied. If data representing different levels of an independent variable are independent (i.e., collected from different entities), then a one-way independent ANOVA (also called a between-groups ANOVA) can be used. When data are related, such as when different entities have provided data for all levels of an independent variable, a one-way repeated measures ANOVA (also

called a within-subjects ANOVA) can be employed. In both cases, the underlying principal is the same: A test statistic F is calculated that is the ratio of systematic variance (variance explained by the independent variable, that is, the experimental manipulation) to unsystematic variance (variance that cannot be explained, or error). If the observed value exceeds the critical value for a small probability (typically .05), we tend to infer that the model is a significant fit of the observed data or, in the case of experiments, that the experimental manipulation has had a significant effect on performance.

An Example

One study looked at the processes underlying obsessive-compulsive disorder by inducing a negative mood, a positive mood, or no mood in people and then asking them to imagine they were going on holiday and to generate as many things as they could that they should check before they left. The data are in Table 1. Three different groups of people participated in this experiment, each group representing a level of the independent variable, mood: negative mood, positive mood, and no mood induced. These data are independent because they came from different people. The dependent variable was the number of items that needed to be checked.

The starting point for ANOVA is to discover how much variability there is in the observed data. To do this, the difference between each observed data point and the grand mean is calculated. These values are then squared and added together to give us the total sum of squared error (SST):

Alternatively, this value can be calculated from the variance of all observations (the grand variance) by multiplying it by the sample size minus 1:

Table 1 Numbers of Things People in Negative or Positive Mood or No Induced Mood Thought They Should Check Before Going on Holiday
	Negative Mood	Positive Mood	No Mood Induced
	7	9	8
	5	12	5
	16	7	11
	13	3	9
	13	10	11
	24	4	10
	20	5	11
	10	4	10
	11	7	7
	7	9	5
x¯	12.60	7.00	8.70
S 2	36.27	8.89	5.57
Grand Mean = 9.43		Grand Variance = 21.43
Source: Davey et al. (2003).
Note:x¯ = mean; S2 = variance.

The degrees of freedom for this value are N – 1, where N is the total number of observations (in this case, 30). For these data, the total degrees of freedom are 29. For the data in Table 1, we get

...