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• When do I use the One-way ANOVA?
• What should I do if my data does not meet the assumption of equal variances across groups?
• Where do I find the One-way ANOVA in SPSS?
• References

The One-way ANOVA (short for One-way ANalysis Of VAriance) is a parametric testing procedure for testing hypotheses about the means of (semi-)continuous data. The ANOVA tests whether the means of 2 or more independent groups are equal.

If you want to test whether the means of 2 or more groups are equal to each other, you can use the unpaired One-way ANOVA. As a rule, the One-way ANOVA is only used when comparing at least 3 groups. For comparing two groups, the One-way ANOVA is equivalent to performing an unpaired t-test.

The One-way ANOVA assumes that the groups come from a multivariate normal distribution (within each group, the variable is normally distributed). It then tests whether the means of these distributions are equal or not. The One-way ANOVA also assumes that the groups come from a distribution with the same variance (spread). With, for example, Levene's Test for equality of variance, you can test whether the variance in the groups can be assumed to be equal. If these conditions are not met, a non-parametric Kruskal Wallis analysis can be considered.

Example of the use of the One-way ANOVA: Suppose you want to test whether the log10 CD4 count is equal for three arms in a study.

Table 1. log 10 CD4 counts in the three treatment arms
Variable*	Treatment arm 1	Treatment arm 2	Treatment arm 3	p-value**
Log 10 CD4 counts	4.5 (1.3)	4.0 (1.5)	4.1 (1.6)	0.66
*Variables are denoted as mean (SD). **Group differences were tested with one-way ANOVA.


We have an animal experimental study in which rats were ventilated using different strategies and either did or did not receive pharmacological pretreatment. Each of the 4 experimental groups included 6 rats. I used the one-way ANOVA when the test for equal variances (Bartlett's test for equal variances) did not produce a significant value. If the p-value of this test was <0.05, I applied a log transformation to the data and ran the ANOVA analysis on this. If the p-value after transformation was still well below 0.05, I used the Kruskal-Wallis analysis. According to the checklist provided by GraphPad Prism, this would be statistically justified. However, I have a question about one analysis. Bartlett's test gives a value of p<0.0001 when the data are not transformed, and a value of 0.0318 after log transformation. Both the ANOVA and Kruskal-Wallis indicate a statistically significant difference between the different experimental groups, but the post-hoc analyses differ considerably, with the post-hoc analysis for the non-parametric test (Dunn's test) showing almost no differences between the groups. Now, the explanation provided with the program says that you should look critically at whether to reject the ANOVA if Bartlett's test does not come out far below 0.05, since with small numbers the Kruskal-Wallis is not such a powerful test. Can I indeed do this, because when I look at the graphs, the post-hoc analysis of the ANOVA (the Newman-Keuls test) gives a result that fits the visual data much better?

Your reasoning is correct. Tests for equal variances are sometimes strict (particularly with small numbers). A p-value of 0.04 is close to the threshold at which you may accept it. I would indeed look at the raw (log) values and consider what you see there. With small numbers, outliers in particular can lead to different results. So that is something you could look into further. If you do indeed have a large outlier, you should check whether, once it is removed, the ANOVA of the ln value is still significant. Non-parametric tests are indeed often stricter (and thus less likely to reach significance) than parametric tests, but are also much less influenced by outliers.

You can find the One-way ANOVA in SPSS under Analyze->Compare Means.

• A short guide to running ANOVA:s in SPSS. By Jörgen Ripa - explains how one-way ANOVAs, as well as more advanced ANOVAs, can be specified in SPSS. Very useful for better understanding ANOVA SPSS syntax. (The original link, http://www.teorekol.lu.se/statistics_for_biologists/SPSS_ANOVA_guide.pdf, no longer works.)

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