One-way ANOVA (analysis of variance) is a statistical procedure for testing the null hypothesis that several univariate samples are taken from populations with the same mean. The samples are assumed to be close to normally distributed and have similar variances. If the sample sizes are equal, these two assumptions are not critical. If the assumptions are strongly violated, the nonparametric Kruskal-Wallis test should be used instead.
ANOVA table
Equations for the entries in the ANOVA table are given in the Past manual.
Random effects (Model II) ANOVA
For balanced, one-way ANOVA, the ANOVA table, F value and p value are the same for fixed-effects and random-effects ANOVA, so you can use the results for either type of ANOVA. If your design is of the random-effects type (i.e. the factor levels are taken at random from a larger population instead of being fixed by experiment), then you should also report the given variance due to random errors and the variance due to difference between groups (can become negative). The intraclass correlation coefficient ICC gives the proportion of variance due to group differences.
Omega squared
The omega squared is a measure of effect size, varying from 0 to 1.
Levene's test
Levene's test for homogeneity of variance (homoskedasticity), that is, whether variances are equal as assumed by ANOVA, is also given. Two versions of the test are included. The original Levene's test is based on means. This version has more power if the distributions are normal or at least symmetric. The version based on medians has less power but is more robust to non-normal distributions. Note that this test can be used also for only two samples, giving an alternative to the F test for two samples described above.
Unequal-variance (Welch) ANOVA
If Levene's test is significant, meaning that you have unequal variances, you can use the unequal-variance (Welch) version of ANOVA, with the F, df and p values given.
Bayes factor
The Jeffrey-Zellner-Siow (JZS) Bayes Factor is reported in favour of the alternative, i.e. it quantifies the evidence for the hypothesis of unequal means. It is calculated according to Rouder et al. (2012).
Effect size (confidence intervals for the pairwise differences in means)
This table gives 95% confidence intervals for the differences in group means. If a confidence interval does not include zero, the difference may be considered significant, and it is marked in pink. These are “multiple-t” or “simultaneous” confidence intervals, which are wider than the confidence intervals for each pairwise difference in isolation.
Analysis of residuals
The “Residuals” tab shows properties of the residuals, in order to evaluate some assumptions of ANOVA such as normal and homoskedastic distribution of residuals.
The Shapiro-Wilk test for normal distribution is given, together with several common plots of residuals (normal probability, residuals vs. group means, and histogram).
Tukey’s pairwise post-hoc tests
If the ANOVA shows significant inequality of the means (small p), you can go on to study the given table of "post-hoc" pairwise comparisons, based on the Tukey-Kramer test. The Studentized Range Statistic Q is given in the lower left triangle of the array, and the probabilities p(equal) in the upper right.
If the sample sizes are not equal, their harmonic mean is used for n. Significance is estimated according to Copenhaver & Holland (1988). There is also an option to use the slightly less accurate method of Lund & Lund (1983) which was used in older versions of Past.
Kruskal-Wallis
The Kruskal-Wallis test is a non-parametric ANOVA, comparing the medians of several univariate groups (given in columns). It can also be regarded as a multiple-group extension of the Mann-Whitney test (Zar 1996). It does not assume normal distribution, but does assume equal-shaped distribution for all groups. The null hypothesis is
H0: The samples are taken from populations with equal medians.
With G the number of groups, the p value is approximated from Hc using the chi-square distribution with G-1 degrees of freedom. This is less accurate if any ng<5.
Mann-Whitney pairwise post-hoc tests
Mann-Whitney pairwise test p values are given for all Np=G(G-1)/2 pairs of groups. The asymptotic approximation described under the Mann-Whitney module is used. If samples are very small, it may be useful to run the exact test available in that module instead. Four different views are available for the symmetric table:
- Raw p values, uncorrected significance: The p values from each individual pairwise test, marked in pink if p<0.05, not corrected for multiple testing.
- Raw p values, sequential Bonferroni significance: The p values from each individual pairwise test are shown uncorrected for multiple testing. Significance (pink marking) is assessed by first evaluating the smallest p value, with Bonferroni correction for Np pairs. If significant (pNp<0.05) the next smallest p value is significant if p(Np-1)<0.05, etc.
- Bonferroni corrected p values: The values shown are p’ = pNp. Marked as significant if p’<0.05.
- Mann-Whitney U: The test statistics.
Dunn’s post-hoc
The Dunn’s post hoc test (Dunn 1964) is a pairwise test often carried out after a significant Kruskal-Wallis test. It is an alternative to the pairwise Mann-Whitney. See pairwise Mann-Whitney above for the various options in the table output (raw p values, raw p values with sequential Bonferroni, Bonferroni corrected p values, and the z statistics).
References
Copenhaver, M.D., Holland, B. 1988. Computation of the distribution of the maximum studentized range statistic with application to multiple significance testing of simple effects. Journal of Statistical Computation and Simulation 30:1-15.
Dunn, O.J. 1964. Multiple comparisons using rank sums. Technometrics 6:241-252.
Lund, R.E., Lund, J.R. 1983. Algorithm AS 190: Probabilities and upper quantiles for the studentized range. Journal of the Royal Statistical Society C 32:204-210.
Rouder, J.N, Morey, R.D., Speckman, P.L., Province, J.M. 2012. Default Bayes factors for ANOVA designs. Journal of Mathematical Psychology 56:356-374.
Zar, J.H. 1996. Biostatistical analysis. 3rd ed. Prentice Hall.