The T-test tutorial page provides a good background for understanding ANOVA ("Analysis of Variance"). Like the two-sample t-test, ANOVA lets us test hypotheses about the mean (average) of a dependent variable across different groups.
While the t-test is used to compare the means between two groups, ANOVA is used to compare means between 3 or more groups.
There are several varieties of ANOVA, such as one-factor (or one-way) ANOVA, two-factor (or two-way) ANOVA, and so on, and also repeated measures ANOVA. The factors are the independent variables, each of which must be measured on a categorical scale - that is, levels of the independent variable must define separate groups.
One-Way ANOVA Example
One-factor ANOVA, also called one-way ANOVA is used when the study involves 3 or more levels of a single independent variable. For example we might look at average test scores for students exposed to one of three different teaching techniques (three levels of a single independent variable).
The null hypothesis for ANOVA is that the mean (average value of the dependent variable) is the same for all groups. The alternative or research hypothesis is that the average is not the same for all groups.
The ANOVA test procedure produces an F-statistic, which is used to calculate the p-value. As described in the topic on Statistical Data Analysis if p < .05, we reject the null hypothesis. We can then conclude that the average of the dependent variable is not the same for all groups.
With ANOVA, if the null hypothesis is rejected, then all we know is that at least 2 groups are different from each other. In order to determine which groups are different from which, post-hoc t-tests are performed using some form of correction (such as the Bonferroni correction) to adjust for an inflated probability of a Type I error.
SPSS Anova Statistical Analysis
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Interpret the results
The p-value for the paint hardness ANOVA is less than 0.05. This result indicates that the mean differences between the hardness of the paint blends is statistically significant. The engineer knows that some of the group means are different.
The engineer uses the Tukey comparison results to formally test whether the difference between a pair of groups is statistically significant. The graph and the table that include the Tukey simultaneous confidence intervals show that the confidence interval for the difference between the means of Blend 2 and 4 is 3.114 to 15.886. This range does not include zero, which indicates that the difference between these means is significant. The engineer can use this estimate of the difference to determine whether the difference is practically significant.
The confidence intervals for the remaining pairs of means all include zero, which indicates that the differences are not significant.
The low predicted R2 value indicates that the model generates imprecise predictions for new observations. The imprecision may be due to the small size of the groups. Thus, the engineer should be wary about using the model to make generalizations beyond the sample data.