Kempthorne uses the randomization-distribution and the assumption of * unit treatment additivity* to produce a * derived linear model* , very similar to the textbook model discussed previously. [30] The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies. [31] However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. [32] [33] In the randomization-based analysis, there is * no assumption* of a * normal* distribution and certainly * no assumption* of * independence* . On the contrary, * the observations are dependent* !

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).
ANOVA Statistics
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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Dear Charles,

First of all I wanted to thank you for this really helpful website and resource pack!

As a practice example I used Ex#2 of Basic concepts for ANOVA to perform, Shapiro-Wilk-Test, Levene-Test, and ANOVA. When I do the Shapiro-Wilk-Test on each of the groups I find that groups/methods 2-4 follow a normal distribution but group/method 1 does not. I thought in the case of a non-normal distribution I wasn’t allowed to perform ANOVA. I’m not very advanced in statistics, so I would really appreciate your help.

Many thanks!