[1] The ANOVA tests the null hypothesis, which states that samples in all groups are drawn from populations with the same mean values.
A higher ratio therefore implies that the samples were drawn from populations with different mean values.
[1] Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908).
When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t2.
[2] ANOVA is a relatively robust procedure with respect to violations of the normality assumption.
[3] The one-way ANOVA can be generalized to the factorial and multivariate layouts, as well as to the analysis of covariance.
[clarification needed] It is often stated in popular literature that none of these F-tests are robust when there are severe violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts.
[4] Furthermore, it is also claimed that if the underlying assumption of homoscedasticity is violated, the Type I error properties degenerate much more severely.
[6] He showed that under the usual departures (positive skew, unequal variances) "the F-test is conservative", and so it is less likely than it should be to find that a variable is significant.
However, as either the sample size or the number of cells increases, "the power curves seem to converge to that based on the normal distribution".
Tiku (1971) found that "the non-normal theory power of F is found to differ from the normal theory power by a correction term which decreases sharply with increasing sample size.
The general conclusion from these studies is that the consequences of such violations are less severe than previously thought.
Although these conclusions should not entirely discourage anyone from being concerned about the normality assumption, they have increased the overall popularity of the distribution-dependent statistical tests in all areas of research.
Calculations of the means and the variance are performed as part of the hypothesis test.
The commonly used normal linear models for a completely randomized experiment are:[10] or where The index
Given the summary statistics, the calculations of the hypothesis test are shown in tabular form.
In a more complex experiment, where the experimental units (or environmental effects) are not homogeneous, row statistics are also used in the analysis.
The null hypothesis, denoted H0, for the overall F-test for this experiment would be that all three levels of the factor produce the same response, on average.
Begin by centering the data in each group The within-group sum of squares is the sum of squares of all 18 values in this table The within-group degrees of freedom is Thus the within-group mean square value is Step 5: The F-ratio is The critical value is the number that the test statistic must exceed to reject the test.
One would not accept the null hypothesis, concluding that there is strong evidence that the expected values in the three groups differ.
After performing the F-test, it is common to carry out some "post-hoc" analysis of the group means.