**One-way ANOVA**

**Theory**

When the effect that one factor have on one dependent variable is studied, one-way ANOVA is used to compare the means of several different groups. It is a generalization of Student's t-test which compares means of two groups. The null hypotheses that is tested with an ANOVA is that there is no difference between the group means, and a low p-value indicates that the nullhypothesis should be rejected. If, e.g. the effects of various blood pressure drugs are studied, the effects of the drugs can be compared with an ANOVA where given drug is the studied factor and blood pressure is the dependent variable. If the ANOVA gives a low p-value, it would indicate that there is indeed a difference in effect on blood pressure between the studied drugs.

The analysis is based on the assumption that the data in each group is drawn **independently** from a **normal distribution**, and that all group distributions **share a common variance**. This is illustrated in the figure below where the data is shown as histograms within each group. The curves show the typical bell shaped form of a normal distribution, and since each curve have the same width, the groups have the same variance. If the samples are independent of each other, it means that the result for one sample does not depend on the results of the other samples. These assumptions must be fulfilled or otherwise, the result from the ANOVA might be misleading.

The ANOVA will only tell you whether there is a significant difference of means between the groups, but not which of the groups that differ from each other. If the ANOVA results in a p-value below the threshold value (e.g. <0.05), you can do a post hoc test to see if there is a significant difference between pairs of groups. GenEx offers three different post hoc tests: *Tukey-Kramer's*, *Bonferroni's*, and *Dunnett's test*. They should be used as follows.

*Tukey-Kramer's test*is appropriet when all or many pairwise comparisons are of intrest.*Bonferroni's test*is appropriet when a small selected number of pairwise comparisons are of intrest.*Dunnett's test*is appropriet when all groups should be compared against one control group.

**How to**

Enter the data in the Data editor together with the classification columns. The data can include several different classification columns, but only one will be used in the one-way ANOVA. Do **not** use zero (0) in the classification columns!

To analyse your data, press the *One-way ANOVA* button in the *Statistics *tab in to top of the main window.

This will open the analysis in the Control panel where you choose the genes that you want to analyze and which one of the classification columns that should be used to divide the data into groups. You can also choose whether to do a post hoc test or not.

The different post hoc tests all require additional information for the analysis. All tests will produce confidence intervals for each pair wise comparison, so the confidence level must be specified or left at its default value of 95%. Both *Bonferroni's* and *Dunnett's test* is available as 1 and 2 sided test where 2 sided is the default. *Bonferroni's test* require that at least one pair wise comparison is chosen from the *Comparisons *list, and *Dunnett's test* require that one control group is chosen from the *Control group* list. If the number of specified pair wise comparisons is large in *Bonferroni's test*, it might be better to perform *Tukey-Kramer's test* instead.

To see the results, press the *Run *button down at the right. The results are presented as one ANOVA table for each gene, with sums of squares (*SS*), degrees of freedom (*df*), mean sums of squares (*MS*), F-statistics (*F*), and *p-value*. If several genes are tested at once, you will be warned that you are performing multiple tests and be suggested a p-value to use as a threshold to keep the overall significance at 0.05. The suggested value is the idāk corrected p-value.

If a post hoc test is chosen, an additional window with the pair wise comparisons will be shown. There is one result table for each gene including an confidence interval (of specified confidence level) for the difference between the groups (*CI low-high*), estimated difference between the groups (*diff*), a test statistic, and a *p-value*. A p-value below the threshold value indicates that there is a significant difference between those groups. The family error rate is controlled for within the analysis of **one** gene, but if more than one gene is tested, a message box will warn that multiple tests are perform and suggest a corrected p-value in the same way as for the ANOVA table. No exact p-values are calculated in Dunnett's test, but it is stated if the p-value is *>=0.05*, *<0.05* (0.1<= p-value <0.5), or *<0.01*.

Warning: Do not use 0 (zero) in the classification columns that defines the groups.