Two-way ANOVA
(This feature is only available in GenEx Pro/Enterprise)
Theory
When the effect that two factor have on one dependent variable is studied, two-way ANOVA is used to compare how these effects the means of several different groups. The null hypotheses that is tested with an ANOVA is that there is no difference between the group means, that there is no effect of either factor and no interaction, and a low p-value indicates that the nullhypothesis should be rejected. The difference of making a two factor design instead of two one factor design, where the effects of one factor is studied one by one, is that a two-factor design can show how the two factors interact with each other. E.g. the effects of various blood pressure drugs could be studied in both females and males, and the studied factors are given drug and gender while the depended variable is the blood pressure. A two-way ANOVA can answer the questions:
The analysis will result in three p-values, one for each hypothesis, and a low p-value indicates that that effect (main/interaction) is indeed significant on the blood pressure. The interaction between two factors can be easily visualized by plotting the dependent variable against one of the factors with one line for each level of the other factor. This type of plot is called interaction plot (see figure). If there is no significant interaction effect, the lines will be parallel.
The analysis is based on the assumption that the data in each group is drawn independently from a normal distribution that all 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), one 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:
Post hoc test for two-factor designs are more complex than if only one factor is studied at a time, if there is a significant interaction effect. Then it becomes harder to separate the two main effects as the will depend on eachother. For this reason, the post hoc tests for two-factor designs are done for one selected factor but on each level of the other factor. If there is no significant interaction effect, it is better to perform two separate one-way ANOVAs and do the post hoc test for one factor at a time.
How to
Enter the data in the Data editor together with the classification columns. The data can include several different classification columns, but only two will be used in the two-way ANOVA. Do not use number 0 (zero) in the classification columns.
To analyse your data, press the Two-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 two 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, and if you want to see an interaction plot.
The different post hoc tests all require additional information for the analysis. As mentioned above, the post hoc test is done for one selected factor for each level of the other factor. The selected factor is chosen from the Do post test for drop-down list. 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 a 1 sided 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.
To see the results, click 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 choosen, 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) - CI (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 you 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, or <0.01.
As can be seen in the ANOVA table for gene A, there is no significant interaction effect. This is clearly illustrated in the interaction plot that shows that the curves are parallell.
Warning: Do not use 0 (zero) in the classification columns that defines the groups.