You can interpret the results of two-way ANOVA by looking at the P values, and especially at multiple comparisons. Many scientists ignore the ANOVA table. But if you are curious in the details, this page explains how the ANOVA table is calculated. Show
Interpreting two-way ANOVA resultsI entered data with two rows, three columns, and three side-by-side replicates per cell. No missing values. So 18 values entered in all. Prism file. I analyzed the data four ways: assuming no repeated measures, assuming repeated measures with matched values stacked, assuming repeated measures with matched values spread across a row, and with repeated measures in both directions. The tables below are color coded to explain these designs. Each color within a table represents one subject. The colors are repeated between tables, but this means nothing. Powerpoint file Two-way ANOVA tableHere are the ANOVA tables for the four conditions. These values are all reported by Prism. I rearranged and renamed a bit so the four can be shown on one table (Excel file). How to report two-way ANOVA results in a tableSum-of-squaresFocus first on the sum-of-squares (SS) column with no repeated measures:
Now look at the SS columns for the analyses of the same data but with various assumptions about repeated measures.
Degrees of freedomNow look at the DF values.
Details on how the SS and DF are computed can be found in Maxwell and Delaney (reference below). Table 12.2 on page 576 explains the ANOVA table for repeated measures in both factors. But note they use the term "A x B x S" where we say "Residual". Table 12.16 on page 595 explains the ANOVA table for repeated measures in one factor. They say "B x S/A" where Prism says "residual", and say "S/A" where Prism says "subject". Mean squaresEach mean square value is computed by dividing a sum-of-squares value by the corresponding degrees of freedom. In other words, for each row in the ANOVA table divide the SS value by the df value to compute the MS value. F ratioEach F ratio is computed by dividing the MS value by another MS value. The MS value for the denominator depends on the experimental design.
P valuesEach F ratio is computed as the ratio of two MS values. Each of those MS values has a corresponding number of degrees of freedom. So the F ratio is associated with one number of degrees of freedom for the numerator and another for the denominator. Prism reports this as something like: F (1, 4) = 273.9 Calculating a P value from F and the two degrees of freedom can be done with a free web calculator or with the =FDIST(F, dfn, dfd) Excel formula Multiple comparisonsThe details of multiple comparisons are explained in this document. What are the df values for the AxB interaction?The results of a two-factor analysis of variance produce df = 1, 30 for the F-ratio for factor A, and df = 2, 30 for the F-ratio for the AxB interaction.
What does an AxB interaction mean in a factorial ANOVA?What does an A X B interaction mean in a two-way ANOVA? The affect of factor A depends on the level of factor B. In the dark-fears study, the main effects seemed to be due to the interaction. Factorial experiments.
Which of the following is the correct way to calculate the degrees of freedom within treatments?df(within) = the sum of the df of each treatment group, so 9 x (8-1) = 63. df(between-treatments) = 9-1 = 8.
What is the implication of a significant AxB interaction?In a two-factor ANOVA, what is the implication of a significant A x B interaction? Neither of the two main effects can be significant. The significance of the interaction has no implications for the main effects.
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