Analysis of Variance 2-Way ANOVA MARE 250 Dr. Jason Turner

Two-way ANOVA - procedure to test the equality of population means when there are two fixed factors Requires that the number of observations for each combination of the factor levels be the same (balanced) If only the additive model (the model without an interaction term) is to be fit, then one or both factors can be random The two-way ANOVA procedure does not support multiple comparisons Two-Way – ANOVA

For Example… One-Way ANOVA – means of urchin #’s from each distance (shallow, middle, deep) are equal Response – urchin #, Factor – distance Two-Way ANOVA – means of urchin’s from each distance collected with each quadrat (0.25, 0.5) are equal Response – urchin #, Factors – distance, quadrat Two-Way – ANOVA

The two-way ANOVA procedure does not support multiple comparisons To compare means using multiple comparisons, or if your data are unbalanced – use a General Linear Model General Linear Model - means of urchin #’s and species #’s from each distance (shallow, middle, deep) are equal Responses – urchin #, Factor – distance, quadrat Unbalanced…No Problem! Or multiple factors… General Linear Model - means of urchin #’s and species #’s from each distance (shallow, middle, deep) are equal Responses – urchin #, Factor – distance, quadrat, transect Two-Way – ANOVA

If the effect of a fixed factor is significant, then the level means for the factor are significantly different from each other If the effect of an interaction term is significant, then the effects of each factor are different at different levels of the other factor(s) For this reason, it does not make sense to try and interpret the individual effects of terms which are involved in significant higher-order interactions Two-Way – ANOVA Results

Two-Way ANOVA : Analysis of Variance Table Source DF SS MS F P Distance Quadrat Interaction Error Total Two-Way – ANOVA Results

For the urchin analysis, the results COULD indicate the following: The effect of Distance (p = 0.008) is significant This indicates that urchin populations numbers were significantly different a different distances from shore The effect of Quadrat type (p = 0.010) is significant This indicates quadrat type had a significant impact upon the number of urchins collected The interaction between Distance and Quadrat (p = 0.249) is not significant This means that the distance and quadrat size results were not influencing the other Thus, it is okay to interpret the individual effects of either factor alone

Use interactions plots to assess the two-factor interactions in a design Evaluate the lines to determine if there is an interaction: If the lines are parallel, there is no interaction The greater the lines depart from being parallel, the greater the degree of interaction Interactions

Interactions Plots

For Friday Two-Way ANOVA with Urchin data: Was last week a complete waste of time – or did we get some workable data?