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 factors 2-Sample T-Test (1R, 1F, 2 Levels) One-Way ANOVA (1R, 1F, >2 Levels) Two-Way ANOVA (1R, 2F, >1 Level) 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 (¼m, ½m) are equal Response – urchin #, Factors – distance, quadrat Two-Way – ANOVA

SeaWall Deep Intermed. Shallow Factor 1 Location (S, M, D) Factor 2 Quad Size (¼m, ½m)

Two-Way – ANOVA SeaWall Deep Intermed. Shallow Factor 1 Location (S, M, D) Factor 2 Quad Size (¼m, ½m)

Two-Way – ANOVA SeaWall Deep Intermed. Shallow Factor 1 Location (S, M, D) Factor 2 Quad Size (¼m, ½m) INTERACTION Factor 1 X Factor 2 Location X Quad Size

If the effect of a fixed factor is significant, then the level means for that factor are significantly different from each other (just like a one-way ANOVA) 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) Two-Way – ANOVA Results

Two-Way – ANOVA Results

Two-Way – ANOVA Results Urchins Location Quad Size

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

For the urchin analysis, the results indicate the following: The effect of Location (p = 0.008) is significant This indicates that urchin populations numbers were significantly different a different distances from shore The effect of Quad Size (p = 0.010) is significant This indicates quadrat type had a significant effect 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

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

For the urchin analysis, the results indicate the following: The effect of Location (p = 0.008) is significant This indicates that urchin populations numbers were significantly different a different distances from shore The effect of Quad Size (p = 0.010) is significant This indicates quadrat type had a significant effect upon the number of urchins collected The interaction between Distance and Quadrat (p = 0.009) is not significant This means that the distance and quadrat size results WERE INFLUENCING the other Thus, the individual Factors must be analyzed 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 If the lines cross, there IS Interaction The greater the lines depart from being parallel, the greater the degree of interaction Interactions

Interactions Plots

Why is there interaction? Because we get a different answer regarding #Urchins by Location (S,M,D) when using different Quadrats (¼m, ½m)

Interactions Plots Why is there interaction? Because we get a different answer regarding #Urchins by Quad Size (¼m, ½m) at different Locations (S,M,D)

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

Two-Way ANOVA is a statistical test – there is a parametric (Two-Way ANOVA) and nonparametric version (Friedman’s) There are 3 ways to run a Two-Way ANOVA in minitab: 1) Two-Way ANOVA – for parametric (normal) balanced (equal n among levels) data 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not 3) Friedman – nonparametric (not normal) data Two-Way – ANOVA

1) Two-Way ANOVA – for parametric (normal) balanced (equal n among levels) data - See examples of Two-Way ANOVA above * Note – Two-Way ANOVA program cannot run Multiple Comparisons Tests (Tukey) Two-Way – ANOVA

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA Location Quad Size

2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA Location*Quad Size

3) Friedman – nonparametric (not normal) data Two-Way – ANOVA

3) Friedman – nonparametric (not normal) data Two-Way – ANOVA