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

Analysis of Variance2-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

Two-Way – ANOVA

SeaWall

Deep

Intermed.

Shallow

Factor 1Location(S, M, D)

Factor 2Quad Size(¼m, ½m)

Two-Way – ANOVA

SeaWall

Deep

Intermed.

Shallow

Factor 1Location(S, M, D)

Factor 2Quad Size(¼m, ½m)

Two-Way – ANOVA

SeaWall

Deep

Intermed.

Shallow

Factor 1Location(S, M, D)

Factor 2Quad Size(¼m, ½m)

INTERACTIONFactor 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 – ANOVAResults

Two-Way – ANOVAResults

Two-Way – ANOVAResults

Urchins

Location

Quad Size

Two-Way ANOVA : Analysis of Variance Table

Source DF SS MS F P Location 1 228.17 228.167 8.99 0.008Quadsize 2 308.33 154.167 6.07 0.010Interaction 2 76.33 38.167 1.50 0.249Error 18 457.00 25.389Total 23 1069.83

Two-Way – ANOVAResults

For the urchin analysis, the results indicate the following:

The effect of Location (p = 0.008) is significantThis indicates that urchin populations numbers were significantly different a different distances from shore

The effect of Quad Size (p = 0.010) is significantThis indicates quadrat type had a significant effect upon the number of urchins collected

The interaction between Distance and Quadrat (p = 0.249) is not significantThis 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 1 228.17 228.167 8.99 0.008Quadsize 2 308.33 154.167 6.07 0.010Interaction 2 76.33 38.167 1.50 0.009Error 18 457.00 25.389Total 23 1069.83

Two-Way – ANOVAResults

For the urchin analysis, the results indicate the following:

The effect of Location (p = 0.008) is significantThis indicates that urchin populations numbers were significantly different a different distances from shore

The effect of Quad Size (p = 0.010) is significantThis indicates quadrat type had a significant effect upon the number of urchins collected

The interaction between Distance and Quadrat (p = 0.009) is not significantThis means that the distance and quadrat size results WERE INFLUENCING the other

Thus, the individualFactors 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 interactionIf the lines cross, there IS Interaction

The greater the lines depart from being parallel, the greater the degree of interaction

Interactions

Interactions Plots

Interactions Plots

Interactions PlotsWhy is there interaction?

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

Interactions PlotsWhy 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) data2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not3) 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