Factorial Designs & Interaction Effects

Outline

  1. Factorial ANOVA

  2. Posthocs

  3. Unbalanced Designs

Is the world additive?

  • Until now, we have assumed factors combine additively

  • BUT - what if the effect of one factor depends on another?

  • This is an INTERACTION and is quite common

  • Yet, challenging to think about, and visualize

Intertidal Grazing!

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Do grazers reduce algal cover in the intertidal?

Experiment Replicated on Two Ends of a gradient

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What happens if you fit this data using * instead of + in the linear model?

The Data

Humdrum Linear Model

sqrtarea ~ height + herbivores

Sum Sq Df F value Pr(>F)
height 88.97334 1 0.3213845 0.5728570
herbivores 1512.18349 1 5.4622243 0.0227309
Residuals 16887.47793 61 NA NA

Residuals Look Weird

Group Residuals Look Odd

Pattern in Fitted v. Residuals

Nonlinearity seen in the Tukey Test!

Test stat Pr(>|t|)
height NA NA
herbivores NA NA
Tukey test -3.317 0.001

(Note: This test is typically used when there is no replication within blocks)

Factorial Blocked Experiment

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Factorial Design

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Note: You can have as many treatment types as you want (and then 3-way, 4-way, etc. interactions)

Problem: Categorical Predictors are Not Additive!

You can only see this if you have replication of treatments (grazing) within blocks (tide height)

The Model For a Factorial ANOVA/ANODEV

\[y_{ijk} = \beta_{0} + \sum \beta_{i}x_{i} + \sum \beta_{j}x_{j} + \sum \beta_{ij}x_{ij} + \epsilon_{ijk}\]

\[\epsilon_{ijk} \sim N(0, \sigma^{2} ), \qquad x_{i} = 0,1\]


> - Note the new last term

> - Deviation due to treatment combination

The General Linear Model

\[\boldsymbol{Y} = \boldsymbol{\beta X} + \boldsymbol{\epsilon}\]

  • \(\boldsymbol X\) can have Nonlinear predictors

  • e.g., It can encompass A, B, and A*B

How do you Fit an Interaction Effect?

graze_int <- lm(sqrtarea ~ height + herbivores + herbivores:height, 
                data=algae)


graze_int <- lm(sqrtarea ~ height*herbivores, data=algae)

No More Pattern in Fitted v. Residuals

Other Assumptions are Met

F-Tests for Interactions

\(SS_{Total} = SS_{A} + SS_{B} + SS_{AB} +SS_{Error}\)

\(SS_{AB} = n\sum_{i}\sum_{j}(\bar{Y_{ij}} - \bar{Y_{i}}- \bar{Y_{j}} - \bar{Y})^{2}\), df=(i-1)(j-1)


MS = SS/DF, e.g, \(MS_{W} = \frac{SS_{W}}{n-k}\)

\(F = \frac{MS_{AB}}{MS_{Error}}\) with DF=(j-1)(k-1),n - 1 - (i-1) - (j-1) - (i-1)(j-1)

ANOVA shows an Interaction Effect

Sum Sq Df F value Pr(>F)
height 88.97334 1 0.3740858 0.5430962
herbivores 1512.18349 1 6.3579319 0.0143595
height:herbivores 2616.95555 1 11.0029142 0.0015486
Residuals 14270.52238 60 NA NA

What does the Interaction Coefficient Mean?

What does the Interaction Coefficient Mean?

Estimate Std. Error t value Pr(>|t|)
heightlow 32.91450 3.855532 8.536955 0.0000000
heightmid 22.48360 3.855532 5.831516 0.0000002
herbivoresplus -22.51075 5.452546 -4.128484 0.0001146
heightmid:herbivoresplus 25.57809 7.711064 3.317064 0.0015486

Outline

  1. Factorial ANOVA

  2. Posthocs

  3. Unbalanced Designs

Post-hoc Tests!

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Posthocs and Factorial Designs

  • Must look at simple effects first

    • The effects of individual treatment combinations


  • Main effects describe effects of one variable in the complete absence of the other

    • Useful only if one treatment CAN be absent

Posthoc Comparisons Within Blocks

 contrast     estimate       SE df t.ratio p.value
 minus - plus 9.721701 3.855532 60   2.521  0.0144

Results are averaged over the levels of: height

Posthoc Comparisons of Blocks

 contrast   estimate       SE df t.ratio p.value
 low - mid -2.358142 3.855532 60  -0.612  0.5431

Results are averaged over the levels of: herbivores

Posthoc with Simple Effects Model

 contrast                estimate       SE df t.ratio p.value
 low,minus - mid,minus  10.430905 5.452546 60   1.913  0.0605
 low,minus - low,plus   22.510748 5.452546 60   4.128  0.0001
 low,minus - mid,plus    7.363559 5.452546 60   1.350  0.1819
 mid,minus - low,plus   12.079843 5.452546 60   2.215  0.0305
 mid,minus - mid,plus   -3.067346 5.452546 60  -0.563  0.5758
 low,plus - mid,plus   -15.147189 5.452546 60  -2.778  0.0073

Posthoc with Simple Effects Model

Outline

  1. Factorial ANOVA

  2. Posthocs

  3. Unbalanced Designs

Oh no! I lost a replicate (or two)




algae_unbalanced <- algae[-c(1:5), ]

Type of Sums of Squares Matters

Type I

Df Sum Sq Mean Sq F value Pr(>F)
height 1 151.8377 151.8377 0.6380017 0.4278712
herbivores 1 1384.0999 1384.0999 5.8158020 0.0192485
height:herbivores 1 2933.5934 2933.5934 12.3265653 0.0008998
Residuals 55 13089.4237 237.9895 NA NA


Type II

Sum Sq Df F value Pr(>F)
height 77.87253 1 0.3272099 0.5696373
herbivores 1384.09995 1 5.8158020 0.0192485
height:herbivores 2933.59337 1 12.3265653 0.0008998
Residuals 13089.42369 55 NA NA

Enter Type III

Sum Sq Df F value Pr(>F)
(Intercept) 14188.804 1 59.619447 0.0000000
height 1175.967 1 4.941256 0.0303521
herbivores 4242.424 1 17.826097 0.0000915
height:herbivores 2933.593 1 12.326565 0.0008998
Residuals 13089.424 55 NA NA

Compare to type II

Sum Sq Df F value Pr(>F)
height 77.87253 1 0.3272099 0.5696373
herbivores 1384.09995 1 5.8158020 0.0192485
height:herbivores 2933.59337 1 12.3265653 0.0008998
Residuals 13089.42369 55 NA NA

What’s Going On: Type I, II, and III Sums of Squares

Type I Sums of Squares:
    SS for A calculated from a model with A + Intercept versus just Intercept

    SS for B calculated from a model with A + B + Intercept versus A + Intercept

    SS for A:B calculated from a model with A + B + A:B +Intercept versus A + B + Intercept


This is fine for a balanced design. Variation evenly partitioned.

What’s Going On: Type I, II, and III Sums of Squares

Type II Sums of Squares:
    SS for A calculated from a model with A + B + Intercept versus B + Intercept

    SS for B calculated from a model with A + B + Intercept versus A + Intercept

    SS for A:B calculated from a model with A + B + A:B +Intercept versus A + B + Intercept


Interaction not incorporated in assessing main effects

What’s Going On: Type I, II, and III Sums of Squares

Type III Sums of Squares:
    SS for A calculated from a model with A + B + A:B + Intercept versus B + A:B + Intercept

    SS for B calculated from a model with A + B + A:B + Intercept versus A + A:B + Intercept

    SS for A:B calculated from a model with A + B + A:B +Intercept versus A + B + Intercept

Each SS is the unique contribution of a treatment
very conservative

Which SS to Use?

  • Traditionally, urged to use Type III

  • What do type III models mean?
    • A + B + A:B v. B + A:B

  • Interactions the same for all, and if A:B is real, main effects not important

  • Type III has lower power for main effects

  • Type II produces more meaningful results if main effects are a concern - which they are!

Many Treatments

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