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!

Do grazers reduce algal cover in the intertidal?

Experiment Replicated on Two Ends of a gradient

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

The Data

Humdrum Linear Model

sqrtarea ~ height + herbivores

Residuals Look Weird

Group Residuals Look Odd

Pattern in Fitted v. Residuals

Nonlinearity seen in the Tukey Test!

           Test stat Pr(>|Test stat|)    
Tukey test   -3.3171        0.0009097 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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

Factorial Blocked Experiment

Factorial Design

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

ANOVA shows an Interaction Effect

What does the Interaction Coefficient Mean?

What does the Interaction Coefficient Mean?

Outline

1. Factorial ANOVA
   
   
2. Posthocs
   
   
3. Unbalanced Designs

Post-hoc Tests!

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     9.72 3.86 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    -2.36 3.86 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    10.43 5.45 60  1.913  0.0605 
 low minus - low plus     22.51 5.45 60  4.128  0.0001 
 low minus - mid plus      7.36 5.45 60  1.350  0.1819 
 mid minus - low plus     12.08 5.45 60  2.215  0.0305 
 mid minus - mid plus     -3.07 5.45 60 -0.563  0.5758 
 low plus - mid plus     -15.15 5.45 60 -2.778  0.0073 

Posthoc with Simple Effects Model

We are often interested in something simpler…

Simpler VIsualization of an Interaction Effect

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

Type II

Enter Type III

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

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

	Type I	Type II
Test for A	A v. 1	A + B v. B	A + B + A:B v B + A:B
Test for B	A + B v. A	A + B v. A	A + B + A:B v A + A:B
Test for B	A + B v. A	A + B v. A	A + B + A:B v A + B

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!

Non-Least Squares Approaches

#Likelihood
intertidal_glm <- glm(sqrtarea ~ height * herbivores, 
                family = gaussian(link = "identity"),
                data=algae)

#Bayes
library(brms)
intertidal_banova <- brm(sqrtarea ~ height * herbivores, 
                 family = gaussian(link = "identity"),
                 data=algae,
                 file = "intertidal_brms")

Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   "-I/usr/local/opt/llvm/include"   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
namespace Eigen {
^
/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
namespace Eigen {
               ^
               ;
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
#include <complex>
         ^~~~~~~~~
3 errors generated.
make: *** [foo.o] Error 1

Variance Paritioning Gets More Interesting!

Many Treatments