Many Treatments: Multi-Way ANOVA

Effects of Stickleback Density on Zooplankton

Where Would You Place Replicates?

Randomization

Costs? Benefits?

Blocked Design

Incorporates Gradient
n=1 per block

Randomized Controlled Blocked Design

Randomization within blocks

Effects of Stickleback Density on Zooplankton

Units placed across a lake so that 1 set of each treatment was ’blocked’ together

Treatment and Block Effects

The Steps of Statistical Modeling

1. What is your question?
2. What model of the world matches your question?
3. Build a test
4. Evaluate test assumptions
5. Evaluate test results
6. Visualize

Multiway ANOVA

• Many different treatment types
  • 2-Way ANOVA is for Treatment and block
  • 3-Way for, e.g., Sticklebacks, Nutrients, and block
  • 4-way, etc., all possible
    
    
• Assumes treatments are fully orthogonal
  • Each type of treatment type A has all levels of treatment type B
  • E.g., Each stickleback treatment is present in each block
    
    
• Experiment is balanced for simple effects
  • Simple effect is the unique combination of two or more treatments
    
  • Balance implies the sample size for each treatment combination is the same

Model for Multiway ANOVA/ANODEV

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

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

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

Assumptions of Multiway Anova

• Independence of data points

• Normality within groups (of residuals)

• No relationship between fitted and residual values

• Homoscedasticity (homogeneity of variance) of groups

• Additivity of Treatments

The Usual Suspects of Assumptions

Group Residuals

Tukey’s Test of Non-additivity:

• Our model is \(y_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij}\)
  
  
• But, if A and B are non-additive, results are incorrect.
  
  
• We don’t have the DF with n=1 per treatment combination to calculate an interaction, so…
  
  
• Assume a model of \(y_{ij} = \mu + \alpha_i + \beta_j + \lambda\alpha_i\beta_j\)
  
  
• We can then test for \(SS_{AB}\) using \(\lambda\alpha_i\beta_j\)

Test stat  Pr(>|t|) 
    0.474     0.635 

Hypotheses for Multiway ANOVA/ANODEV

TreatmentHo: \(\mu_{i1} = \mu{i2} = \mu{i3} = ...\)


Block Ho: \(\mu_{j1} = \mu{j2} = \mu{j3} = ...\)

i.e., The variane due to each treatment type is no different than noise

We Decompose Sums of Squares for Multiway ANOVA

\(SS_{Total} = SS_{Between A} + SS_{Between B} + SS_{Within}\)

• Factors are Orthogonal and Balanced, so, Model SS can be split
  
  
• F-Test using Mean Squares as Before

F-Test

How to evaluate effects of each treatment

1. Examine means estimates
   
   
2. Evaluate treatment after parcelling out effect of other treatment
   
   
3. Evaluate treatment at the median or mean level of other treatment

Evaluating Treatment Effects

Parcelling Out Second Treatment

Component-Residual Plots take examine unique effect of one treatment after removing influence of the other.

Median Value of Second Treatment

Comparison of Differences at Average of Other Treatment

 contrast       estimate        SE df t.ratio p.value
 control - high     1.64 0.2894823  8   5.665  0.0012
 control - low      1.02 0.2894823  8   3.524  0.0190
 high - low        -0.62 0.2894823  8  -2.142  0.1424

Results are averaged over the levels of: block 
P value adjustment: tukey method for comparing a family of 3 estimates 

What if my design is unbalanced?

Uh oh… a cat ate my treatment!

This is a big problem

Sums of Squares are not ‘unbalanced’ - more information goes into one than the other

Suddenly, order matters…

Entering Treatment First

zooplankton ~ treatment + block

zooplankton ~ block + treatment

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

What’s Going On: Type I and II 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


Each SS is the unique contribution of a treatment
If the design is balanced, no different than type I

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
Test for B	A + B v. A	A + B v. A

Sequential SS v. Marginal SS

Type II SS to the Rescue

Type II SS to the Rescue

Compare to zooplankton ~ block + treatment:

Beyond 2-Way ANOVA

Latin Squares Design

Accomodates multiple gradients

Latin Squares

Every row and column contains one replicate of a treatment.
Can be generalized to n gradients \[y_{ijkl} = \beta_{0} + \sum \beta_{i}x_{i} + \sum \beta_{j}x_{j} + \sum \beta_{k}x_{k} +\epsilon_{ijkl}\]

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

Many Treatments