Many Treatments: Multi-Way ANOVA



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Effects of Stickleback Density on Zooplankton



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Where Would You Place Replicates?

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Randomization

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Costs? Benefits?

Blocked Design

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Incorporates Gradient
n=1 per block

Randomized Controlled Blocked Design

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Randomization within blocks

Effects of Stickleback Density on Zooplankton



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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 \(\lambda\alpha_i\beta_j\)

Tukey’s Test of Non-additivity:

           Test stat Pr(>|Test stat|)
treatment                            
block                                
Tukey test    0.4742           0.6354

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

term df sumsq meansq statistic p.value
treatment 2 6.857 3.429 16.366 0.001
block 4 2.340 0.585 2.792 0.101
Residuals 8 1.676 0.209 NA NA

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

term estimate std.error statistic p.value
treatmentcontrol 3.42 0.3126766 10.9378182 0.0000043
treatmenthigh 1.78 0.3126766 5.6927826 0.0004582
treatmentlow 2.40 0.3126766 7.6756619 0.0000588
block2 0.00 0.3737200 0.0000000 1.0000000
block3 -0.70 0.3737200 -1.8730599 0.0979452
block4 -1.00 0.3737200 -2.6757998 0.0281084
block5 -0.30 0.3737200 -0.8027399 0.4453163

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 0.289 8 5.665 0.001
control - low 1.02 0.289 8 3.524 0.019
high - low -0.62 0.289 8 -2.142 0.142

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
Df Sum Sq Mean Sq F value Pr(>F)
treatment 2 4.171 2.086 18.046 0.002
block 4 1.749 0.437 3.783 0.060
Residuals 7 0.809 0.116 NA NA
zooplankton ~ block + treatment
Df Sum Sq Mean Sq F value Pr(>F)
block 4 1.878 0.469 4.062 0.052
treatment 2 4.043 2.021 17.490 0.002
Residuals 7 0.809 0.116 NA NA

What’s Going On: Type I and II 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


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

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

Sum Sq Df F value Pr(>F)
treatment 4.043 2 17.490 0.002
block 1.749 4 3.783 0.060
Residuals 0.809 7 NA NA

Compare to zooplankton ~ treatment + block:

Df Sum Sq Mean Sq F value Pr(>F)
treatment 2 4.171 2.086 18.046 0.002
block 4 1.749 0.437 3.783 0.060
Residuals 7 0.809 0.116 NA NA

Type II SS to the Rescue

Sum Sq Df F value Pr(>F)
treatment 4.043 2 17.490 0.002
block 1.749 4 3.783 0.060
Residuals 0.809 7 NA NA

Compare to zooplankton ~ block + treatment:

Df Sum Sq Mean Sq F value Pr(>F)
block 4 1.878 0.469 4.062 0.052
treatment 2 4.043 2.021 17.490 0.002
Residuals 7 0.809 0.116 NA NA

Non-Least Squares Approaches

Variance Paritioning Gets More Interesting!

Beyond 2-Way ANOVA

Latin Squares Design

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Accomodates multiple gradients

Latin Squares

Col 1 Col 2 Col 3 Col 4
Row 1 A B C D
Row 2 B C D A
Row 3 C D A B
Row 4 D A B C



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

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