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Two-way ANOVAs and interactions Randomized block designs Fixed, random, mixed-model ANOVAs

ANOVA lecture. Two-way ANOVAs and interactions Randomized block designs Fixed, random, mixed-model ANOVAs Factorial vs. nested designs Formal design notation Split-plot designs Repeated measures. Two-way ANOVA. Just like one-way ANOVA, except subdivides the treatment SS into:

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Two-way ANOVAs and interactions Randomized block designs Fixed, random, mixed-model ANOVAs

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  1. ANOVA lecture • Two-way ANOVAs and interactions • Randomized block designs • Fixed, random, mixed-model ANOVAs • Factorial vs. nested designs • Formal design notation • Split-plot designs • Repeated measures

  2. Two-way ANOVA Just like one-way ANOVA, except subdivides the treatment SS into: • Treatment 1 • Treatment 2 • Interaction 1&2

  3. Two-way ANOVA Suppose we wanted to know if moss grows thicker on north or south side of trees, and we look at 10 aspen and 10 fir trees: • Aspect (2 levels, so 1 df) • Tree species (2 levels, so 1 df) • Aspect x species interaction (1df x 1df = 1df) • Error? k(n-1) = 4 (10-1) = 36

  4. Additive effect: Alder Fir South North Interactions Combination of treatments gives non-additive effect

  5. Interactions Combination of treatments gives non-additive effect Anything not parallel! South South North North

  6. Group exercise:Is this still an additive effect? Log (y) South North

  7. Careful! If you log-transformed your variables, the absence of interaction is a multiplicative effect: log (a) + log (b) = log (ab) y Log (y) South South North North

  8. Ecological rule #1: the world is not uniform! Medium patch Poor patch Good patch

  9. 3 options in assigning treatments: • Randomly assign • Systematic • Randomized block Poor patch Medium patch Good patch

  10. 1. Randomly assign Poor patch Medium patch Good patch Statistically robust Pros? Cons? With small n, chance of all in a bad patch

  11. 2. Systematic Poor patch Medium patch Good patch No clumping possible Pros? Cons? Violates random assumption of statistics…but is this so bad?

  12. 3. Randomized block Poor patch Medium patch Good patch BLOCK B BLOCK C BLOCK A

  13. Pro: Can remove between-block variance from error variance…may increase power of test Error Error Block Treatment Treatment

  14. Con: Blocks use up degrees of freedom Error Error Block Treatment Treatment

  15. 3. Randomized block BLOCK B BLOCK C BLOCK A • Note: • Do not have to know if patches differ in quality • Must have all treatment combinations represented in each block

  16. Analysis of blocked design? Input “block” just like another factor. A specific example: Treatment 1 2 2 Treat Block Treat x block Block 1

  17. Group exercise: In groups of 3, find 3 different scales of blocks in your transect lab. Which models have replication?

  18. Factor – a variable of interest e.g. temperature Level – a particular value / state of a factor e.g. hot, cold In this example, temperature is a factor with two levels.

  19. Fixed factor Either (1) The investigator chooses the levels of the factor for some purpose. Eg. Ambient CO2 vs. double CO2 OR (2) The levels used represent all possible levels. Eg. Biological sex: Male, female

  20. Random factor The levels of the factor are chosen randomly from a universe of possible levels. Eg. We want to look at whether butterfly collectors differ in their diversity estimates for 4 plots. We select 5 collectors “randomly”. Eg. We use three breeding lines of fruit flies as blocks in a genetics experiment. Blocks are typically random effects!

  21. Formal notation Af6 is a fixed factor called A with 6 levels Br5 is a random factor called B with 5 levels

  22. Fixed-effects ANOVA (Model I) • All factors are fixed • Random-effects ANOVA (Model II) • All factors are random • Mixed-model ANOVA (Model III) • Contains both fixed and random effects, e.g. randomized block!

  23. Two-way factorial ANOVA How to calculate “F” Random effect (factors A & B random) Mixed model (A fixed, B random) Fixed effect (factors A & B fixed) Factor A MS A Error MS MS A MS A x B MS A MS A x B Factor B MS B MS A x B MS B Error MS MS B Error MS A x B MS A x B Error MS MS A x B Error MS MS A x B Error MS

  24. Factorial design: All levels of one factor crossed by all levels of another factor, ie. all possible combinations are represented. If you can fill in a table, it’s factorial! Double CO2 Ambient CO2 Pea plant Bean plant Corn plant

  25. Formal notation Af6 x Br5 tells us that this is a factorial design with factor A “crossed” with factor B

  26. Group exercise (new groups of 3) Experimental design handout

  27. ANOVA Example: formal notation Example 1 Ecologists: Er10 Papers: Pf2 Example 2: Populations: Ur2 Herbivory: Hf2 [Plants: Pr5] Example 3: Light: Lf3 Nutrients: Nf3 Blocks: Br3 [Replicates: Rr2]

  28. No fertilizer Nitrogen fertilizer Phosphorus fertilizer Strain A Strain B Strain C Strain D Strain E Strain F Nested design In this example, strain type is “nested within” fertilizer. Fertilizer is often called “group”, strain “subgroup” The nested factor is always random

  29. Grand mean Variance: Group No fertilizer Nitrogen fertilizer Phosphorus fertilizer Strain A Strain B Strain C Strain D Strain E Strain F

  30. Grand mean Variance: Group No fertilizer Nitrogen fertilizer Phosphorus fertilizer Variance: Subgroup within a group Strain A Strain B Strain C Strain D Strain E Strain F

  31. Grand mean Variance: Group No fertilizer Nitrogen fertilizer Phosphorus fertilizer Variance: Subgroup within a group Strain A Strain B Strain C Strain D Strain E Strain F Variance: Among all subgroups

  32. Nested ANOVA: “A” Subgroups nested within “B” Groups, with n replicates In our example, A=2, B=3 and n=2 df F Groups MS Groups MS Subgroups within groups B-1 Subgroups within groups B(A-1) MS Subgroups within groups MS Among all subgroups Among all subgroups AB(n-1) Total ABn-1

  33. Split plot design An experiment replicated within an experiment! 4 Main plots, e.g. greenhouses Elevated CO2 Ambient CO2

  34. Main plot CO2 MS maintreat F Main plot error MS mainerror Split plot design An experiment replicated within an experiment!

  35. Split plot design An experiment replicated within an experiment! 4 Main plots, e.g. greenhouses Elevated CO2 Ambient CO2

  36. Split plot design An experiment replicated within an experiment! 3 5 6 2 4 1 6 3 2 5 1 4 1 5 6 3 2 4 5 3 6 4 2 1 Subplots with six different nutrient concentrations

  37. Split plot design An experiment replicated within an experiment! Subplot nutrient MS subtreat F nutrient x CO2MS subinteract F Subplot error MS suberror

  38. Split plot design An experiment replicated within an experiment! Main plot CO2 MS maintreat F Main plot error MS mainerror Subplot nutrient MS subtreat F nutrient x CO2MS subinteract F Subplot error MS suberror

  39. Repeated measures ANOVA Multiple observations on same individual. ANOVA analysis possible (treat individuals as blocks) but requires certain assumptions of independence! Several more robust alternatives (before-after differences, MANOVA)

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