Experimental Design & Factorial Treatments

Experimental Design: Factorial Designs and Treatments

Introduction

• Apologies for absence due to illness.
• Focus on factorial designs and treatments this week.
• Goal: Understand experimental designs, replicates, treatment structures, blocking structures, and their implementation/interpretation in R.

Treatment Designs vs. Experimental Designs

• Treatment Designs: Selection of treatments/factors and their levels.
  • Example: Fire treatment (one factor) with levels like burnt/unburnt or unburnt/low severity/high severity.
• Experimental Designs: How treatments are allocated to experimental units and measured.
  • Includes blocking structures and replication levels (plot, animal, etc.).
• So far, we've primarily focused on one-factor designs (one-way ANOVA).

Factorial Treatment Designs

• Involve more than one factor, leading to multi-way ANOVAs.
• Efficient way to design experiments compared to multiple one-way ANOVAs.
• Example: Two-factor design with bird feeding:
  • Factors: Diet (Diet 1, Diet 2) and Sex (Male, Female).
  • Measure food consumption.
  • Replication: Three males and three females.

Hypotheses in Factorial Designs

• With two factors, you can address three questions:
  • Diet: Is there a difference in food consumption between diets, regardless of sex?
  • Sex: Are males or females eating more, regardless of diet?
  • Interaction: Does the effect of diet on food consumption depend on the sex of the bird?

Interaction Plots

• Plotting two factors together can reveal interactions.
• Example: Plotting males (blue) and females (red) for two diets.
  • If error bars diverge for one diet but not the other, it suggests a potential interaction.

Rice Example: Three Varieties and Five Nitrogen Levels

• Experiment: Growing three rice varieties under five nitrogen levels with a randomized control block design (four blocks).
• 15 plots per block (5 nitrogen levels x 3 rice varieties).
• Replication: Four blocks.
• Measure yield in tons per hectare.

Data Analysis and Interpretation

• Plotting data with both nitrogen and variety allows for interaction analysis.
• Parallel lines indicate no interaction.
• Diverging lines suggest potential interactions.

Three Potential Hypotheses

• Focus depends on research questions; may prioritize interaction or main effects.
• Interaction: Is there an interaction between nitrogen and rice variety (i.e., does the response to nitrogen differ by variety)? Interaction term is key.
• Nitrogen Main Effect: Regardless of variety, does adding more nitrogen increase yield? Only examined if the interaction is not significant.
• Variety Main Effect: Regardless of nitrogen level, do some rice varieties produce better yields? Only examined if the interaction is not significant.

Model Equation

• Model equation incorporates blocking effect, variety effect, nitrogen effect, and their interaction:
  Observed\ Data = Overall\ Mean + Blocking\ Effect + Variety\ Effect + Nitrogen\ Effect + (Variety \times Nitrogen) + Random\ Error
• In R, the treatment structure can be written as variety + nitrogen + variety:nitrogen or as the shortcut variety * nitrogen.

Effects and Effect Sizes

• Effects: Differences between means.
• Effect Size: Difference between a group's mean and the overall mean.
• Variety Effect: Mean(Variety) - Mean(Experiment)
• Nitrogen Effect: Mean(Nitrogen) - Mean(Experiment)
• Interaction Effect: Mean(Interaction) - Mean(Experiment)
• Effect sizes indicate how different a variable is from zero and are useful for communicating results (e.g., to farmers).

ANOVA Table

• Degrees of freedom:
  • Blocks: Blocks - 1
  • Varieties: Varieties - 1
  • Nitrogen: Nitrogen\ Levels - 1
  • Interaction: df(Variety) \times df(Nitrogen)
  • Residual degrees of freedom: Blocks \times (Treatment\ Combinations - 1)
• Total treatment degrees of freedom: just simply add up our main effects and our interaction term
• Degrees of freedom calculations are cross-checked for accuracy.

Hypothesis Testing

• Interaction:
  • Null Hypothesis: No interaction between variety and nitrogen on yield.
  • Alternative Hypothesis: There is an interaction.
• Main Effects (if interaction is non-significant):
  • Variety:
    • Null Hypothesis: All varieties have the same mean yield.
    • Alternative Hypothesis: At least one variety has a different mean yield.
  • Nitrogen:
    • Null Hypothesis: All nitrogen levels have the same mean yield.
    • Alternative Hypothesis: At least one nitrogen level has a different mean yield.

R Coding and ANOVA Table Interpretation

• Use the anova function in R with the model equation.
• Key Trick: Interpret ANOVA tables from the bottom up.
• Start with the highest-order interaction term.
• If the interaction is significant, do not interpret main effects in isolation.
• If the interaction is not significant, proceed to examine main effects.
• Post-hoc tests are used to identify differences between levels within a significant main effect.

Interaction Plots: Visualizing Interactions

• Different scenarios:
  • No significant effects: lines overlap.
  • Main effect A significant: A1 is different from A2, lines for B are on top of each other.
  • Main effect B significant: lines separate from each other.
  • Both main effects, no interaction: lines are separated by the b variables and that slope of the line is pretty much the same between those two lines. They're kind of just tracking each other. They're in parallel with each other.
  • Significant interaction: Lines are not parallel, slopes differ, suggesting interaction.

Interaction Plots in R

• Use eMeans package for better interaction plots.
• interaction.plot function in base R provides an alternative.

Three-Factor (Three-Way) Designs

• Example: Manipulating Nitrogen, Phosphorus, and Potassium (NPK) fertilizer.
• Three main effects (one for each fertilizer).
• Three two-way interactions (NP, NK, PK).
• One three-way interaction (NPK).
• Degrees of freedom are calculated similarly to two-way designs.

Maize Example

• Factors: Fertilizer (four levels), Weed (present/absent), Maize Variety (A/B).
• Blocking design with two blocks.
• Replication: Two (number of blocks).
• Model equation:
  Observed\ Data = Overall\ Mean + Block + N + P + K + (N \times P) + (N \times K) + (P \times K) + (N \times P \times K) + Error
• R code: yield ~ block + fertilizer * weed * variety.

ANOVA Table Interpretation for Three-Factor Design

• Start from the bottom (three-way interaction).
• If the three, the variety of maize, whether it's infected with witchweed or not, and the different levels of fertilizer aren't interacting to influence our yield proceed to two-way interactions.
• Significant two-way interactions: Delve into post-hoc comparisons.

Graphing Three-Way Interactions

• Use eMeans to visualize three-way interactions.
• If three-way interaction is not significant, assess two-way interactions through plots and post-hoc tests.