There are five possible outcomes for a factorial experiment with two EVs \(X_1, X_2\).
2024-02-01
Outcome Scenarios
and Interaction Plots
1. Neither \(X_1\) nor \(X_2\) affect \(Y\)
lm(Y ~ block + X1 + X2 + X1:X2)
- All ANOVA terms non-significant
- Most concise description of data: grand mean
Y ~ block
“Yield is unaffected by sow rate or variety.”
2. \(X_2\) affects \(Y\), but \(X_1\) does not
lm(Y ~ block + X1 + X2 + X1:X2)
- ANOVA: only \(X_2\) significant
- Most concise description of data:
Y ~ block + X2
“Yield is affected by variety, but not by sow rate”.
3. \(X_1\) affects \(Y\), but \(X_2\) does not
lm(Y ~ block + X1 + X2 + X1:X2)
- ANOVA: only \(X_1\) significant
- Most concise description of data:
Y ~ block + X1
“Yield is affected by sow rate, but not by variety”.
4. \(X_1\) and \(X_2\) affect \(Y\) additively
lm(Y ~ block + X1 + X2 + X1:X2)
- ANOVA: \(X_1\) and \(X_2\) significant,
but interaction termnot . - Most concise description of data:
Y ~ block + X1 + X2
“Yield is independently affected by sow rate and by variety; the effects are additive”.
5. \(X_1\) and \(X_2\) affect \(Y\) non-additively
lm(Y ~ block + X1 + X2 + X1:X2)
- ANOVA:
interaction term significant . - Regardless of significance of main effects!!
- Most concise description of data: as in model eqn.
“Yield is affected by sow rate and by variety; the effect of sow rate depends on variety and vice versa”.
Order of Interpretation
lm(Y ~ block + X1 + X2 + X1:X2)
Always start with with the interaction: from bottom of ANOVA table up!
- No significant interaction, but two significant main effects: two simple stories
- Significant interaction: one more complicated story.
If you have a significant interaction, don’t attempt to interpret the significance of the main effects in the ANOVA:
The significant interaction tells you that