Hypotheses of Multi-way ANOVA

In the context of a two-way ANOVA, three pairs of hypotheses are formulated:

  1. Main Effect of Factor A (e.g., Temperature)
    • Null Hypothesis (\(H_{0A}\)): The means of the dependent variable are equal across the levels of factor A.
    • Alternative Hypothesis (\(H_{A1}\)): At least one level of factor A has a different mean.
  2. Main Effect of Factor B (e.g., Nutrient Level)
    • Null Hypothesis (\(H_{0B}\)): The means of the dependent variable are equal across the levels of factor B.
    • Alternative Hypothesis (\(H_{A2}\)): At least one level of factor B has a different mean.
  3. Interaction Effect Between Factor A and B
    • Null Hypothesis (\(H_{0AB}\)): There is no interaction between factors A and B.
    • Alternative Hypothesis (\(H_{A3}\)): There is an interaction between factors A and B.

Applying Hypotheses to our Simulated Data

Let’s illustrate these hypotheses with our simulated growth_data_2 (from Section 62).

# Fitting a two-way ANOVA model (using growth_data_2 from Section 62)
anova_model_2 <-
  aov(Growth ~ Temperature * Nutrient, data = growth_data_2)

# Viewing the summary
summary(anova_model_2)
                     Df Sum Sq Mean Sq F value   Pr(>F)    
Temperature           1   6054    6054   97.84 2.54e-16 ***
Nutrient              1     88      88    1.43    0.235    
Temperature:Nutrient  1   6054    6054   97.84 2.54e-16 ***
Residuals            96   5940      62                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Let’s interpret these specific ANOVA results:

  1. Temperature: The p-value for Temperature is 2.54e-16, which is less than the standard significance level of 0.05. This suggests that we reject the null hypothesis \(H_{0A}\) and conclude that there is a significant main effect of Temperature on Growth.

  2. Nutrient: The p-value for Nutrient is 0.235, which is greater than 0.05. This indicates that we fail to reject the null hypothesis \(H_{0B}\) and conclude that there is no significant main effect of Nutrient on Growth.

  3. Temperature:Nutrient: The p-value for the interaction term is 2.54e-16, which is less than 0.05. This suggests that we reject the null hypothesis \(H_{0AB}\) and conclude that there is a significant interaction effect between Temperature and Nutrient on Growth.

Note: Interestingly, the Sum Sq for Temperature and the interaction term (Temperature:Nutrient) are the same (6054). This suggests that all the variability explained by Temperature is due to its interaction with Nutrient. In other words, the effect of Temperature on Growth is entirely dependent on the level of Nutrient. This is probably due to the way we simulated the data.

ANOVA Table Interpretation

Let’s break down each component of the ANOVA table and discuss how to interpret them:

Degrees of Freedom (Df)

  • For a main effect, Df is the number of levels of the factor minus 1.

  • For an interaction effect, Df is the product of the Dfs of the involved factors.

Sum of Squares (Sum Sq)

  • For a main effect, Sum Sq represents the total variability in the response variable accounted for by different levels of the factor.

  • For an interaction effect, Sum Sq represents the variability in the response variable due to the different combinations of levels of the involved factors, beyond their main effects.

Mean Squares (Mean Sq)

  • For a main effect, Mean Sq is the Sum Sq divided by its respective Df, representing the average variability in the response variable caused by different levels of the factor.

  • For an interaction effect, Mean Sq is computed similarly (Sum Sq/Df), but represents the average variability due to different combinations of levels of the involved factors, beyond their main effects.

Note: For interaction terms, the Mean Sq tells us about the additional variance explained by considering the interaction between factors, beyond the variance explained by their main effects. If the interaction is significant, it suggests that the effect of one factor changes across the levels of the other factor.

F value

  • For both main and interaction effects, the F value is calculated as the Mean Sq of the factor (or interaction term) divided by the Mean Sq of the Residuals. It represents the ratio of variability explained by the factor to the unexplained variability.

P-value (Pr(>F))

  • For both main and interaction effects, a small p-value (typically ≤ 0.05) indicates that the factor (or interaction) has a significant effect on the response variable.

Residuals

  • Residuals reflect the difference between the observed values and those predicted by the model. The residuals’ Df is total Df minus the sum of Dfs for all factors and interactions, and its Sum Sq and Mean Sq represent the unexplained variability in the response.

In the context of our specific results:

  • The significant Temperature x Nutrient interaction (low p-value and high F value) implies that the effect of Temperature on Growth is not constant but varies depending on the Nutrient level.

  • The identical Sum Sq for Temperature and the interaction suggests that all the variability explained by Temperature is due to its interaction with Nutrient.

  • The Mean Sq for the interaction (6054) is much higher than the Mean Sq for the residuals (62), indicating that the interaction accounts for a large portion of the total variability in Growth.

Linking Hypotheses to Interpretation

  • Significant Temperature effect: influences Growth (reject \(H_{0A}\)).

  • No significant Nutrient effect: insufficient evidence of its influence on Growth (fail to reject \(H_{0B}\)).

  • Significant Interaction: Temperature and Nutrient jointly affect Growth (reject \(H_{0AB}\)).