Performing a Type-III sum of squares (SS) ANOVA in R with contrast codes is crucial for analyzing data, especially when dealing with unbalanced designs or specific hypotheses about factor levels. This guide will walk through what Type-III SS ANOVA is, how to prepare data, different contrast codes, using the car package for Type-III SS ANOVA, interpreting the results, and performing post-hoc tests.
What is Type-III Sum of Squares?
Type-III sum of squares tests sums of squares for each main effect and interaction in a model after accounting for other effects. This approach is advantageous in unbalanced designs (when groups have different sizes) and when interested in specific hypotheses about factor levels.
Now we will discuss step-by-step How One Do a Type-III SS ANOVA in R Programming Language.
Step 1: Preparing the Data for ANOVA in R
- First, load the necessary packages.
- Then create a sample dataset.
# Load required packages
library(car)
library(tidyverse)
# Create sample data
set.seed(123)
data <- expand.grid(A = factor(1:3), B = factor(1:2), rep = 1:10)
data$response <- rnorm(nrow(data), mean = rep(1:6, each = 10), sd = 0.5)
Step 2: Types of Contrast Codes
Contrast codes specify how factor levels are compared:
- Treatment (default in R): Compares each level to a reference level.
- Helmert: Compares each level to the mean of subsequent levels.
- Sum (effects): Compares each level to the overall mean.
# Define contrast codes
contrasts(data$A) <- contr.helmert(3)
contrasts(data$B) <- contr.helmert(2)
Step 3:Performing Type-III SS ANOVA
Fit a linear model and perform the ANOVA.
# Fit linear model
model <- lm(response ~ A * B, data = data)
# Conduct Type-III SS ANOVA
anova_results <- Anova(model, type = "III")
print(anova_results)
Output:
Anova Table (Type III tests)
Response: response
Sum Sq Df F value Pr(>F)
(Intercept) 748.84 1 213.5371 <2e-16 ***
A 0.58 2 0.0825 0.9209
B 1.36 1 0.3878 0.5361
A:B 0.07 2 0.0093 0.9908
Residuals 189.37 54
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
- Intercept: The intercept is highly significant, with a very low p-value (<2e-16), meaning the overall mean of the response variable is significantly different from zero.
- Factor A: The sum of squares (0.58) and the F-value (0.0825) for factor A suggest that A does not have a significant effect on the response variable, as indicated by the high p-value (0.9209).
- Factor B: Similarly, factor B has a sum of squares of 1.36 and an F-value of 0.3878, with a high p-value (0.5361), indicating it is not significant.
- Interaction (A): Interaction between A and B has a sum of squares of 0.07, an F-value of 0.0093, and a very high p-value (0.9908), indicating no significant interaction effect.
- Residuals: Residuals sum of squares (189.37) reflects the unexplained variance in the response variable, with 54 degrees of freedom.
Step 4: Performing Post-Hoc Tests
After ANOVA, conduct post-hoc tests to explore significant differences between factor levels.
# Post-hoc tests using Tukey's HSD
library(multcomp)
posthoc <- glht(model, linfct = mcp(A = "Tukey"))
summary(posthoc)
Output:
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
Fit: lm(formula = response ~ A * B, data = data)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
2 - 1 == 0 0.03171 0.59219 0.054 0.998
3 - 1 == 0 0.22237 0.59219 0.376 0.925
3 - 2 == 0 0.19066 0.59219 0.322 0.945
(Adjusted p values reported -- single-step method)
- Indicates that p-values have been adjusted for multiple comparisons using the single-step method. This adjustment helps control the family-wise error rate.
- None of the pairwise comparisons among the levels of factor A are statistically significant, as indicated by the high p-values (all > 0.05).
- The differences between the means of the levels of factor A are small and not significantly different from zero.
- It suggests that there are no significant differences between the levels of factor A in this dataset.
Conclusion
Doing a Type-III sum of squares ANOVA in R with contrast codes helps analyze data when groups are uneven or when testing specific ideas about factor levels while considering other variables. In this case, the factors and their interactions were not significant, but the method works well for many types of data. Using these steps allows for a better understanding and accurate analysis of complex data.
