ANOVA Analysis Guide
Compare means across multiple groups simultaneously using Analysis of Variance and understand when group differences are statistically significant.
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Introduction
ANOVA (Analysis of Variance) is a statistical technique for comparing means across three or more groups. Despite its name focusing on "variance," ANOVA actually tests whether group means are significantly different.
While a t-test compares two means, ANOVA extends this to multiple groups without inflating the Type I error rate that would occur from multiple t-tests.
When to Use ANOVA
ANOVA is appropriate when:
- • Comparing means of 3+ groups
- • Independent variable is categorical
- • Dependent variable is continuous
- • Groups are independent
Example Scenarios
- • Comparing test scores across 4 teaching methods
- • Testing drug effectiveness across 3 dosage levels
- • Comparing sales across 5 different regions
- • Analyzing crop yields with different fertilizers
How ANOVA Works
ANOVA partitions the total variance in the data into two components:
Between-Group Variance (SSB)
Variation due to differences between group means. This is what we want to be large.
SSB = Σnᵢ(x̄ᵢ - x̄)²
Sum of squared deviations of group means from grand mean
Within-Group Variance (SSW)
Variation due to individual differences within each group. This represents "noise."
SSW = ΣΣ(xᵢⱼ - x̄ᵢ)²
Sum of squared deviations within each group
The Logic
If group means are truly different, between-group variance should be much larger than within-group variance. ANOVA tests this ratio using the F-statistic.
The F-Statistic
F-Statistic Formula
MSB (Mean Square Between)
SSB / (k - 1), where k = number of groups
MSW (Mean Square Within)
SSW / (N - k), where N = total sample size
Interpreting F
- • F ≈ 1: Group means are similar (null hypothesis likely true)
- • F > 1: Group means differ more than expected by chance
- • Larger F values provide stronger evidence against H₀
- • Compare F to critical value or calculate p-value
ANOVA Assumptions
1. Independence
Observations are independent within and between groups. Each measurement should not influence others.
2. Normality
Data in each group should be approximately normally distributed. ANOVA is robust to minor violations with large samples.
3. Homogeneity of Variances
Variances should be roughly equal across groups. Test with Levene's test. Use Welch's ANOVA if violated.
Post-Hoc Tests
ANOVA tells you if groups differ, but not which groups differ. Post-hoc tests identify specific pairwise differences.
Tukey's HSD
Most common choice. Controls family-wise error rate. Good when comparing all pairs.
Bonferroni
Conservative approach. Divides α by number of comparisons. Good for few specific comparisons.
Scheffé
Most conservative. Good for complex comparisons beyond simple pairs.
Games-Howell
Use when variances are unequal. Does not assume homogeneity of variance.
Types of ANOVA
One-Way ANOVA
One independent variable with 3+ levels.
Example: Comparing test scores across 4 different schools.
Two-Way ANOVA
Two independent variables. Tests main effects and interaction.
Example: Effects of both teaching method AND gender on test scores.
Repeated Measures ANOVA
Same subjects measured multiple times. Accounts for within-subject correlation.
Example: Measuring anxiety before, during, and after treatment.
Summary
Key Takeaways
- 1.ANOVA compares means across 3+ groups using variance ratios.
- 2.F = MSB/MSW tests if between-group variance exceeds within-group variance.
- 3.Key assumptions: independence, normality, equal variances.
- 4.Significant ANOVA requires post-hoc tests to identify which groups differ.
- 5.Choose the right ANOVA type based on your research design.