ANOVA and 3 Important Assumptions of It

Introduction

Analysis of Variance (ANOVA) and its variants are foundational techniques in inferential statistics used to compare means across groups and evaluate complex relationships between variables.

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Assumptions of Analysis of Variance

Before applying ANOVA, it is crucial to verify the underlying statistical assumptions. Violating these can compromise the validity of the results.

1. Independence of Observations

Each observation must be statistically independent of the others. This is typically ensured through proper randomization in experimental design. For instance, if one participant’s response is influenced by another’s, this assumption is violated.

Example: In a drug efficacy study, if participants are in close contact (e.g., same household), their responses might not be independent due to shared environmental factors.

2. Normality

The distribution of the dependent variable within each group should be approximately normal. This is assessed through visual tools (histograms, Q-Q plots) or statistical tests (Shapiro-Wilk, Kolmogorov-Smirnov). ANOVA is robust to mild deviations from normality, especially with large sample sizes due to the Central Limit Theorem.

Implication: Non-normal data may bias the F-statistic, increasing Type I or II errors.

3. Homogeneity of Variance (Homoscedasticity)

This assumption requires that the variances of the dependent variable are equal across groups. Levene’s test is commonly used to test this assumption. If violated, researchers may opt for Welch’s ANOVA or transform the data.

Types of Variances

Example: In comparing test scores across different schools, if one school has very high variability in scores and others do not, the assumption is not met.

One-Way ANOVA

One-Way ANOVA evaluates whether the means of three or more unrelated groups differ significantly on a single dependent variable.

Summary Table

Example Use Case

A researcher wants to compare mean blood pressure levels among patients following three different diets. Here, diet is the independent variable with three levels, and blood pressure is the dependent variable.

Post-hoc Tests

If the ANOVA is significant, post-hoc tests (e.g., Tukey’s HSD, Bonferroni) identify which groups differ.

Repeated Measures ANOVA

Repeated Measures ANOVA is used when the same subjects are measured multiple times under different conditions or over time. It accounts for the correlation between repeated observations.

Assumption: Sphericity

Sphericity refers to the equality of variances of the differences between treatment levels. Violations are common and can be corrected using Greenhouse-Geisser or Huynh-Feldt adjustments.

Example Use Case

Testing cognitive performance of students at three time points: pre-intervention, post-intervention, and follow-up.

One Way and Two Way ANOVA

Two-Way ANOVA

Two-Way ANOVA evaluates the effect of two independent variables on a single dependent variable, including their interaction.

Main and Interaction Effects

• • Main effect: Influence of each factor individually.
  • Interaction effect: Combined effect of factors; whether the effect of one variable depends on the level of another.

Example Use Case

Investigating how teaching method (traditional vs. online) and gender affect academic performance.

Interpretation: A significant interaction might suggest that one method works better for one gender.

ANCOVA and MANCOVA

Analysis of Covariance (ANCOVA)

ANCOVA combines regression and ANOVA by adjusting the dependent variable for the effect of one or more covariates—continuous variables not of primary interest but that influence the outcome.

Purpose

• • Reduce error variance
  • Control for confounding variables

Example Use Case

Studying the effect of different therapy techniques on depression scores while controlling for pre-treatment depression levels.

Multivariate Analysis of Variance (MANOVA)

MANOVA extends ANOVA to multiple dependent variables. It assesses the joint multivariate effect of the independent variable(s) on the dependent variables.

When to Use

• • When multiple dependent variables are correlated.
  • To avoid inflated Type I error due to multiple ANOVAs.

Statistical Tests

• • Wilks’ Lambda
  • Pillai’s Trace
  • Hotelling’s Trace

Example Use Case

Evaluating the impact of a training program on employee performance, satisfaction, and retention simultaneously.

Conclusion

Understanding the assumptions and appropriate applications of ANOVA and its advanced variants—Repeated Measures ANOVA, Two-Way ANOVA, ANCOVA, and MANOVA—is essential for conducting rigorous and interpretable research. Each method serves a specific purpose in addressing increasingly complex data scenarios. Violating assumptions can lead to misleading conclusions, underscoring the importance of diagnostic testing and model selection.

References

Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage Publications.

Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics (6th ed.). Pearson.

Howell, D. C. (2012). Statistical Methods for Psychology (8th ed.). Cengage Learning.

Montgomery, D. C. (2017). Design and Analysis of Experiments (9th ed.). Wiley.

Keppel, G., & Wickens, T. D. (2004). Design and Analysis: A Researcher’s Handbook (4th ed.). Pearson Education.