Analysis of Variance (ANOVA) - Comparing More Than Two Means

Introduction

• Focus on analysis comparing more than two means.

• Reminder about available resources (online quizzes, Q&A sessions, and email consultations).

• Transition from t-test (for two groups) to F-test for more than two groups.

Importance of ANOVA

• Comparison Context: e.g., Comparing lifespan of patients from four hospitals.

• Multiple comparisons (6 pairs) increase complexity and potential error in significance levels.

  • Type I Error: Probability of incorrectly rejecting the null hypothesis.

  • 6 pair tests yield an overall Type I error of 0.265 versus the desired 0.05 significance level.

What is ANOVA?

• Definition: One-Way Analysis of Variance (ANOVA) used when comparing means of multiple groups.

• Similarities to T-Test: Same general assumptions for normality and equal variances, with consideration for independence of observations.

Hypotheses in ANOVA

• Null Hypothesis (H0): All group population means are the same.

• Alternative Hypothesis (H1): At least one group population mean differs from the others.

Components of Variation in ANOVA

• Two main components of variation:

  • Between-group Variation: Difference due to groups.

  • Within-group Variation: Random or unexplained variation within groups.

• Example illustrated with severe vs mild disease groups highlights domination of differences or masking effects of variation.

Example of Therapy Impact on Anxiety Levels

• Study Design: Three groups of students (A, B, C) receive different hours of therapy (5, 10, 15 hours) respectively.

• Analysis includes assessment of anxiety post-therapy; balanced data from equal groups.

• Box Plot Representation: Helps visualize differences; group B higher than C and A.

Conducting ANOVA (Calculation Overview)

• Total Variation: Variance calculation similar to previous modules, using an overall mean for all observations (150 students).

• Partitioning Total Variation:

  • Between-group Sum of Squares (SSB)

  • Within-group Sum of Squares (SSW)

• Degrees of Freedom: Different for each sum of squares component (calculated depending on group sizes).

• Conversion to Mean Squares:

  • Total mean square = Total SS / (n-1)

  • Between Mean Square = SSB / (k-1)

  • Within Mean Square = SSW / (n-k)

Statistical Ratio in ANOVA

• F-statistic: Ratio of between-group variation to within-group variation (F = SSB/MSW).

• Indicates the strength of the difference between groups; larger values suggest significant group differences.

• Distribution: F-statistic follows an F-distribution, dependent on degrees of freedom from SSB and SSW.

Results Interpretation

• Example yields F-statistic (e.g., 18.