WEEK 10 - One Way ANOVA

In this week, we will explore the principles of One Way ANOVA, focusing on its application in comparing means across multiple groups. We will cover the assumptions of the test, how to interpret results, and practical examples to illustrate its use in real-world scenarios. Additionally, we will discuss the importance of post-hoc tests to determine which specific group means are significantly different from one another.

Page 2: Introduction to ANOVA

• Purpose of ANOVA:

  • Understand why ANOVA is used in statistics.

• Content Outline:

  • Why ANOVA?

  • Multiple Testing

  • Basic Principles of ANOVA

  • Types of ANOVA

  • F-ratio

  • Post-hoc Tests

Page 3: Learning Outcomes

• By the end of the lecture, students should be able to:

  • Explain multiple testing.

  • Define and differentiate types of ANOVA.

  • Describe F-ratio.

  • Describe different types of post-hoc tests and appropriate applications.

Page 4: Limitations of T-tests

• T-tests Comparison:

  • T-tests compare means but are limited to comparing only 2 means.

• Why T-tests can only use one independent variable?

  • Discuss issues with using multiple t-tests for analysis.

  • Impact of multiple testing on results.

Page 5: Type I and II Errors

• Statistical Sampling:

  • Each statistical test involves drawing a sample from a population.

• Type I Error (α):

  • Incorrectly rejecting the null hypothesis (H0).

• Type II Error (β):

  • Failing to detect an effect that the alternative hypothesis (HA) proposes.

Page 6: Error Overview

• Study Finding Summary:

  • Effect Present vs. Absent.

• Decision Tree:

  • Outcomes leading to Type I and II errors in hypothesis testing.

Page 7: Effect Size

• Definition of Effect Size (d):

  • Relationship between sample means (µ1 and µ2).

• Implications:

  • Larger effect size increases statistical power and reduces Type II error probability.

Page 8: Error in Significance Testing

• Type I Error Example:

  • False alarm scenario leading to unnecessary actions.

• Type II Error Example:

  • Failure to detect a true condition leading to harmful consequences.

Page 9: Definition of Power

• Statistical Power (1 - β):

  • Ability to detect a true relationship or difference.

  • Probability of rejecting the null hypothesis when it is indeed false.

Page 10: The Concept of Power

• Power Calculation:

  • Defined as P(rejecting H0 | H0 is false).

• Relationship with Type II Error:

  • Type II Error probability is denoted as β, and its complement indicates test power.

Page 11-12: Null and Alternative Hypotheses

• H0: The null hypothesis (µ=µ1).

• HA: The alternative hypothesis (µ≠µ1).

• Importance of identifying true vs. false states in applying these hypotheses.

Page 13: Understanding Effect Size (d)

• Definition:

  • Effect size reflects the magnitude of difference between null hypothesis and alternative hypothesis.

• Application:

  • Useful in quantifying relationships in studies.

Page 14-15: Interpreting Effect Sizes

• Categorization of Effect Sizes:

  • Effect sizes can be trivial, small, medium, or large.

• Example:

  • Comparison between different age groups to illustrate effect size differences.

Page 16: Values of Effect Size & Interpretation

• Table Overview:

  • Different measures for effect size (Cohen's d, Rank-biserial, Eta squared).

  • Interpretation of trivial to large values in different statistical contexts.

Page 17: Implications of Effect Size

• Larger effect sizes are associated with:

  • Greater statistical power.

  • Reduced likelihood of Type II errors.

Page 18: Consequences of Errors

• Type I Error:

  • Physical and psychological costs associated with mistaken conclusions.

• Type II Error:

  • Consequences due to not recognizing significant conditions or changes.

Page 19-20: Multiple Testing Issues

• Definition:

  • Increased Type I error occurrence from repeated statistical tests.

• Consequences of Multiple t-tests:

  • Cumulative Type I error and overall familywise error rate.

Page 21-22: Need for ANOVA in Multiple Testing

• Need for single comprehensive tests:

  • ANOVA effectively evaluates group differences without increasing Type I errors.

Page 23: Types of ANOVA

• Overview of ANOVA Types:

  • One-way independent ANOVA.

  • Two-way independent ANOVA (considering two independent variables).

  • One-way repeated measures ANOVA.

Page 24: Basic Principles of ANOVA

• Testing Group Means:

  • Null hypothesis states all means are equal.

  • Possible outcomes upon rejection of null hypothesis.

Page 25: Variability in ANOVA

• Total Variability Calculation:

  • Total sum of squares divided into between-groups and within-groups variability.

• Focus on F-ratio for analysis.

Page 26: Understanding F-ratio

• F-ratio Calculation:

  • Formula: between-groups mean square variance / within-groups mean square variance.

• Significance of F-ratio:

  • Indicates the effectiveness of the independent variable on the dependent variable.

Page 27-36: Working Example

• Example Study:

  • Testing synthetic gibberellins on plant growth across three concentrations.

  • Step-by-step calculations of heights and means to reach conclusion through ANOVA.

Page 37-39: Post-hoc Tests

• Purpose of Post-hoc Tests:

  • Determine which specific group means differ.

• Types of Post-hoc Tests:

  • Multiple comparison procedures (e.g., Tukey HSD).

  • Homogenous subset tests.

Page 40-42: Using JASP for Post-hoc Analysis

• Overview of JASP features:

  • Various post-hoc tests and corrections to maintain control over Type I error.

• Guidelines for selecting post-hoc tests based on data composition.

Page 43: Conclusion

• UPM Contact Information:

  • Thank you message and institutional branding as Universiti Putra Malaysia.