Last modified: February 05, 2025

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ANOVA and the Analysis of Variance for Multiple Groups

Scenario: Peer Assessment and Student Learning

Does peer assessment enhance student learning?

Formulating the Hypotheses

To statistically assess the impact of the different treatments, we set up the following hypotheses:

I. Null Hypothesis (H0): All group means are equal. That is, there is no significant difference in final exam scores between the three treatment groups.

H0:μ1=μ2=μ3

II. Alternative Hypothesis (HA): At least one group mean is different, indicating that the treatments have different effects on student learning.

HA:At least one μj differs

Comparing Two Groups: The Two-Sample t-Test

When comparing only two groups, the two-sample t-test is appropriate. It tests whether there is a significant difference between the means of two independent groups.

I. t-Test Formula

The t-statistic is calculated as:

t=y1y2SEy1y2

where:

II. Standard Error Calculation

Assuming equal variances:

SEy¯1y¯2=s2n1+s2n2

where:

III. Limitations

ANOVA: Analysis of Variance for Multiple Groups

To compare three or more groups, we use Analysis of Variance (ANOVA). ANOVA tests for significant differences among group means by analyzing variances.

Key Concepts:

ANOVA Hypotheses:

The ANOVA Methodology

ANOVA partitions the total variability in the data into components attributable to various sources.

Total Sum of Squares (SSTotal)

The Total Sum of Squares measures the total variability in the data:

SSTotal=i=1N(yiy¯)2

where:

1. Treatment Variance (Between-Groups Variability)

Sum of Squares for Treatments (SST) measures the variability between the group means and the overall mean:

SST=j=1knj(y¯jy¯)2

where:

Treatment Mean Square (MST) is the average treatment variability:

MST=SSTk1

Degrees of freedom for treatments: dfTreatment=k1.

2. Error Variance (Within-Groups Variability)

Sum of Squares for Error (SSE) measures the variability within the groups:

SSE=j=1ki=1nj(yijy¯j)2

where:

Error Mean Square (MSE) is the average within-group variability:

MSE=SSENk

where:

3. Calculating the F-Statistic

The F-statistic compares the between-groups variance to the within-groups variance:

F=MSTMSE

where:

F-Distribution

Example: Peer Assessment Data

Data Summary

Suppose we have the following data:

ANOVA Table Construction

ANOVA Table summarizes the calculations:

Source df Sum of Squares (SS) Mean Square (MS) F
Treatment k1 SST MST=SSTk1 F=MSTMSE
Error Nk SSE MSE=SSENk
Total N1 SSTotal

Given Example:

Source df Sum of Squares Mean Square F
Treatment 2 98.4 49.2 2.57
Error 38 723.8 19.05
Total 40 822.2

Calculations

SSTotal:

SSTotal=SST+SSE=98.4+723.8=822.2

Degrees of Freedom:

Mean Squares:

F-Statistic:

F=MSTMSE=49.219.052.58

P-Value:

Decision

Interpretation of ANOVA Results

Understanding the F-Statistic

P-Value Interpretation

Conclusion for the Example

The One-Way ANOVA Model

Statistical Model

The one-way ANOVA model expresses each observation as:

yij=μj+ϵij

where:

Alternative Representation

We can express the model in terms of the overall mean μ and group effects τj:

yij=μ+τj+ϵij

where:

Assumptions of the Model

  1. Normality: The residuals ϵij are normally distributed.
  2. Independence: Observations are independent.
  3. Homogeneity of Variance: The variances within each group are equal (σ2 is constant across groups).

Assumptions of the F-Test

For the results of the ANOVA F-test to be valid, certain assumptions must be met:

1. Equal Variances (Homogeneity of Variance)

2. Independence of Observations

3. Normality of Residuals

Post-ANOVA Testing

If the ANOVA F-test leads us to reject H0, indicating that not all group means are equal, we may want to identify which groups differ.

Performing multiple t-tests increases the risk of Type I error (false positives). To control for this, we use:

1. Bonferroni Correction

Adjusts the significance level:

α=αNumber of Comparisons

Example: For 3 groups, there are 3×22=3 comparisons.

2. Tukey's Honest Significant Difference (HSD) Test

Steps for Pairwise Comparisons

I. Calculate the Standard Error for Differences:

SEy¯iy¯j=MSE(1ni+1nj)

II. Compute the t-Statistic for Each Pair:

t=yiyjSEyiyj

III. Determine the Adjusted p-Values:

Use the adjusted significance level or critical values from Tukey's distribution.

Practical Application Steps

1. Conduct ANOVA

2. Check Assumptions

3. Interpret Results

Table of Contents

    ANOVA and the Analysis of Variance for Multiple Groups
    1. Scenario: Peer Assessment and Student Learning
      1. Formulating the Hypotheses
      2. Comparing Two Groups: The Two-Sample t-Test
    2. ANOVA: Analysis of Variance for Multiple Groups
      1. The ANOVA Methodology
    3. Total Sum of Squares (SSTotal)
      1. 1. Treatment Variance (Between-Groups Variability)
      2. 2. Error Variance (Within-Groups Variability)
      3. 3. Calculating the F-Statistic
    4. Example: Peer Assessment Data
      1. Data Summary
      2. ANOVA Table Construction
      3. Calculations
      4. Decision
    5. Interpretation of ANOVA Results
      1. Understanding the F-Statistic
      2. P-Value Interpretation
      3. Conclusion for the Example
    6. The One-Way ANOVA Model
      1. Statistical Model
      2. Alternative Representation
      3. Assumptions of the Model
    7. Assumptions of the F-Test
      1. 1. Equal Variances (Homogeneity of Variance)
      2. 2. Independence of Observations
      3. 3. Normality of Residuals
    8. Post-ANOVA Testing
      1. 1. Bonferroni Correction
      2. 2. Tukey's Honest Significant Difference (HSD) Test
      3. Steps for Pairwise Comparisons
    9. Practical Application Steps
      1. 1. Conduct ANOVA
      2. 2. Check Assumptions
      3. 3. Interpret Results