Use this calculator to determine the Intraclass Correlation Coefficient (ICC) based on your ANOVA data. Input the sum of squares and degrees of freedom for both between-groups and within-groups to compute the ICC, which measures the reliability or agreement within clusters/groups in your study.

ANOVA Intraclass Correlation (ICC) Calculator

In a one-way random effects model, the ICC is calculated as: $$ ICC = \frac{MSB – MSW}{MSB + (k-1) \cdot MSW}, $$ where \(MSB\) is the between‑subjects mean square, \(MSW\) is the within‑subjects mean square, and \(k\) is the number of ratings per subject.

* Enter the values for MSB, MSW, and \(k\) (with \(k \ge 2\)).

Step 1: Enter ANOVA Parameters

e.g., 10

e.g., 5

e.g., 3 (must be at least 2)

Formula: $$ ICC = \frac{MSB – MSW}{MSB + (k-1)\cdot MSW} $$

ANOVA Intraclass Correlation (ICC) Calculator (In-Depth Explanation)

ANOVA Intraclass Correlation (ICC) Calculator (In-Depth Explanation)

The Intraclass Correlation Coefficient (ICC) is a measure used to evaluate the reliability or consistency of measurements made by different observers measuring the same quantity. In a one-way random effects ANOVA model, the ICC quantifies how much of the total variance in the data is due to differences between subjects versus measurement error or differences within subjects.

Table of Contents

  1. Overview of ICC
  2. Key Concepts and Terminology
  3. The ICC Formula
  4. Step-by-Step Calculation Process
  5. Practical Examples
  6. Common Applications
  7. Conclusion

1. Overview of ICC

The Intraclass Correlation Coefficient (ICC) assesses the degree of agreement or consistency among ratings or measurements. In a one-way random effects model, the ICC is particularly useful when each subject is measured by different raters (or repeated measurements) selected at random. A high ICC indicates that most of the variance is due to differences between subjects, while a low ICC suggests that variability within subjects (measurement error or inconsistency) is high.

2. Key Concepts and Terminology

The calculation of ICC in a one-way random effects model involves several key components:

  • Between-Subjects Mean Square (\(MS_{between}\)): Represents the variance among the group (subject) means.
  • Within-Subjects Mean Square (\(MS_{within}\)): Represents the variance within each subject (error or measurement variance).
  • Number of Ratings per Subject (\(k\)): The number of independent measurements or raters for each subject.

3. The ICC Formula

In a one-way random effects ANOVA model, the ICC is calculated as:

\( \text{ICC} = \frac{MS_{between} - MS_{within}}{MS_{between} + (k - 1) \, MS_{within}} \)

Where:

  • \(MS_{between}\) is the between‐subjects mean square, reflecting the variance among subjects.
  • \(MS_{within}\) is the within‐subjects mean square, reflecting the variance due to error or differences within each subject.
  • \(k\) is the number of ratings per subject.

This formula shows that as the variability between subjects increases relative to the variability within subjects, the ICC approaches 1, indicating high reliability. Conversely, if the within-subject variance is high, the ICC will be lower.

4. Step-by-Step Calculation Process

  1. Collect Data:

    Gather the ratings or measurements for each subject. Ensure you have at least \(k\) ratings per subject.

  2. Compute Group Means:

    Calculate the mean for each subject (group) and the overall mean of all ratings.

  3. Calculate \(MS_{between}\):

    Compute the between‐subjects sum of squares and divide by its degrees of freedom (\(k_{\text{groups}} - 1\)).

  4. Calculate \(MS_{within}\):

    Compute the within‐subjects sum of squares and divide by its degrees of freedom (total observations - number of groups).

  5. Substitute into the ICC Formula:
    \( \text{ICC} = \frac{MS_{between} - MS_{within}}{MS_{between} + (k - 1) \, MS_{within}} \)
  6. Interpret the Result:

    A high ICC (close to 1) indicates that most of the variance is due to differences between subjects, while a low ICC indicates that the measurements are inconsistent.

5. Practical Examples

Example: Evaluating Measurement Consistency

Scenario: Assume you have 4 ratings per subject (\(k = 4\)) for 10 subjects. Based on your ANOVA, you compute:

  • \(MS_{between} = 50\)
  • \(MS_{within} = 20\)

Calculation:

\( \text{ICC} = \frac{50 - 20}{50 + (4 - 1) \times 20} = \frac{30}{50 + 60} = \frac{30}{110} \approx 0.273 \)

The ICC in this example is approximately 0.273, suggesting relatively low reliability or consistency among the ratings.

6. Common Applications

  • Psychology and Social Sciences: Assessing the consistency of ratings in surveys and observational studies.
  • Medical Research: Evaluating the reliability of diagnostic tests or observer ratings.
  • Quality Control: Measuring the repeatability of measurements in manufacturing processes.
  • Educational Assessment: Determining the consistency of scoring by multiple examiners.

7. Conclusion

The ANOVA Intraclass Correlation (ICC) Calculator is a powerful tool for quantifying the reliability and consistency of measurements or ratings in a one-way random effects model. By using the formula:

\( \text{ICC} = \frac{MS_{between} - MS_{within}}{MS_{between} + (k - 1) \, MS_{within}} \)

you can assess how much of the total variance is due to true differences between subjects as opposed to measurement error. This metric is invaluable in fields ranging from psychology and education to medical research and quality control, ensuring that data-driven decisions are based on reliable and consistent measurements.