F‑Square Effect Size Confidence Interval Calculator

For multiple regression, the F‑Square effect size is defined as $$ f^2=\frac{R^2}{1-R^2}, $$ with corresponding noncentrality parameter $$ \lambda=f^2\,(n-p-1). $$

The overall F‑statistic is given by $$ F_{obs}=\frac{(R^2/p)}{((1-R^2)/(n-p-1))}. $$

* Enter the sample size \( n \), number of predictors \( p \), observed \( R^2 \) (between 0 and 1), and desired confidence level (in %). Note: \( n > p+1 \) and \( R^2 < 1 \).

Step 1: Enter Parameters

e.g., 100

e.g., 5

e.g., 0.3

e.g., 95

Formulas:
\( f^2=\frac{R^2}{1-R^2} \) and \( \lambda=f^2\,(n-p-1) \).
Confidence limits for \( f^2 \) are obtained by inverting the noncentral F CDF.

F‑Square Effect Size Confidence Interval Calculator – Educational Guide

F‑Square Effect Size Confidence Interval Calculator

Welcome to our F‑Square Effect Size Confidence Interval Calculator! This tool is designed for multiple regression analysis to compute the F‑Square effect size (f²) and its confidence interval. It helps you quantify the incremental effect of additional predictors and assess the precision of your effect size estimate.

Table of Contents

What is F‑Square (f²)?

In multiple regression, F‑Square (f²) is an effect size measure that represents the incremental change in explained variance when additional predictors are included. It is calculated from the difference in R² values between a full model and a reduced model.

  • Incremental Variance: The increase in R² due to new predictors.
  • Effect Size: A standardized measure indicating the practical significance of the added predictors.
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Calculation Formula

Cohen’s f² is computed using the formula:

$$f^2 = \frac{R^2_{\text{full}} – R^2_{\text{reduced}}}{1 – R^2_{\text{full}}}$$

Where:

  • \(R^2_{\text{full}}\): R² for the full model with additional predictors.
  • \(R^2_{\text{reduced}}\): R² for the reduced model without the additional predictors.
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Confidence Interval Method

The confidence interval for Cohen’s f² is typically derived using methods based on the noncentral F‑distribution. This approach involves:

  • Estimating the noncentrality parameter from the observed f² value.
  • Using the inverse cumulative distribution function of the noncentral F‑distribution to determine the lower and upper bounds.

These methods provide a range that likely contains the true effect size with a specified level of confidence (commonly 95%).

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Key Concepts

  • Hierarchical Regression: The process of adding predictors in steps to assess their incremental impact.
  • R² (Coefficient of Determination): Represents the proportion of variance explained by the model.
  • Effect Size (f²): A measure of the practical significance of the change in R².
  • Noncentral F‑Distribution: Used to derive confidence intervals for effect sizes when the null hypothesis is not central.
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Step-by-Step Calculation Process

  1. Obtain R² Values:

    Retrieve the R² for both the full model (\(R^2_{\text{full}}\)) and the reduced model (\(R^2_{\text{reduced}}\)).

  2. Calculate f²:

    Use the formula:

    $$f^2 = \frac{R^2_{\text{full}} – R^2_{\text{reduced}}}{1 – R^2_{\text{full}}}$$

  3. Select Confidence Level:

    Choose the desired confidence level (e.g., 95%).

  4. Compute Confidence Interval:

    Apply a method based on the noncentral F‑distribution to estimate the lower and upper bounds of Cohen’s f².

  5. Interpret the Interval:

    The interval indicates the range within which the true effect size is likely to lie, giving insight into the precision of your estimate.

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Practical Examples

Example: Estimating f² and Its Confidence Interval

Scenario: A reduced model yields \(R^2_{\text{reduced}} = 0.40\) and a full model yields \(R^2_{\text{full}} = 0.55\).

  1. Calculate f²:

    \( \Delta R^2 = 0.55 – 0.40 = 0.15 \)

    $$f^2 = \frac{0.15}{1 – 0.55} = \frac{0.15}{0.45} \approx 0.33$$

  2. Compute Confidence Interval:

    Using a noncentral F‑distribution method, suppose the 95% confidence interval for f² is estimated as [0.25, 0.42].

  3. Interpretation:

    With 95% confidence, the true effect size lies between 0.25 and 0.42, indicating a medium-to-large incremental effect.

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Interpreting the Results

The F‑Square Effect Size Confidence Interval Calculator provides both the observed Cohen’s f² and its confidence interval. A narrower confidence interval suggests a more precise estimate of the incremental effect size, while a wider interval indicates greater uncertainty.

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Applications

This calculator is particularly useful in:

  • Hierarchical Regression: Evaluating the added value of new predictors.
  • Social Sciences: Assessing the impact of additional variables in behavioral research.
  • Economics & Business: Measuring incremental improvements in forecasting models.
  • Educational Research: Determining the effect size of interventions on academic outcomes.
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Advantages

  • User-Friendly: Simple input fields for entering R² values and selecting a confidence level.
  • Quick Computation: Rapidly calculates both the effect size and its confidence interval.
  • Educational: Enhances understanding of incremental effect sizes in multiple regression analysis.
  • Informed Decision-Making: Provides quantitative insights to support model evaluation and improvement.
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Conclusion

Our Cohen’s f² Confidence Interval Calculator is an essential tool for researchers and analysts performing hierarchical regression. By computing both the observed effect size and its confidence interval, you gain valuable insights into the practical significance of additional predictors. This, in turn, supports more informed decisions in model building and interpretation. For further assistance or additional analytical resources, please explore our other calculators or contact our support team.

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