Cohen’s f² Confidence Interval Calculator

Compute the observed effect size and its confidence interval using the formula: $$ f^2=\frac{R^2_{\text{full}}-R^2_{\text{base}}}{1-R^2_{\text{full}}}. $$

* Input sample size \( n \), baseline and full-model \( R^2 \), number of predictors in each model, and desired confidence level (e.g., 0.95). Note that \( R^2_{\text{full}}>R^2_{\text{base}} \) and \( n > k_{\text{full}}+1 \).

Step 1: Enter Model Parameters

e.g., 150

e.g., 0.20

e.g., 3

e.g., 0.35 (must exceed baseline \( R^2 \))

e.g., 5 (must be greater than \( k_{\text{base}} \))

e.g., 0.95

Formulas:
Observed effect size: $$ f^2=\frac{R^2_{\text{full}}-R^2_{\text{base}}}{1-R^2_{\text{full}}} $$
F‑statistic: $$ F=\frac{(R^2_{\text{full}}-R^2_{\text{base}})/(k_{\text{full}}-k_{\text{base}})}{(1-R^2_{\text{full}})/(n-k_{\text{full}}-1)}. $$

Cohen’s f² Confidence Interval Calculator - Educational Guide

Cohen’s f² Confidence Interval Calculator

Welcome to our Cohen’s f² Confidence Interval Calculator! This tool not only computes the observed effect size (Cohen’s f²) from your regression analysis but also calculates its confidence interval. By quantifying the incremental effect size and its uncertainty, you can better assess the practical significance of additional predictors in hierarchical models.

Table of Contents

What is Cohen’s f²?

Cohen’s f² is an effect size measure used in multiple regression analysis, particularly in hierarchical regression. It quantifies the incremental change in explained variance when additional predictors are added to a model.

  • Observed Effect Size: The calculated value of f² based on your model's \(R^2\) values.
  • Incremental Variance: The increase in \(R^2\) from adding new predictors.
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Calculation Formula

The effect size (Cohen’s f²) is computed using the following formula:

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

Where:

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

To compute the confidence interval for Cohen’s f², statistical methods such as noncentral F-distribution approaches can be applied. These methods involve:

  • Estimating the noncentrality parameter from your observed effect size.
  • Inverting the F-distribution to obtain lower and upper bounds.

The exact method may vary based on the software or algorithm used, but the goal is to determine the range within which the true effect size lies with a specified confidence level.

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Key Concepts

  • Hierarchical Regression: A method of entering predictors in steps to assess their incremental contribution.
  • Coefficient of Determination (\(R^2\)): The proportion of variance in the dependent variable explained by the model.
  • Effect Size (f²): A standardized measure of the incremental impact of added predictors.
  • Confidence Interval: The range within which the true effect size is expected to lie with a given level of confidence (e.g., 95%).
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Step-by-Step Calculation Process

  1. Obtain \(R^2\) Values:

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

  2. Calculate Cohen’s f²:

    Use the formula:

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

  3. Determine the Confidence Level:

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

  4. Compute the Confidence Interval:

    Apply an appropriate method (often based on the noncentral F-distribution) to obtain the lower and upper bounds of Cohen’s f².

  5. Interpret the Interval:

    The resulting interval reflects the precision of your observed effect size estimate.

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Practical Examples

Example: Estimating Cohen’s f² and Its Confidence Interval

Scenario: Suppose your reduced model has \(R^2_{\text{reduced}} = 0.40\) and your full model has \(R^2_{\text{full}} = 0.55\).

  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 an appropriate method, the confidence interval might be estimated as [0.25, 0.42].

  3. Interpretation:

    With the chosen confidence level (e.g., 95%), you can be confident that the true effect size lies between 0.25 and 0.42.

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Interpreting the Results

The calculator provides both the observed Cohen’s f² effect size and its confidence interval. A narrow interval indicates a precise estimate, while a wider interval suggests more uncertainty. These metrics help you understand the practical significance of adding new predictors to your regression model.

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Applications

This calculator is useful in:

  • Hierarchical Regression: Evaluating the incremental effect of additional predictors.
  • Social Sciences & Psychology: Quantifying the impact of new variables on behavioral models.
  • Economics & Business: Assessing improvements in forecasting models when extra predictors are included.
  • Educational Research: Measuring the effect size of interventions or new factors in academic performance studies.
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Advantages

  • User-Friendly: Simple interface for entering \(R^2\) values and selecting the confidence level.
  • Quick Computation: Rapidly calculates both the effect size and its confidence interval.
  • Educational: Enhances understanding of effect size and its importance in model evaluation.
  • Informed Decisions: Provides a quantitative basis for assessing the practical significance of added predictors.
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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 incremental impact of additional predictors. This facilitates more informed decision-making in model building and data interpretation. For further assistance or additional analytical resources, please explore our other calculators or contact our support team.

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