R‑Squared Confidence Interval Calculator

For a regression model with sample size \( n \) and \( p \) predictors, the observed \( R^2 \) is converted to Cohen’s \( f^2=\frac{R^2}{1-R^2} \) with noncentrality parameter \(\lambda = f^2\,(n-p-1)\).

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

* Enter sample size \( n \), number of predictors \( p \), observed \( R^2 \) (0–1), 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

Conversion formulas: $$ f^2=\frac{R^2}{1-R^2},\quad \lambda=f^2\,(n-p-1),\quad R^2=\frac{\lambda}{\lambda+n-p-1}. $$

R‑Squared Confidence Interval Calculator – Educational Guide

R‑Squared Confidence Interval Calculator

Welcome to our R‑Squared Confidence Interval Calculator! This tool is designed for regression models with sample size n and k predictors. It converts the observed R² into Cohen’s f², using the corresponding noncentrality parameter, to compute a confidence interval for your model’s explanatory power.

Table of Contents

What is an R‑Squared Confidence Interval?

The R‑Squared Confidence Interval provides a range that is likely to contain the true coefficient of determination (R²) for your regression model. It accounts for sampling variability and offers insight into the reliability of your model’s explanatory power.

  • R²: The proportion of variance in the dependent variable explained by the predictors.
  • Confidence Interval: A range within which the true R² is expected to lie with a specified level of confidence (e.g., 95%).
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Conversion to Cohen’s f²

To facilitate the computation of the confidence interval, the observed R² is first converted to Cohen’s f² using the following formula:

$$f^2 = \frac{R^2}{1 – R^2}$$

This conversion is useful because Cohen’s f² provides a standardized effect size that can be related to the noncentral F‑distribution.

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Calculation Formula for the Confidence Interval

Once Cohen’s f² is obtained, its confidence interval is derived using the noncentrality parameter (δ) associated with the F‑distribution. While the exact computation often relies on numerical methods or specialized software, the conceptual formula is:

$$\delta = f^2 \times (n – k – 1)$$

The confidence interval for R² is then back‐calculated from the confidence limits for f².

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Key Concepts

  • R² (Coefficient of Determination): Indicates how well the model explains the variability of the outcome.
  • Cohen’s f²: A measure of effect size derived from R².
  • Noncentral F‑Distribution: Used to determine the confidence limits when the null hypothesis is not central.
  • Sample Size (n) and Predictors (k): Influence the degrees of freedom and, subsequently, the precision of the confidence interval.
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Step-by-Step Calculation Process

  1. Input Your Model Parameters:

    Enter the observed R², the sample size (n), and the number of predictors (k).

  2. Convert R² to Cohen’s f²:

    Compute f² using:

    $$f^2 = \frac{R^2}{1 – R^2}$$

  3. Calculate the Noncentrality Parameter:

    Determine δ using:

    $$\delta = f^2 \times (n – k – 1)$$

  4. Compute the Confidence Interval for f²:

    Use a method based on the noncentral F‑distribution to estimate the lower and upper bounds for f².

  5. Back-Calculate R² Confidence Interval:

    Convert the f² interval back to an R² interval using:

    $$R^2 = \frac{f^2}{1 + f^2}$$

  6. Review the Results:

    The resulting confidence interval indicates the range within which the true R² is likely to fall.

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Practical Examples

Example: Estimating R² Confidence Interval

Scenario: Suppose your regression model has an observed \(R^2 = 0.50\) with a sample size \(n = 100\) and \(k = 5\) predictors.

  1. Convert to f²:

    $$f^2 = \frac{0.50}{1 – 0.50} = 1.0$$

  2. Compute Noncentrality Parameter:

    $$\delta = 1.0 \times (100 – 5 – 1) = 94$$

  3. Estimate f² Confidence Interval:

    Using a noncentral F‑distribution approach (via tables or software), suppose the 95% confidence interval for f² is estimated as [0.80, 1.20].

  4. Convert Back to R²:

    $$R^2_{\text{lower}} = \frac{0.80}{1 + 0.80} \approx 0.44$$
    $$R^2_{\text{upper}} = \frac{1.20}{1 + 1.20} \approx 0.55$$

  5. Interpretation:

    With 95% confidence, the true R² lies between approximately 0.44 and 0.55.

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Interpreting the Results

The F‑Square Effect Size Confidence Interval Calculator provides the confidence interval for your model’s R² by converting it to Cohen’s f² and using the noncentral F‑distribution. A narrow interval indicates a precise estimate of your model’s explanatory power, while a wider interval indicates greater uncertainty.

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Applications

This calculator is valuable in:

  • Multiple Regression Analysis: Assessing the reliability of the model’s R².
  • Research Studies: Planning studies with adequate power and interpreting model fit.
  • Social Sciences, Economics, and Health Sciences: Quantifying how well predictors explain variability in outcomes.
  • Model Evaluation: Comparing competing models based on their explanatory power.
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Advantages

  • User-Friendly: Intuitive input for R², sample size, and number of predictors.
  • Comprehensive Analysis: Provides both an observed effect size and its uncertainty through a confidence interval.
  • Educational: Enhances understanding of the relationship between R² and Cohen’s f².
  • Practical: Supports informed decision-making in model evaluation and study planning.
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Conclusion

Our R‑Squared Confidence Interval Calculator is an essential tool for researchers and analysts. By converting the observed R² to Cohen’s f² and computing its confidence interval using the noncentral F‑distribution, you gain valuable insights into the precision and reliability of your regression model’s explanatory power. For further assistance or additional analytical resources, please explore our other calculators or contact our support team.

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