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Cohen’s f² Confidence Interval Calculator for Multiple Regression

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Cohen’s f² Confidence Interval Calculator

Cohen’s f² Confidence Interval Calculator

For multiple regression with sample size \( n \), \( p \) predictors and observed \( R^2 \), Cohen’s \( f^2 \) is given by $$ f^2=\frac{R^2}{1-R^2}. $$

The noncentral F statistic is $$ F_{obs}=\frac{(R^2/p)}{((1-R^2)/(n-p-1))}, $$ with noncentrality parameter $$ \lambda=f^2\,(n-p-1). $$

* 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

Confidence interval for \( f^2 \) is obtained by inverting the noncentral F CDF: $$ \lambda=f^2\,(n-p-1), \quad f^2=\frac{\lambda}{n-p-1}. $$

Confidence Intervals for Cohen’s f²

Understanding and Calculating Confidence Intervals for Cohen’s f²

Cohen’s f² effect size measures the impact of predictors in a multiple regression model. It represents the proportion of variance explained by the predictors compared to the unexplained variance. Calculating confidence intervals for f² helps assess precision and compare different models.

Formula for Cohen’s f²

The effect size f² is calculated using:

f² = R² / (1 - R²)

Where:

  • R²: Proportion of variance explained by the predictors.
  • 1 - R²: Proportion of unexplained variance.

Steps to Calculate Confidence Intervals

  1. Calculate the F-statistic: 
    F = (R² / k) / ((1 - R²) / (N - k - 1))
    • k: Number of predictors.
    • N: Total sample size.
  2. Find critical F-values: Use the degrees of freedom:
    • df₁ = k
    • df₂ = N - k - 1
    Lookup critical F-values from a table or statistical software.
  3. Convert F-values to f²: 
    f²_lower = (F_lower * k) / (N - k - 1 - F_lower * k)
    f²_upper = (F_upper * k) / (N - k - 1 - F_upper * k)

Example

Calculate a 95% confidence interval for f²:

  • R² = 0.25
  • k = 3 predictors
  • N = 100 samples

Step 1: Compute F-statistic: 

F = (0.25 / 3) / ((1 - 0.25) / (100 - 3 - 1)) = 10.42

Step 2: Find critical F-values (from table or software):

  • F_lower = 2.7
  • F_upper = 9.84

Step 3: Calculate confidence bounds: 

f²_lower = (2.7 * 3) / (100 - 3 - 1 - 2.7 * 3) = 0.086
f²_upper = (9.84 * 3) / (100 - 3 - 1 - 9.84 * 3) = 0.473

Result: Confidence interval for f²: [0.086, 0.473]

Conclusion

Confidence intervals for Cohen’s f² provide insights into the precision of effect size estimates and help compare the performance of regression models.