Cohen’s f² Effect Size Confidence Intervals

Determine the 90%, 95%, and 99% confidence intervals for Cohen’s f² effect size in a multiple regression study.

To calculate these intervals, you will need the following:

f²: The observed effect size.
k: The number of predictor variables.
N: The total sample size.

Knowing the confidence intervals for f² can help evaluate and compare the performance of different regression models in analytics studies.

Cohen’s f² Confidence Interval Calculator for Multiple Regression

R² (Coefficient of Determination): Sample Size (n): Number of Predictors (k): Confidence Level (%):

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

Calculate the F-statistic:
```
F = (R² / k) / ((1 - R²) / (N - k - 1))
```
- k: Number of predictors.
- N: Total sample size.
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.

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.