Cohen’s f² Effect Size from R² Calculator

Determine the practical significance of a multiple regression model by calculating Cohen’s f² from the coefficient of determination (R²).

Effect Size Calculator

What are Cohen’s Guidelines for Calculating Effect Size using R Squared?

In statistical analysis, particularly in multiple regression, R-squared (R²) tells you the proportion of variance in the dependent variable that is predictable from the independent variable(s). While R² is a valuable measure of model fit, it doesn’t directly quantify the *magnitude* or *practical significance* of the effect. This is where Cohen’s guidelines for calculating effect size using r squared come into play.

Jacob Cohen introduced f-squared (f²) as an effect size measure to accompany R². It provides a standardized way to assess whether the proportion of variance accounted for by the regression model is small, medium, or large. This helps researchers move beyond statistical significance (p-values) and evaluate the real-world importance of their findings. Calculating f² is a critical step in power analysis and interpreting the substantive impact of a model.

The Cohen’s f² Formula and Explanation

The formula to convert R² into Cohen’s f² is simple and direct. It assesses the ratio of explained variance to unexplained variance.

f² = R² / (1 – R²)

This calculation allows us to interpret the effect size according to established benchmarks. For more complex models, you might be interested in our Multiple Correlation Coefficient Calculator.

Variable Explanations for the f² Formula

Variable	Meaning	Unit	Typical Range
f²	Cohen’s f-squared effect size	Unitless ratio	0 to ∞ (practically rarely exceeds 1.0)
R²	Coefficient of Determination	Unitless ratio	0 to 1

Practical Examples

Example 1: A Small Effect

A social scientist runs a multiple regression model to predict job satisfaction from several workplace factors. The model yields an R² of 0.04.

• Input (R²): 0.04
• Calculation: f² = 0.04 / (1 – 0.04) = 0.04 / 0.96 ≈ 0.042
• Result: The f² value is approximately 0.042. According to Cohen’s guidelines (small ≥ 0.02, medium ≥ 0.15, large ≥ 0.35), this represents a small effect size. The model has statistical significance, but its practical impact is minor.

Example 2: A Large Effect

An engineer models the compressive strength of a new concrete mix. The regression model results in an R² of 0.28. Understanding this is crucial, just as it is in statistical power analysis.

• Input (R²): 0.28
• Calculation: f² = 0.28 / (1 – 0.28) = 0.28 / 0.72 ≈ 0.389
• Result: The f² value is approximately 0.389. This is above the threshold for a large effect size (≥ 0.35), indicating that the predictors in the model have a substantial and practically significant impact on concrete strength.

How to Use This Cohen’s f² Calculator

This tool simplifies the process of determining your model’s effect size.

1. Enter R-Squared (R²): In the input field, type the R² value obtained from your regression analysis. This value must be between 0 and 1.
2. View Real-Time Calculation: The calculator automatically computes the f² value and displays it as you type.
3. Interpret the Result: The tool provides the numerical f² value and a qualitative interpretation (Small, Medium, or Large) based on Cohen’s guidelines for calculating effect size using r squared.
4. Visualize the Effect: The dynamic bar chart instantly compares your result to Cohen’s benchmarks, offering a clear visual context for your model’s effect size.

Key Factors That Affect Cohen’s f²

The calculated f² value is entirely dependent on R², so factors that influence R² will directly impact the effect size. For a different perspective on variance, consider using an ANOVA calculator.

• Strength of Predictor-Outcome Relationship: Stronger linear relationships between independent and dependent variables lead to a higher R² and thus a larger f².
• Number of Predictors: Adding more predictors to a model can inflate R² (though not always adjusted R²), which can in turn increase the f² value. It’s important to avoid overfitting.
• Measurement Error: High levels of error in measuring variables can weaken observed relationships, lowering R² and resulting in a smaller calculated effect size.
• Sample Homogeneity: If the sample is very homogeneous (i.e., has low variability in the variables), it can be harder to detect strong relationships, potentially leading to a lower R².
• Linearity: R² and f² measure the strength of *linear* relationships. If the true relationship is non-linear, the R² will be artificially low, underestimating the true effect size.
• Outliers: Influential outliers can either inflate or deflate the R², which will have a direct and sometimes misleading effect on the f² value.

Frequently Asked Questions (FAQ)

1. What is the difference between R² and f²?

R² measures the proportion of variance explained by your model (e.g., “15% of the variance in sales is explained by advertising spend”). f², on the other hand, is a standardized measure of the magnitude of this effect, telling you if that 15% is considered small, medium, or large in practical terms. It’s about significance vs. magnitude.

2. Can f² be used for simple linear regression?

Yes, while f² is most commonly discussed in the context of multiple regression, the formula works perfectly for simple linear regression (one predictor), where R² is simply the square of the Pearson correlation coefficient (r).

3. Why are the units unitless?

Both R² and f² are ratios of variances. Since the units in the numerator (explained variance) and denominator (unexplained variance) are the same, they cancel out, leaving a pure, unitless number that can be compared across different studies and contexts.

4. What is a “good” f² value?

It’s context-dependent. In fields like particle physics, a small effect size might be a monumental discovery. In social sciences, achieving a medium effect might be considered a very strong result. Cohen’s guidelines (0.02, 0.15, 0.35) are general rules of thumb, not absolute laws.

5. How does sample size affect f²?

Sample size does not appear in the f² formula, so it doesn’t directly change the f² value for a given R². However, larger sample sizes provide more precise and stable estimates of R², meaning you can be more confident in your resulting f² calculation. This is a core concept in sample size calculation.

6. What if my R² value is very close to 1?

If R² is, for example, 0.99, then f² = 0.99 / (1 – 0.99) = 99. The f² value can grow very large as R² approaches 1. This indicates an extremely large effect where the model explains almost all the variance.

7. Are there alternatives to Cohen’s f²?

Yes, other effect size measures for regression include Eta squared (η²) and Omega squared (ω²), which are common in ANOVA contexts but serve a similar purpose. However, f² is the standard for power analysis in multiple regression as per Cohen’s framework.

8. Where do Cohen’s guidelines come from?

Jacob Cohen proposed these benchmarks in his influential 1988 book, “Statistical Power Analysis for the Behavioral Sciences.” They were based on his observations of effect sizes commonly found in behavioral science research and were intended to provide a conventional frame of reference.