# Cohen's f2: Definition, Criterion and Example

In a multiple regression model where both independent and dependent variables are continuous, one of the most common method for calculating the effect size of each of the variables or construct is Cohen’s f2. Cohen categorized effect size as small, medium or large as shown in table 1. However, these conventions must be used carefully since what is small or trivial is context specific.

For instance, a small effect size (e.g. 0.04) maybe considered large when testing the efficacy of covid-19 vaccine on a patient while the same effect size maybe seen as weak in a study involving the acceptance of a technology by employees or even the attitude of students taking up a particular college course.

Effect size | $f^2$ |
---|---|

Small | $\ge 0.02$ |

Medium | $\ge 0.15$ |

Large | $\ge 0.35$ |

Table 1: Criterion for Cohen’s $f^2$

It come with no surprise that most statistical software do not produce output for Cohen’s $f^2$, but provides a way to calculate it. Mathematically,

$$ f^2 = \frac{R^2}{1-R^2} ,$$

where $R^2$ or R-square (r-square) is the coefficient of determination

Table 2 shows the effect size, Cohen’s $f^2$ criterion used by a marketing firm to measure the overall customer satisfaction of clients using variables such as Quick Service, Service Quality, Competitive Pricing and Good Value.

Variable | R-square | Cohen’s $f^2$ | Effect size |
---|---|---|---|

Quick Service | $0.275$ | $0.38$ | Large |

Service Quality | $0.180$ | $0.22$ | Medium |

Competitive Pricing | $0.315$ | $0.46$ | Large |

Good Value | $0.047$ | $0.05$ | Small |

Table 2: Effect size calculation for a Marketing firm

The calculation for table 2 is shown below:

Quick Service: $ f^2 = \displaystyle\frac{0.275}{1-0.275} = \frac{0.275}{0.725} =0.38$

Service Quality: $ f^2 = \displaystyle\frac{0.180}{1-0.180} = \frac{0.180}{0.820} =0.22$

Competitive Pricing: $ f^2 = \displaystyle\frac{0.315}{1-0.315} = \frac{0.315}{0.685} =0.46$

Good Value: $ f^2 = \displaystyle\frac{0.047}{1-0.047} = \frac{0.047}{0.953} =0.05$

**Alternatively:** You can download the Excel file to automatically calulate and interpret the effect size for you.

## Reference:

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Academic press.