# Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi-Squared Statistic

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## Abstract

**:**

## 1. Introduction

## 2. Effect Sizes for Tests of Independence

#### 2.1. Phi

#### 2.2. Cramér’s V (and Tschuprow’s T)

## 3. Effect Sizes for the Goodness-of-Fit Tests

#### 3.1. Cohen’s w

#### 3.2. Fei

## 4. Simulation Study of the Distributional Form of the Fei Effect Size

## 5. Conclusions

## 6. How to Type the פ Symbol

- By copying the character from https://util.unicode.org/UnicodeJsps/character.jsp?a=05E4 (access date: 9 March 2023) or similar webpages.
- In R, by typing the string “\u05e4”.
- In LaTeX, by typing \char”05e4 and using a Unicode-compatible compiler, such as XeTeX or LuaLaTeX.
- In Microsoft Word, from the Hebrew character of the Symbols window (Insert → Symbol…) or by typing 05e4, followed by Alt + X on the keyboard (Windows only).
- On Windows, using the Character Map application or by holding down Alt and typing +1508 on the numeric keypad.
- On macOS, by enabling the Unicode Hex Input language from System Settings… → Keyboard, then typing Opt + 05e4.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison of true and simulation-based actual effect size פ (Fei) for different expected proportions, true effect sizes, and sample sizes. Histograms represent the distributions of the sample פ’s from the simulated datasets, and the plotted lines represent density functions for (scaled) non-central $\chi $ distributions for the corresponding effect sizes and sample sizes.

Sex | Survived | Died |
---|---|---|

Male | 367 | 1364 |

Female | 344 | 126 |

**Table 2.**Correlation and effect size $\varphi $ (phi) for the survival of Titanic passengers by sex, Titanic dataset from R.

Variable 1 | Variable 2 | r (95% CI) | $\mathit{\varphi}$ (95% CI) |
---|---|---|---|

Sex (male/female) | Survival (survived/died) | −0.46 (−0.49, −0.42) | 0.46 (0.42, 1.00) |

**Table 3.**Effect size Cramér’s V for the survival of Titanic passengers by class/position, Titanic dataset from R.

Class/Position | Survived | Died |
---|---|---|

1st | 203 | 122 |

2nd | 118 | 167 |

3rd | 178 | 528 |

Crew | 212 | 673 |

Type | Product | Cramér’s V (95% CI) | Tschuprow’s T (95% CI) | ||
---|---|---|---|---|---|

Soy | Milk | Meat | |||

Vegan | 47 | 0 | 0 | 1.00 (0.81, 1.00) | 0.84 (0.68, 1.00) |

Not-Vegan | 0 | 12 | 12 |

**Table 5.**Effect size Cohen’s w for variables with different numbers of categories and distributions.

Observed Counts | Expected Proportion | Cohen’s w (95% CI) |
---|---|---|

90/10 | 0.5/0.5 | 0.80 (0.61, 1.00) |

90/10 | 0.35/0.65 | 1.15 (0.99, 1.36) |

5/10/80/5 | 0.25/0.25/0.25/0.25 | 1.27 (1.10, 1.73) |

Observed Counts | Expected Proportion | Fei (95% CI) |
---|---|---|

90/10 | 0.5/0.5 | 0.80 (0.64, 1.00) |

90/10 | 0.35/0.65 | 0.85 (0.73, 1.00) |

5/10/80/5 | 0.25/0.25/0.25/0.25 | 0.73 (0.64, 1.00) |

Test | Table Size | Effect Size |
---|---|---|

${\chi}^{2}$ test for independence | 2-by-2 | $\varphi $ |

Larger than 2-by-2 | $V$ or $T$ (Reduces to $\varphi $ when table is 2-by-2) | |

${\chi}^{2}$ test for goodness-of-fit | 2 classes with uniform null distribution | $w$ |

More than 2 classes and/or non-uniform null distribution | $\u05e4$ (Reduces to $w$ when there are 2 classes with uniform null dist). |

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## Share and Cite

**MDPI and ACS Style**

Ben-Shachar, M.S.; Patil, I.; Thériault, R.; Wiernik, B.M.; Lüdecke, D.
Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi-Squared Statistic. *Mathematics* **2023**, *11*, 1982.
https://doi.org/10.3390/math11091982

**AMA Style**

Ben-Shachar MS, Patil I, Thériault R, Wiernik BM, Lüdecke D.
Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi-Squared Statistic. *Mathematics*. 2023; 11(9):1982.
https://doi.org/10.3390/math11091982

**Chicago/Turabian Style**

Ben-Shachar, Mattan S., Indrajeet Patil, Rémi Thériault, Brenton M. Wiernik, and Daniel Lüdecke.
2023. "Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi-Squared Statistic" *Mathematics* 11, no. 9: 1982.
https://doi.org/10.3390/math11091982