### 6.7.8. Phi Distribution

We should point out that phi is not a distribution as such, but a multi-parameter relationship based on the noncentral F-distribution.

The following two procedures are provided here as an alternative to OC Curves published by Pearson and Hartley (1951).

Here the significance level (α) can be any value between 0 and 1, whereas the OC Curves are limited to α = 0.05 and α = 0.01. Also, the accuracy of calculations here is far higher.

#### 6.7.8.1. Phi Distribution

The user is expected to enter:

· Phi

· Numerator Degrees of Freedom

· Denominator Degrees of Freedom

· Confidence Level

and the program will output the estimated power of the test (1 – β).

Example

Figure B.1d on p. AppB 862 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Phi Distribution and the Phi Distribution option. Enter the following data at the next dialogue:

Sample Size and Power Estimation:

Phi Distribution

Phi = |
3.0000 |

Numerator Degrees of Freedom = |
4.0000 |

Denominator Degrees of Freedom = |
20.0000 |

Confidence Level = |
0.9900 |

Power of the Test = |
0.9899 |

#### 6.7.8.2. Inverse Phi Distribution

The user is expected to enter:

· Power of the Test

· Numerator Degrees of Freedom

· Denominator Degrees of Freedom

· Confidence Level

and the program will output the estimated .

Example

Figure B.1g on p. AppB 865 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Phi Distribution and the Inverse Phi Distribution option. Enter the following data at the next dialogue:

Sample Size and Power Estimation:

Inverse Phi Distribution

Power of the Test = |
0.9050 |

Numerator Degrees of Freedom = |
7.0000 |

Denominator Degrees of Freedom = |
12.0000 |

Confidence Level = |
0.9500 |

Phi = |
1.9980 |