## ANOVA2(NR) Test Examples

#### Analyze data from a two-factor experiment that has no replication

The sample data are contained in a (6 rows by 4 columns) 2D wave dataWave1 where Factor A corresponds to rows and Factor B corresponds to columns:

4.04 | 5.08 | 2.83 | 3.06 |

4.91 | 4.68 | 3.17 | 2.71 |

2.94 | 1.9 | 1.61 | 2.02 |

5.95 | 5.26 | 4.85 | 4.45 |

3.9 | 4.85 | 2.69 | 2.93 |

5.72 | 5.14 | 4.39 | 3.29 |

To run the test execute the command:

StatsANOVA2NRTest/T=1 dataWave1

The results of the operation are displayed in the table ANOVA2(NR):

SS | DF | MS | F | F_{c} | Accept | |||

Total | 36.5039 | 23 | ||||||

Factor B | 11.2881 | 3 | 3.7627 | 14.427 | 3.28738 | 0 | ||

Factor A | 21.3036 | 5 | 4.26072 | 16.3364 | 2.90129 | 0 | ||

Remainder | 3.91216 | 15 | 0.260811 |

The assumption in this test is that there is no interaction between Factor A and Factor B so the operation
computes the results for both factors. H_{0} provides that there is no difference in the means of the populations. SS are the sum of the squares, DF are the degrees of
freedom, MS are the mean square, F are the computed statistics, F_{c} are the critical F values for the corresponding degrees of freedom.
Evidently in this case H_{0} is rejected for both rows and columns.

#### Analysis of data of the same dimensions but having less variation as shown in dataWave2

4.6 | 4.92 | 4.39 | 4.26 | ||

4.52 | 5 | 5.51 | 4.63 | ||

5.06 | 5.7 | 5.72 | 5.43 | ||

4.51 | 5.35 | 4.43 | 4.86 | ||

5.74 | 4.85 | 5.38 | 5.17 | ||

4.89 | 4.74 | 4.66 | 5.4 |

To repeat the test for these data use the command:

StatsANOVA2NRTest/T=1 dataWave2

The resulting table is:

SS | DF | MS | F | F_{c} | Accept | |

Total | 4.71293 | 23 | ||||

Factor B | 0.137833 | 3 | 0.0459444 | 0.303508 | 3.28738 | 1 |

Factor A | 2.30443 | 5 | 0.460887 | 3.04461 | 2.90129 | 0 |

Remainder | 2.27067 | 15 | 0.151378 |

In this case we can't reject H_{0} for Factor B (columns), i.e., the the mean reading is the same for
all columns. The situation is different for rows where F>F_{c} at least for the default 0.05 significance
level. To test with relaxed significance (0.01) execute the command:

StatsANOVA2NRTest/T=1/ALPH=0.01 dataWave2

This results in:

SS | DF | MS | F | F_{c} | Accept | |

Total | 4.71293 | 23 | ||||

Factor B | 0.137833 | 3 | 0.0459444 | 0.303508 | 5.41696 | 1 |

Factor A | 2.30443 | 5 | 0.460887 | 3.04461 | 4.55561 | 1 |

Remainder | 2.27067 | 15 | 0.151378 |

As expected, the change in significance was enough to drive F,sub>c above the value of F for Factor A and so at this significance level we can't
reject H_{0} for Factor A.

#### Handling One Missing Value

In the following examples we use the same data as dataWave1 but we introduce one missing value. To do so we set (row 3, column 2) to NaN.

StatsANOVA2NRTest/T=1/FOMD dataWave3

The operation prints in the history:

Single missing value computed to be: 4.34667

SS bias estimate: 0.572033

The result table contains the following values:

SS | DF | MS | F | F_{c} | Accept | |

Total | 35.7387 | 22 | ||||

Factor B | 11.3438 | 3 | 3.78126 | 12.2375 | 3.34389 | 0 |

Factor A | 20.0691 | 5 | 4.01382 | 12.9901 | 2.95825 | 0 |

Remainder | 4.32585 | 14 | 0.30899 |

The computed datum 4.34667 is used as a substitute for 4.85 which we erased from the original data. The test results are essentially unchanged.