_{1}

^{*}

In this paper, we generalize the proof of the Cochran statistic in the case of an ANOVA two ways structure that asymptotically follows a Chi-2. While construction of homogeneity statistics test usually resorts to the determination of the covariance matrix and its inverse, the Moore-Penrose matrix, our approach, avoids this step. We also show that the Cochran statistic in ANOVA two ways is equivalent to conventional homogeneity statistics test. In particular, we show that it satisfies the invariance property. Finally, we conduct empirical verification from a meta-analysis that confirms our theoretical results.

In ANOVA methodology, it is generally accepted that the error variance is unknown and is the subject of an estimate. However, in practice, these fundamental assumptions are rarely checked, forcing the use of Fisher statistic in the homogeneity test on the mean of the different groups ([

According to the work of [

where

[_{0} of equality of the means of different groups (i, j), the Cochran statistic asymptotically follows a

Despite the existence of some attempts proposed by [

Step 1: The global average is estimated by a linear combination of the individual averages.

Step 2: One assumes that the population variances of each group are unknown and estimated by the variances of the corresponding samples.

Let the vector q be given by:

The covariance matrix is then estimated by

Step 3: One constructs the statistics test

In explicit form,

where

In the case of one way ANOVA, [

Following [

The remainder of the paper is organized as follows: Section 2 presents the main results. Section 3 provides an empirical evaluation of the proposed test; and Section 4 concludes the paper.

In this section, we first show that statistics

Proposition 1.

Proof.

We suppose that

Now let us consider

Let us consider

According to [

where

Proposition 2.

Proof.

In practice, the variance

Based on Slutsky Theorem, the statistic T is asymptotically distributed as a

Proposition 3.

T and C are equivalent.

Proof.

Since,

and since,

We obtain:

Therefore, we get:

Also since,

So that,

Therefore we obtain the equivalence between T and C. And as it was demonstrated that T is asymptotically distributed as a

Defining G by

G verifies the following invariance property.

Theorem 1.

The G statistics is invariant by the choice of the weights and

Proof of Theorem 1.

To prove this theorem, we need the following lemmas.

Lemma 1.

According to [

Proof.

Straightforward. +

Lemma 2.

According to [

Proof.

Straightforward. +

Lemma 3.

For Q in (4), its singular value decomposition is,

Proof.

It is easy to show that

where

Therefore,

Lemma 4.

We have

Proof.

According to Lemma 3,

+

From the above lemmas, we then can provide the proof of Theorem 1.

Proof of Theorem 1.

Therefore,

In this Section, we empirically verify equality between both statistics G and C from a meta-analysis. The data come from the Stael program base. Specifically, we want to compare the effectiveness of three different molecules and, at the same time, we want to appreciate the impact of administration mode of different molecules (orally or intravenously). However, we don’t want to multiply experiments and number of subjects. In total, there are six possible combinations that means 6 series of measures (of different or identical subjects) on which is then measured a relevant quantitative parameter, sensible capture the influence of the decision of the molecules tested). The various combinations of two factors (molecules 3 and 2 modes of treatment) are the factorial design. Here the factor 1 has 3 modes: molecule A, B and C, while the factor 2 admits 2 modalities: Oral and injection.

Thus, from the definition of Cochran statistics C:

Mol. A | Mol. A | Mo. B | Mol. B | Mol. C | Mol. C |
---|---|---|---|---|---|

Oral | Injection | Oral | Injection | Oral | Injection |

10 | 11 | 7 | 8 | 12 | 7 |

12 | 18 | 14 | 9 | 9 | 6 |

8 | 12 | 10 | 10 | 11 | 10 |

10 | 15 | 11 | 9 | 10 | 7 |

6 | 13 | 9 | 11 | 7 | 7 |

13 | 8 | 10 | 13 | 8 | 5 |

9 | 15 | 9 | 7 | 13 | 6 |

10 | 16 | 11 | 14 | 14 | 7 |

9 | 9 | 7 | 15 | 10 | 9 |

8 | 13 | 9 | 12 | 11 | 6 |

Factor 1 | Mol. A | Mol. B | Mol. C |
---|---|---|---|

Nber subjects | 20 | 20 | 20 |

Mean | 11.25 | 10.25 | 8.75 |

Std deviat. | 3.127 | 2.381 | 2.552 |

Median | 10.5 | 10 | 8.5 |

Factor 2 | Oral | Injection |
---|---|---|

Nber subjects | 30 | 30 |

Mean | 9.9 | 10.27 |

Std dvt. | 2.057 | 3.503 |

Median | 10 | 9.5 |

Mol. A | Mol. A | Mol. B | Mol. B | Mol. C | Mol. C | |
---|---|---|---|---|---|---|

Oral | Injection | Oral | Injection | Oral | Injection | |

10 | 11 | 7 | 8 | 12 | 7 | |

12 | 18 | 14 | 9 | 9 | 6 | |

8 | 12 | 10 | 10 | 11 | 10 | |

10 | 15 | 11 | 9 | 10 | 7 | |

6 | 13 | 9 | 11 | 7 | 7 | |

13 | 8 | 10 | 13 | 8 | 5 | |

9 | 15 | 9 | 7 | 13 | 6 | |

10 | 16 | 11 | 14 | 14 | 7 | |

9 | 9 | 7 | 15 | 10 | 9 | |

8 | 13 | 9 | 12 | 11 | 6 | |

Size (n_{ij}) | 10 | 10 | 10 | 10 | 10 | 10 |

Mean (y_{ij}) | 9.5 | 13 | 9.7 | 10.8 | 10.5 | 7 |

Std deviation (s_{ij}) | 2.01 | 3.13 | 2.06 | 2.66 | 2.17 | 1.49 |

Variance ( | 4.06 | 9.78 | 4.23 | 7.07 | 4.72 | 2.22 |

W_{ij} | 2.47 | 1.02 | 2.36 | 1.42 | 2.12 | 4.50 |

h_{ij}(weights) | 0.18 | 0.07 | 0.17 | 0.10 | 0.15 | 0.32 |

y_{ij}*h_{ij} | 1.69 | 0.96 | 1.65 | 1.10 | 1.60 | 2.27 |

Q_{ij} | 0.23 | 3.73 | 0.43 | 1.53 | 1.23 | −2.27 |

Then we determine the G statistics as:

The Moore-Penrose decomposition

Finally, we have

The Moore-Penrose pseudo-inverse matrix of

Therefore, the G statistics is calculated according to the formula,

Finally, we can verify the invariance property of G statistics, compared to _{ }that means that

We then obtain

Returning to the procedure described in the previous Section, the following results were obtained,

And the corresponding Moore-Penrose matrix is

Once again, we can observe that

According to the above results, we observe that_{0} that assume that all groups _{0} of homogeneity between groups is rejected.

The literature generally uses a multi-step method for determining homogeneity statistics test. It is based on a linear combination of individual mean of the sample to estimate the overall mean. Like the G statistic in (6), this approach involves determining a covariance matrix and its Moore-Penrose inverse. However, we show that Theorem 1 generalizes the result of [

PamphileMezui-Mbeng, (2015) A Note on Cochran Test for Homogeneity in Two Ways ANOVA and Meta-Analysis. Open Journal of Statistics,05,787-796. doi: 10.4236/ojs.2015.57078