Determines the orthogonal decomposition of the sample space for a design, using an analysis of the canonical relationships between the projectors derived from two or more model formulae (C.J. Brien).

### Options

`PRINT` = string tokens |
What to print (`decomposition` , `df` , `ecriteria` , `efficiencies` ); default `deco` |
---|---|

`GRANDMEAN` = string tokens |
Add the term for the grand or overall mean to each formula (
`include` , `omit` ) ; default `omit` |

`CRITERIA` = string tokens |
The efficiency criteria to be saved and/or printed (`aefficiency` , `mefficiency` , `sefficiency` , `eefficiency` , `xefficiency` , `order` , `dforth` ); default `aeff` , `eeff` , `orde` |

`FACTORIAL` = scalar |
Limit on the number of factors and variates in each model term default `*` i.e. no limit |

`TOLERANCE` = variate |
Tolerances for zero in various contexts; default 10^{-8} for all of these |

### Parameters

FORMULAE = pointers |
Each pointer contains two or more model formulae whose joint decomposition is required |
---|---|

`ORTHOGONALMETHOD` = string tokens |
Specifies the method to use for each model formula when orthogonalizing a projection matrix to those for terms that occur earlier in the formula (`differencing` , `eigenmethods` ); default `diff` |

PROJECTIONSETS = pointers |
Saves the projection pointers formed from the formulae |

`COMBINEDPROJECTIONSET` = pointers |
Saves the projector pointers that produce the orthogonal decomposition |

`EFFICIENCYFACTORS` = pointers |
Saves the canonical efficiency factors |

`ECRITERIA` = pointers |
Saves the unadjusted efficiency criteria |

`ADJECRITERIA` = pointers |
Saves the adjusted efficiency criteria |

`ADJDF` = pointers |
Saves the adjusted degrees of freedom |

`SAVE` = pointers |
Saves information about the analysis for use by `ACDISPLAY` and `ACKEEP` |

### Description

`ACANONICAL`

forms the decomposition of the sample space of a design, to examine its “anatomy” (Brien 2016a, b), and summarizes this in a decomposition table (Brien & Bailey 2009, Bailey & Brien 2016). This reflects the properties of the design, by showing the confounding between sources from different model formulae. The decomposition table is similar to the skeleton anova table that the `ANOVA`

directive produces based on the `BLOCKSTRUCTURE`

and `TREATMENTSTRUCTURE`

formulae for designs that have first order balance. `ANOVA`

performs a dummy analysis of a randomly-generated variate on which a series of sweeps are performed. On the other hand, `ACANONICAL`

produces the decomposition table by perfoming an eigenanalysis of the canonical relationships between projection matrices corresponding to the sources derived from the terms in the formulae. It is more general than the `ANOVA`

directive in that it can produce a decomposition table for arbitrarily non-orthogonal designs and is not restricted to two formulae. However, for designs with 500 or more observations, the analysis may take in excess of 5 minutes.

The `FORMULAE`

parameter specifies the model formulae for which the decomposition table is to be produced.

The `ORTHOGONALMETHOD`

parameter controls which method to use for orthogonalizing a projection matrix to those for terms that occur before it in a formula. Different methods can be used for different formulae.

The parameters `PROJECTIONSETS`

, `COMBINEDPROJECTIONSET`

, `EFFICIENCYFACTORS`

, `ECRITERIA`

, `ADJECRITERIA`

, `ADJDF`

save information from the analysis. Each of these forms a pointer, whose number of elements is one less than the number of formulae. The first element contains the result of using the projectors from the second formula to decompose those from the first. The second takes that result, and decomposes it according to the projectors from the third formula. This process of refining the current decomposition using the projectors from the next formula is continued until there are no unused formulae.

The `PROJECTIONSETS`

parameter saves a triply-suffixed system of pointers to projector pointers from the pairs of decompositions. Each projection pointer has a `'matrix'`

element containing the projection matrix for a term, and a `'df'`

element containing the degrees of freedom of the projection matrix. Suppose that {*P _{i}*} is a set of projection pointers for a decomposition up to the

*i*th formula, and that this decomposition is to be further refined using the set {

*P*} of projection pointers corresponding to the

_{j}*j*th formula. The projection matrix in each case will be the projector onto the subspace of a {

*P*} projector, pertaining to the subspace of a {

_{i}*P*} projector.

_{j}The `COMBINEDPROJECTIONSET`

parameter saves a pointer, with a single suffix, containing the set of projection pointers whose `'matrix'`

component contain the non-zero projection matrices, that produce the orthogonal decomposition summarized in the decomposition table; see Brien & Bailey (2009, 2010) and Bailey & Brien (2016) for structure-balanced examples. The `'labels'`

of the projection pointers reflect the terms involved in the subspaces that are projected onto by the corresponding projector.

The `EFFICIENCYFACTORS`

parameter saves a pointer, with a pair of suffices, that contains the set of variates containing the canonical efficiency factors for each combination of a matrix from {*P _{j}*}, and a matrix from {

*P*}, with non-zero efficiency factors. The efficiency factors are adjusted for all matrices preceding it in forming the decomposition.

_{i}The `ECRITERIA`

parameter saves a set of matrices, each of which contains one of the unadjusted efficiency criteria, nominated by the option `CRITERIA`

, for one of the decompositions.

The `ADJECRITERIA`

parameter saves a set of matrices, each of which contains one of the adjusted efficiency criteria, nominated by the option `CRITERIA`

, for one of the decompositions.

The `ADJDF`

parameter saves a set of matrices, each of which contains the degrees of freedom of an adjusted projector from {*P _{j}*}, where the matrix from {

*P*} has been adjusted for all those preceding it in {

_{j}*P*}.

_{j}The `SAVE`

parameter saves all the information from the analysis, in a pointer with elements `status`

, `efficiencies`

, `effcriteria`

, `adjeffcriteria`

, `adjdf`

and `combinedset`

.

The `PRINT`

option controls printing, with settings:

`decomposition` |
table summarizing the decomposition, |
---|---|

`df` |
degrees of freedom, |

`ecriteria` |
efficiency criteria (as requested by the `CRITERIA` option), and |

`efficiencies` |
efficiency factors. |

The `GRANDMEAN`

option controls the inclusion or omission of a term for the grand or overall mean for each formula.

The `CRITERIA`

option specifies the efficiency criteria to save or print, with the following settings:

`aefficiency` |
the harmonic mean of the canonical efficiency factors; |
---|---|

`mefficiency` |
the mean of the canonical efficiency factors, |

`sefficiency` |
the variance of the canonical efficiency factors, |

`eefficiency` |
the minimum of the canonical efficiency factors, |

`xefficiency` |
the maximum of the canonical efficiency factors, |

`order` |
the number of unique canonical efficiency factors, and |

`dforth` |
the number of degrees of freedom that are orthogonal. |

The `FACTORIAL`

option can be used to limit on the number of factors and variates in each term.

The `TOLERANCE`

option specifies the values that are small enough to be considered zero. Its setting is a variate with two values. The first is used in determining if elements of structures, usually matrices, are sufficiently close to zero. The second determines if eigenvalues, or quantities derived from them, are sufficiently close to zero.

Options: `PRINT`

, `CRITERIA`

, `FACTORIAL`

, `TOLERANCE`

.

Parameters: `FORMULAE`

, `ORTHOGONALMETHOD`

, `PROJECTIONSET`

, `COMBINEDPROJECTIONSET`

, `EFFICIENCYFACTORS`

, `ECRITERIA`

, `ADJECRITERIA`

, `ADJDF`

, `SAVE`

.

### Method

First of all, the set of projection pointers is obtained for each formula supplied by the `FORMULAE`

parameter; there is one projection pointer for each term in a formula. Then an analysis of the canonical relationships is performed between the sets of projection matrices for the first two formulae. If there is a third formula, the relationships between its projectors and the already established decomposition are formed, and so on until all the formulae have been processed. The core of the analysis is the determination of eigenvalues of the product of pairs of projectors using the results of James & Wilkinson (1971).

However, if the order of balance between two projection matrices is 10 or more, the James &Wilkinson (1971) methods fails to produce an idempotent matrix. Equation 5.3 of Payne & Tobias (1992) is then used to obtain the projection matrices for their joint decomposition; this requires the inversion of a product involving the two projections matrices.

### References

Bailey, R.A. & Brien, C.J. (2016) Randomization-based models for multitiered experiments. I. A chain of randomizations. *The Annals of Statistics*, 44, 1131-1164.

Brien, C. J. (2016a) Multiphase experiments in practice, with an emphasis on nonorthogonal designs. I. A look back. submitted to *The Australian & New Zealand Journal of Statistics*.

Brien, C.J. (2016b) Multiphase experiments in practice, with an emphasis on nonorthogonal designs. II. Developments. submitted to *The Australian & New Zealand Journal of Statistics*.

Brien, C.J. & R.A. Bailey (2009). Decomposition tables for multitiered experiments. I. A chain of randomizations. *The Annals of Statistics*, 36, 4184 – 4213.

Brien, C.J. & R.A. Bailey (2010). Decomposition tables for multitiered experiments. II. Two-one randomizations. *The Annals of Statistics*, 38, 3164 – 3190.

James, A.T. & Wilkinson, G.N. (1971) Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. *Biometrika*, 58, 279-294.

Payne, R.W. & R.D. Tobias (1992). General balance, combination of information and the analysis of covariance. *Scandinavian Journal of Statistics*, 19, 3-23.

### See also

Procedures: `ACDISPLAY`

, `ACKEEP`

.

Commands for: Analysis of variance.

### Example

CAPTION 'ACANONICAL example',\ !t('Partially-balanced incomplete-block design with 2 association',\ 'classes from page 379 of Cochran and Cox (1957) Experimental',\ 'Designs. 2nd ed. Wiley, New York.'); STYLE=meta,plain FACTOR [LEVELS=6; VALUES=4(1...6)] Block & [LEVELS=4; VALUES=(1...4)6] Unit & [LEVELS=6; VALUES=1,4,2,5, 2,5,3,6, 3,6,1,4,\ 4,1,5,2, 5,2,6,3, 6,3,4,1] Treat ACANONICAL [PRINT=decomposition,efficiencies] !p(!f(Block/Unit),!f(Treat)) " define a pseudo-factor for the contrasts partially confounded with blocks " FACTOR [LEVELS=3] Pf; VALUES=NEWLEVELS(Treat; !(1,2,3,1,2,3)) TREATMENTS Treat//Pf BLOCKS Block/Unit ANOVA