Forms the spectral components from the canonical components of a multitiered design, and constrains any negative spectral components to zero (C.J. Brien).

### Options

`PRINT` = string tokens |
Controls printed output (`relationshipsmatrix` , `canonicalcomponentestimates` , `spectralcomponentestimates` , `nconstrainedcomponents` , `all` ); default `spec` |
---|---|

`VPRINT` = string tokens |
Controls the output from the final `REML` refit (`model` , `components` , `effects` , `means` , `stratumvariances` , `monitoring` , `vcovariance` , `deviance` , `Waldtests` , `missingvalues` , `covariancemodels` ); default `*` i.e. none |

`INITIALMETHOD` = string token |
Whether to use the estimates from the unconstrained fit as initial values in constrained fits or the default `REML` initial values (`remldefault` , `unconstrainedanalysis` ); default `unco` |

`MAXCYCLE` = scalar |
Sets a limit on the number of iterations in the `REML` analyses; default 30 |

`TOLERANCE` = scalar |
Tolerance for zero values; default 10^{-10} |

`DPRINT` = string tokens |
Controls output of diagnostic information (`spectralcomponents` , `canonicalcomponents` , `relationshipmatrix` , `all` ); default `*` i.e. none |

### Parameters

`Y` = variates |
Response variates |
---|---|

`CORRESPONDENCE` = matrices |
Upper-triangular matrix giving the spectral components in terms of the canonical components |

`SPECTRALESTIMATES` = variates |
Saves estimates of the spectral components |

`CANONICALESTIMATES` = variates |
Saves estimates of the canonical components |

`NCONSTRAINEDCOMPONENTS` = scalars |
Saves the number of spectral components constrained to zero, returns a missing value if some components could not be constrained |

`EXIT` = scalars |
Exit status of the final `REML` refit |

`SAVE` = REML save structures |
Supplies the save structure from the prior analysis of each `Y` variate; this need not be set, if that was the most recent `REML` analysis |

### Description

Randomization-based models, as described by Brien & Bailey (2006) and Bailey & Brien (2013), include the constraint that the spectral components are non-negative, even if the canonical components are allowed to be negative. While the estimates of the spectral components for two-tiered experiments are guaranteed to be non-negative, this not the case for multitiered experiments. `VSPECTRALCHECK`

forms estimates of the spectral components from the canonical components, or unconstrained variance components, that are estimated from fitting a mixed model using the `REML`

directive. It then checks for negative spectral components and, if any are found, imposes relationships between the canonical components so that the spectral components are constrained to be zero.

`VSPECTRALCHECK`

expects that a mixed model has been fitted using the `VCOMPONENTS`

and `REML`

directives only. It checks that the random model contains only gammas and σ^{2}, and that there are no spline models. In the random model (specified by the `RANDOM`

parameter of `VCOMPONENTS`

), the terms must be ordered so that, for each term, all the terms to which it is marginal follow it. All canonical components should be specified as unconstrained in the preceding `REML`

analysis (this being the default for the `VCOMPONENTS`

directive).

If `VSPECTRALCHECK`

detects a negative spectral component, it redefines the random model, specifying a matrix of constraints using the `RELATIONSHIP`

parameter of `VCOMPONENTS`

. It then refits the model using `REML`

. Because relationships are to be imposed between the canonical components, the standard Fisher-scoring algorithm (option `METHOD=fisher`

in `REML`

) must be used in the refits. The new estimates for the canonical components are extracted after the refit, and these are used to form new estimates of the spectral components. This process continues until all the spectral components are non-negative.

The `Y`

parameter specifies the variate that was analysed by the preceding `REML`

command. The `SAVE`

parameter can supply the corresponding `REML`

save structure; if this is not set, it is assumed that the y-variate is the one analysed in the most recent `REML`

analysis. A warning is given if the `Y`

variate seems to be different from that in the `SAVE`

structure.

The `CORRESPONDENCE`

parameter specifies a matrix giving coefficients of equations specifying the spectral components in terms of the canonical components. It must be a square, upper triangular matrix with rows corresponding to spectral components, and columns to canonical components. The rows and columns are considered to be in the same order as terms in the random model specified previously, by the `VCOMPONENTS`

directive. The upper triangular form implies that the terms in the random model must be ordered, so that each term occurs before any terms to which it is marginal. In particular, the unit term will be in the last row and column of the matrix. The element (*i*, *j*) of this matrix is non-zero if *j*≥*i*, and the term for row *i* is marginal to or equal to the term in column *j*; in this case, it is equal to the number of replicates

of a combination of the levels of the factors in the term in column *j* (see Bailey & Brien 2013, Equation 5).

The `SPECTRALESTIMATES`

and `CANONICALESTIMATES`

parameters save the constrained estimates of the spectral and canonical components, respectively, in variates. The `NOCONSTRAINEDCOMPONENTS`

parameter saves the number of constrained spectral components, in a scalar. The `EXIT`

parameter can specify a scalar to save the exit status of the final `REML`

fit.

Printed output is controlled by the `PRINT`

option with settings:

`relationshipsmatrix` |
to print the matrix of relationships imposed on the canonical components in the `REML` refits, |
---|---|

`canonicalcomponentestimates` |
to print the estimates of the canonical components under the imposed relationships, |

`spectralcomponentestimates` |
to print the estimates of the spectral components without and, if applicable, also with the constraints imposed, |

`noconstrainedcomponents` |
to print the number of constrained components, with missing values indicating that a constraint could not be imposed, and |

`all` |
to print all of the above. |

You can set the `VPRINT`

option to print information from the final `REML`

refit. This operates in the same way as the `PRINT`

option of `REML`

, except that the default is to print nothing. There is also a `DPRINT`

option to print diagnostic information.

The `INITIALMETHOD`

option control how the initial values are calculated for the `REML`

refits. By default, the estimates from the unconstrained fit are used as initial values for the refits. Alternatively, you can set `INITIALMETHOD=remldefault`

, to get `REML`

to form the initial values automatically, in the usual way.

The `MAXCYCLE`

option sets a limit on the number of iterations (default 30). The `TOLERANCE`

option specifies the tolerance for zero. This is used do determine whether a component is small enough to be considered zero, and in the checking of the `Y`

variate against that in the `SAVE`

structure.

Options: `PRINT`

, `VPRINT`

, `INITIALMETHOD`

, `MAXCYCLE`

, `TOLERANCE`

, `DPRINT`

.

Parameters: `Y`

, `CORRESPONDENCE`

, `SPECTRALESTIMATES`

, `CANONICALESTIMATES`

, `NCONSTRAINEDCOMPONENTS`

, `EXIT`

, `SAVE`

.

### Method

Estimates of the canonical components are obtained from a prior `REML`

analysis, and the estimates of the spectral components are obtained using the `CORRESPONDENCE`

matrix. If a spectral component is negative, then relationships between the canonical components, determined from the row in the `CORRESPONDENCE`

matrix for the spectral component, are imposed in a refit of the mixed model by the `REML`

directive. It is possible that some random terms may be removed from the mixed model. After `VSPECTRALCHECK`

has been run, the latest `REML`

analysis will be the one that `VSPECTRALCHECK`

has performed to constrain the components. So, for example, `VDISPLAY`

can be used to display additional information, and `VKEEP`

can be used to save information, in the usual way.

### References

Bailey, R. A. & Brien C. J. (2013). Randomization-based models for multitiered experiments. I. A chain of randomizations. arXiv preprint arXiv:1310.4132: 30.

Brien, C.J. (2015). Randomization inference for randomizations in a chain. Submitted for publication.

Brien, C.J. & Bailey, R.A. (2006). Multiple randomizations. *Journal of the Royal Statistical Society, Series B*, 68, 571-609.

Brien, C.J. & Payne, R.W. (1999). Tiers, structure formulae and the analysis of complicated experiments. *The Statistician*, 48, 41-52.

### See also

Procedure: `AMTIER`

.

Directives: `REML`

, `VCOMPONENTS`

.

Commands for: REML analysis of linear mixed models.

### Example

CAPTION 'VSPECTRALCHECK example','Example from Brien & Payne (1999).';\ STYLE=meta,plain SPLOAD [PRINT=*] '%gendir%/examples/Amtier.gsh' "Set up matrix that relates canonical to spectral components" TEXT spectral_lab; !t('R','RQ','RQC','RQCH','O','J',\ 'OI','OJ','OIS','OIJ','OISJ','OISJP') MATRIX [ROWS=spectral_lab; COLUMNS=spectral_lab] canonical2spectral;\ VALUES=!(192,96,24,12,0,0,0,0,0,0,0,0,\ 0,96,24,12,0,0,0,0,0,0,0,0,\ 0,0,24,12,0,0,0,0,0,0,0,0,\ 0,0,0,12,0,0,0,0,0,0,0,0,\ 0,0,0,0,288,0,96,48,24,16,4,1,\ 0,0,0,0,0,96,0,48,0,16,4,1,\ 0,0,0,0,0,0,96,0,24,16,4,1,\ 0,0,0,0,0,0,0,48,0,16,4,1,\ 0,0,0,0,0,0,0,0,24,0,4,1,\ 0,0,0,0,0,0,0,0,0,16,4,1,\ 0,0,0,0,0,0,0,0,0,0,4,1,\ 0,0,0,0,0,0,0,0,0,0,0,1) VCOMPONENTS [FIXED = Trellis*Method]\ RANDOM = (Rows*(Squares/Columns))/Halfplots - Squares/Columns+\ ((Occasions/Intervals/Sittings)*Judges)/Positions REML [MAXCYCLE=100; METHOD=Fisher; FMETHOD=none] Score VSPECTRALCHECK [PRINT=relation,spectralcomp,nconstr] Score;\ CORRESPONDENCE = canonical2spectral