Analyses an incomplete-block design by `REML`

, allowing automatic selection of random and spatial correlation models (R.W. Payne).

### Options

`PRINT` = string tokens |
Controls what summary output is produced about the models (`deviance` , `aic` , `bic` , `sic` , `dffixed` , `dfrandom` , `change` , `exit` , `best` , `description` ); default `best` , `desc` |
---|---|

`PBEST` = string tokens |
Controls the output from the `REML` analysis with the best model (`model` , `components` , `effects` , `means` , `stratumvariances` , `monitoring` , `vcovariance` , `deviance` , `Waldtests` , `missingvalues` , `covariancemodels` , `aic` , `sic` , `bic` ); default `*` i.e. none |

`PTRY` = string tokens |
Controls the output to present from the `REML` analysis used to try each model (`model` , `components` , `effects` , `means` , `stratumvariances` , `monitoring` , `vcovariance` , `deviance` , `Waldtests` , `missingvalues` , `covariancemodels` , `aic` , `sic` , `bic` ); default `*` i.e. none |

`FIXED` = formula |
Fixed model terms; default `*` i.e. none |

`RANDOM` = formula |
Additional random model terms; default `*` i.e. none |

`CONSTANT` = string token |
How to treat the constant term (`estimate` , `omit` ); default `esti` |

`FACTORIAL` = scalar |
Limit on the number of factors or covariates in each fixed term; default 3 |

`REPLICATES` = factor |
Replicate factor |

`BLOCKS` = factor |
Block factor; no default (must be specified) |

`ROWS` = factor |
Row factor for spatial analysis |

`COLUMNS` = factor |
Column factor for spatial analysis |

`ROWCOORDINATES` = variate or factor |
Row coordinates for fitting trends and spatial models if the design is irregular; if unset, these are defined from the levels of the `ROWS` factor |

`COLCOORDINATES` = variate or factor |
Column coordinates for fitting trends and spatial models if the design is irregular; if unset, these are defined from the levels of the `COLUMNS` factor |

`PLOTFACTOR` = factor |
Factor numbering the plots in the design; if unset, a local factor is defined automatically |

`PTERMS` = formula |
Terms (fixed or random) for which effects or means are to be printed; default `*` implies all the fixed terms |

`PSE` = string token |
Standard errors to be printed with tables of effects and means (`differences` , `estimates` , `alldifferences` , `allestimates` , `none` ); default `diff` |

`MVINCLUDE` = string tokens |
Whether to include units with missing values in the explanatory factors and variates and/or the y-variates (`explanatory` , `yvariate` ); default `*` i.e. omit units with missing values in either explanatory factors or variates or y-variates |

`VCONSTRAINTS` = string token |
Whether to constrain variance components to be positive (`none` , `positive` ); default `none` |

`RSTRATEGY` = string token |
Strategy for selecting the random model (`all` , `allfeasible` , `optimal` , `automatic` , `full` ); default `allf` |

`METHOD` = string token |
Criterion to choose the best random model (`aic` , `sic` , `bic` ); default `sic` |

`TRYSPATIAL` = string token |
Whether to try spatial models (`always` , `ifregular` ); default `*` i.e. no spatial models |

`TRYTRENDS` = string token |
Whether to see whether row and column trends are needed in the fixed model (`yes` , `no` ); default `no` |

`SPATIALFACTOR` = factor |
Factor to use to define the term for a two-dimensional power-distance model; if unset, a local factor is defined automatically |

### Parameters

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

`BESTMODEL` = pointers |
Saves a model-definition structure for the best model for each y-variate |

`EXIT` = scalars |
Exit status of the best model for each y-variate |

`SAVE` = REML save structures |
Save structure from the analysis of the best model for each y-variate |

### Description

`VABLOCKDESIGN`

analyses data from an incomplete-block design by `REML`

. An *incomplete-block design* is one where the blocks each have too few units to contain one of each of the treatments. In the context of a `REML`

analysis, the treatment factors are usually the fixed factors. So this is to say that the blocks are unable each to contain a unit with every combination of levels of the fixed factors.

Some designs are *resolvable*. The blocks can then be grouped together into subsets in which each treatment is replicated once. These groupings of blocks thus form replicates, which may be useful while the experiment is taking place. For example, if several operators are needed to make observations in a field trial, it is usual to get each one to observe the plots of a complete replicate. Then any operator differences will be included in the between-replicate variation, and will not add to the variability of the treatment estimates. Of course it can be useful to include a replicate factor even if the “replicates” are not exact, e.g. if some of the treatments do not occur at every level of the replicate factor.

The `RSTRATEGY`

option selects the strategy to use to determine the random model, with the following settings.

`all` |
fits the full random model (replicates and blocks within replicates if a replicate factor has been specified, or just blocks if there are no replicates). |
---|---|

`allfeasible` |
tries to fit the full random model. If this is not possible, it tries models removing one, and then two random terms, until successful. |

`optimal` |
tries all feasible random models. |

`full` |
synonym of `all` . |

`automatic` |
synonym of `optimal` . |

The full random model should reflect the way in which the treatments were randomized onto the experiment, so it is generally best to use this. The default of `RSTRATEGY=allfeasible`

, will do this if possible, or use a simpler random model if `REML`

is unable to fit the full model. Note: `VABLOCKDESIGN`

regards a model as successful, if the `REML`

directive returns an exit status of zero (i.e. successful fitting) and there are no bound or aliased variance parameters.

The `BLOCKS`

option must specify the block factor, and the replicate factor (if any) is specified by the `REPLICATES`

option. If you want to fit spatial covariance models, you must specify row and column factors, using by the `ROWS`

and `COLUMNS`

options respectively. If the replicates are adjacent to each other in the field and you want to fit spatial covariance models across the whole field, rather than within each replicate, you should define the levels of the row and column factors to run across the experiment. Otherwise they should be defined within replicates (i.e. using the same numbers within each replicate). The spatial models will then be fitted within replicates.

You can use the `ROWCOORDINATES`

and `COLCOORDINATES`

options to specify variates or factors giving the actual positions of the plots in the field. These are needed if you want to fit row or column trends (i.e. covariates) in the fixed model, or to fit a power-distance covariance model when the plots are on an irregular grid. If the levels of the `ROWS`

and `COLUMNS`

factors are defined across the whole experiment rather than within replicates, their values are used as defaults if `ROWCOORDINATES`

and `COLCOORDINATES`

are not set. Their values are also used as defaults if `ROWCOORDINATES`

or `COLCOORDINATES`

are set to variates or factors with no values; the variates or factors are then defined to contain those values.

The `PLOTFACTOR`

option allows you to specify a factor to index the plots, which is used to define the null random model (i.e. the one with no block or replicate effects), or to include a random term for measurement error when fitting covariance models. If this is not set, a local factor called `plots`

is set up automatically.

The `FIXED`

option specifies the fixed terms to be fitted in the analysis. The default fixed model consists of just the constant term, which then becomes the grand mean. The constant term can be omitted by setting option `CONSTANT=omit`

, provided a fixed model has been specified. The `FACTORIAL`

option sets a limit on the number of factors and variates allowed in each fixed term (default 3); any term containing more than that number is deleted from the model. The `RANDOM`

option allows you to you specify any extra random terms to include (in addition to replicates and blocks-within-replicates). The `VCONSTRAINTS`

option allows you to constrain the variance components to be positive; by default they are not constrained.

The `TRYSPATIAL`

option indicates whether to try fitting spatial models, with settings:

`always` |
always tries to fit them, |
---|---|

`ifregular` |
fit them only if the plots are on a regular grid. |

With the default, `TRYSPATIAL=*`

, no spatial models are fitted. For a regular grid, `VAROWCOLUMNDESIGN`

tries models with order 1 auto-regressive structures on the rows and/or the columns of the design, provided there are more than four rows or columns, respectively. For an irregular grid, if there are more than four rows and more four columns, it tries an anisotropic power-distance model using city-block distance. Otherwise, if there is only one dimension with more than four coordinates, it tries an isotropic power-distance model.

The `SPATIALFACTOR`

option allows you to specify a factor to use to define the term required for a two-dimensional power-distance model. If this is not set, a local factor called `RowColumn2d`

is used.

You can set option `TRYTRENDS=yes`

to see whether row and column trends (i.e. covariates) are needed in the fixed model. By default this is not done.

The `MVINCLUDE`

option controls whether units with missing values in the explanatory factors and variates and/or the y-variate are included in the analysis, as in the `REML`

directive.

The `METHOD`

option specifies how to assess the random (and spatial) models

`aic` |
uses their Akaike information coefficients, |
---|---|

`sic` or `bic` |
uses their Schwarz (Bayesian) information coefficients (default). |

The `PRINT`

option specifies the summary output to be produced about the models. The settings are mainly the same as those of the `VRACCUMULATE`

procedure (which is used to store and then print details of the analyses). There is an extra setting, `description`

, to provide a description of the model and strategy. There is also a setting, `best`

, to print the description of the best model. By default, `PRINT=best,description`

.

The `PBEST`

option specifies the output to be produced from the `REML`

analysis with the best model. Similarly, the `PTRY`

option indicates what output should be produced for each candidate random model when it is tried. Their settings are mainly the same as those of the `PRINT`

option of the `REML`

directive. There are also extra settings `aic`

and `sic`

(with a synonym `bic`

) to print the Akaike and Schwarz (Bayesian) information coefficients, respectively. The default for both these options is to produce no output.

The `PTERMS`

option operates as in `REML`

, to specify the terms whose means and effects are printed by `PBEST`

and `PTRY`

; the default is all the fixed terms. Likewise, the `PSE`

option controls the type of standard error that is displayed with the means and effects; the default is to give a summary of the standard errors of differences.

The `Y`

parameter specifies the response variate. A model-definition structure for the best model can be saved, in a pointer, by the `BESTMODEL`

parameter; the `VMODEL`

procedure can use this to define the model (using the `VCOMPONENTS`

and `VSTRUCTURE`

directives) so that you can reanalyse it yourself using the `REML`

directive. Alternatively, you can save the `REML`

save structure from the analysis with the best model using the `SAVE`

parameter. The `EXIT`

parameter allows you to save a code from `REML`

, giving the “exit status” of the fit (zero if successful).

Options: `PRINT`

, `PBEST`

, `PTRY`

, `FIXED`

, `RANDOM`

, `CONSTANT`

, `FACTORIAL`

, `REPLICATES`

, `BLOCKS`

, `ROWS`

, `COLUMNS`

, `ROWCOORDINATES`

, `COLCOORDINATES`

, `PLOTFACTOR`

, `PTERMS`

, `PSE`

, `MVINCLUDE`

, `VCONSTRAINTS`

, `RSTRATEGY`

, `METHOD`

, `TRYSPATIAL`

, `TRYTRENDS`

, `SPATIALFACTOR`

.

Parameters: `Y`

, `BESTMODEL`

, `EXIT`

, `SAVE`

.

### Method

Model definition structures are defined for the various candidate models. (Run the example to see those that are considered for a resolvable block design.) The `VARANDOM`

procedure is used to fit them, with the `VRACCUMULATE`

procedure storing the necessary details for the best one to be selected.

### See also

Directives: `REML`

, `VCOMPONENTS`

, `VSTRUCTURE`

.

Procedures: `VAOPTIONS`

, `VARANDOM`

, `VARECOVER`

, `VAROWCOLUMNDESIGN`

, `VASERIES`

, `VALINEBYTESTER`

, `VFMODEL`

, `VFSTRUCTURE`

.

Commands for: REML analysis of linear mixed models.

### Example

CAPTION 'VABLOCKDESIGN example',\ !t('Resolvable incomplete-block design',\ '(5x5 simple lattice from Cochran & Cox 1957, page 406).');\ STYLE=meta,plain FACTOR [NVALUES=50; LEVELS=2] Replicates & [LEVELS=5] Blocks,Plots,A,B & [LEVELS=25; VALUES=\ (1...25),(1,6...21),(2,7...22),(3,8...23),(4,9...24),(5,10...25)] Variety GENERATE Replicates,Blocks,Plots & [TREATMENT=Variety; REPLICATES=Replicates; BLOCKS=Blocks] A,B READ Yield 6 7 5 8 6 16 12 12 13 8 17 7 7 9 14 18 16 13 13 14 14 15 11 14 14 24 13 24 11 8 21 11 14 11 23 16 4 12 12 12 17 10 30 9 23 15 15 22 16 19: VABLOCKDESIGN [PRINT=best,description,deviance,aic,bic,dfrandom; PTRY=*;\ FIXED=Variety; REPLICATES=Replicates; BLOCKS=Blocks;\ RSTRATEGY=optimal] Y=Yield; BESTMODEL=bestmodel; SAVE=savebest VDISPLAY [PRINT=model,components,wald] savebest