Defines the variance-components model for `REML`

.

### Options

`FIXED` = formula |
Fixed model terms; default `*` |
---|---|

`ABSORB` = factor |
Defines the absorbing factor (appropriate only when `REML` option `METHOD=Fisher` ); 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 |

`CADJUST` = string token |
What adjustment to make to covariates before analysis (`mean` , `none` ); default `mean` |

`RELATIONSHIP` = matrix |
Defines relationships constraining the values of the components; default `*` |

`SPLINE` = formula |
Defines random cubic spline terms to be generated: each term must contain only one variate, if there is more than one factor in a term, separate splines are calculated for each combination of levels of the factors |

`EXPERIMENTS` = factor |
Factor defining the different experiments in a multi-experiment (meta-) analysis |

### Parameters

`RANDOM` = formula |
Random model terms and the residual variance |
---|---|

`INITIAL` = scalars |
Initial values for each component |

`CONSTRAINTS` = string tokens |
How to constrain each variance component and the residual variance (`none` , `positive` , `fixrelative` , `fixabsolute` ); default `none` |

### Description

The `VCOMPONENTS`

directive specifies the linear mixed model to be fitted by subsequent `REML`

statements. There are usually two parts of the mixed model to be defined, namely the fixed and random model terms. In addition, it is possible to specify terms to generate cubic splines.

Random effects are used to describe the effects of factors where the values present in the experiment can be considered as a random selection of values from some large homogeneous population. Inference about this population can then be made, for example estimation of its variance. Predictions of random effects may also be of interest. Fixed effects are used to describe treatments imposed in an experiment where the effect of those specific choices of treatment are of interest. Cubic spline terms can be included to investigate smooth non-linear departures from the model specified.

For example, consider a split-plot experiment used to assess the effects on yield of three oat varieties with four levels of nitrogen application. Here specific levels of nitrogen application have been used and the aim is to estimate the effects of these levels; so they would be considered as fixed effects in the model, as would the three oat varieties. However, the effects of the actual blocks and plots in the experiment are not of interest in themselves, but they do provide a means of estimating the variability of the more general population of blocks and plots in order to get an estimate of background variation against which to compare the fixed effects. Blocks and plots would therefore be defined as random effects. In this case, the fixed effects correspond to the effects used as treatments in `ANOVA`

and the random effects would correspond to the blocking factors in `ANOVA`

.

In general, both the fixed and random parts of the model are constructed from several factors or variates. The structure of both parts is specified using model formulae and can contain both factors and variates with the usual adding, crossing or nesting operators.

The fixed terms in the model are defined by a model formula supplied using the `FIXED`

option, and the random model terms are defined by a model formula supplied by the `RANDOM`

parameter. Thus, for example, the model for the split-plot experiment described above would be specified by

`VCOMPONENTS [FIXED=Nitrogen*Variety]\`

` RANDOM=Block/Wplot/Subplot`

where `Nitrogen`

and `Variety`

are factors indicating the treatments applied to each unit, and `Block`

, `Wplot`

and `Subplot`

are factors indicating the block, wholeplot (within block) and subplot (within wholeplot) to which each unit belongs.

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. If the random model is unset, only a single source of variation (the residual component called `*units*`

) is used.

When covariates are included in the fixed or random models, by default they are automatically centred before analysis. However, you can set option `CADJUST=none`

to specify that the uncentred covariates are to be used instead.

The `FACTORIAL`

option is used to set a limit on the number of factors and variates allowed in each fixed term; any term containing more than that number is deleted from the model.

The `SPLINE`

option can be used to generate cubic spline terms to be fitted as part of the random model. The smoothing parameter is estimated by REML and the fitted spline is interpreted as a BLUP (best linear unbiased predictor). If a term consists of a single variate, for example `SPLINE=X`

, a cubic spline will be generated using all distinct covariate values present as knots, with weighting for replicate points. If factors are included, for example `SPLINE=N.V.X`

where `N`

and `V`

are factors, separate cubic splines will be generated for each level of the combined factor `N.V`

. The knot points for the splines will be generated as the set of distinct values in `X`

, and the same knot points will be used at each level of `N.V`

.

The `EXPERIMENTS`

option is used when a combined (or *meta*-) analysis is being defined over several experiments. The setting is a factor identifying the experiment to which each unit belonged. When this option has been set, a different residual term will be set up for each level of the `EXPERIMENTS`

factor. In the simplest case, this means that a different residual variance will be used for each experiment. For more complex cases, different correlated error models can be applied to separate experiments using the `VRESIDUAL`

directive. In either case, the factors and variates for the separate experiments should be concatenated into structures which run over all the experiments. For example, consider an experiment set up at two sites to compare a set of 24 varieties in four replicates. In one site the experiment was laid out as a grid of eight rows by 12 columns, in the other a grid of 16 rows by six columns was used. In these circumstances, a single set of factors (of length 192) can be used to specify the design, using factors to describe variety, rows and columns, plus a factor `expt`

defining the allocation of units to experiments. Note that the factor `row`

will have 16 levels and `col`

will have 12 levels. The restriction of site 1 to 8 rows and site 2 to 6 columns is specified using the `VRESIDUAL`

directive. Where some factors differ between experiments, these should be defined on the units relevant to the appropriate experiment(s) and missing elsewhere.

For random terms, initial values for the ratio of variance components to the error variance (the gamma ratios) are supplied using the `INITIAL`

parameter, and you can impose general constraints on the variance components using the `CONSTRAINTS`

parameter, or equality constraints between components using the `RELATIONSHIP`

option. By default, all the gamma ratios have initial values of one. The `CONSTRAINTS`

parameter can request that any variance component should be held positive or fixed at its initial value. The default setting, `none`

, allows the variance components to become negative, provided the overall estimated variance-covariance matrix for the data remains positive definite. The `RELATIONSHIP`

option can be used to define linear relationships between the variance components, for example that component A should be constrained to be twice component B.

Covariance models for random terms, including unknown parameters to be estimated, can be specified using the `VSTRUCTURE`

directive.

The `ABSORB`

option allows you to specify a factor from either the fixed or the random model to act as an absorbing factor for the model. Note that the absorbing factor is ignored for the AI algorithm with sparse matrix methods: that is, this option is relevant only when the `METHOD`

option of `REML`

is set to `Fisher`

. The absorbing factor is used to divide the model terms into two groups; this partition is then used in calculations during the fitting process to reduce the size of the matrices that have to be inverted and stored. Use of an absorbing factor can therefore save computing time and data space. However, although exactly the same model is fitted when an absorbing factor is used, some of the standard errors are unavailable. A good choice of absorbing factor might be a factor with a large number of levels, or any factor whose effects and standard errors are not of interest.

Options: `FIXED`

, `ABSORB`

, `CONSTANT`

, `FACTORIAL`

, `CADJUST`

, `RELATIONSHIP`

, `SPLINE`

, `EXPERIMENTS`

.

Parameters: `RANDOM`

, `INITIAL`

, `CONSTRAINTS`

.

### Action with `RESTRICT`

You can restrict the set of units to be included in the analysis by restricting any of the factors or variates in the fixed and random models defined by `VCOMPONENTS`

, or by restricting any of the y-variates in the subsequent `REML`

statement. However, if more than one of these vectors is restricted, all must be restricted to the same set of units.

### See also

Directives: `REML`

, `VSTRUCTURE`

, `VRESIDUAL`

, `VPEDIGREE`

, `VSTATUS`

.

Procedures: `FCONTRASTS`

, `FDIALLEL`

, `VFMODEL`

, `VRACCUMULATE`

, `VRMETAMODEL`

.

Commands for: REML analysis of linear mixed models.

### Example

" Example REML-1: Incomplete block analysis (from Cochran & Cox, Table 10.3, p. 406) " FILEREAD [NAME='%gendir%/examples/REML-1.DAT']\ Reps,Blocks,Treats,Yield; FGROUPS=3(yes),no VCOMPONENTS [FIXED=Treats] RANDOM=Reps/Blocks CALCULATE Log = LOG(Yield) REML [PRINT=components,effects,means,stratum,vcov,monitor; PSE=diff;\ PTERMS=Reps/Blocks; METHOD=Fisher; MAXCYCLE=2] Yield,Log; SAVE=Syield,Slog VAIC " Plot residuals for each analysis" VPLOT [GRAPHICS=high; SAVE=Syield] fitted VPLOT [GRAPHICS=high; SAVE=Slog] fitted