Defines the variance-components model for
FIXED = formula
|Fixed model terms; default
ABSORB = factor
|Defines the absorbing factor (appropriate only when
* i.e. none
CONSTANT = string token
|How to treat the constant term (
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 (
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
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 (
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
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
Variety are factors indicating the treatments applied to each unit, 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.
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.
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
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
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
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.
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.
Commands for: REML analysis of linear mixed models.
" 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