Analyses a field trial using the Linear Variance Neighbour model (D.B. Baird).
|Controls printed output (
||Indicates which version of the LV model to use (
||Number between 0 and 1 which defines the value for the variance parameter λ (if
||Specifies which estimator of residual variance to use to calculate the sed’s of treatment effects (
||Defines the precision to which the variance parameter λ should be estimated; default 0.01|
||Y-values (usually plot yields) row by row|
||Treatment factor for each y-variate|
||Block factor, defining groups of plots to be de-trended independently|
||Unit-within-block factor, defining the order of plots within each block|
||Saves the estimated treatment effects from each analysis|
||Saves the estimated standard errors of differences between treatments|
||Saves the estimated white noise component|
||Saves the estimated trend component|
||Saves the estimated variance components: the tuning parameter λ, and either the variance of the random walk innovations (λ<0.9) or the white noise variance (λ≥0.9)|
LVARMODEL analyses a field trial, whose plots are in lines of equal length, using the Linear Variance (LV) Neighbour analysis (Williams 1986). The LV model is equivalent to the extended First Difference model of Besag & Kempton (1986). The model allows for local trends within a row, and the analysis attempts to remove these trends by using a form of smoothing. In the full LV model, the degree of smoothing is estimated from the data; alternatively the reduced model, corresponding to the ordinary First Difference (FD) model of Besag & Kempton (1986), applies a full linear de-trending to the data.
The LV model specifies the data as the sum of three components: the treatment effects, a trend component which is a random walk process, and a residual white noise component. The full Linear Variance plus Incomplete Block model of Williams (1986) has an additional random component for incomplete blocks, These can be fitted as a fixed effect, by treating each block as a separate line of plots.
The variable to be analysed (normally a plot yield) is specified in a variate, using the
Y parameter. The factor defining the treatments on the plots is specified using the
TREATMENTS parameter. The
BLOCKS parameter specifies the block factor, defining the groups of plots that are to be de-trended separately, and the
UNITS parameter specifies the units-within-blocks factor defining the order of the plots within each block. For example, if the plots are on a rectangular grid and trends are to be removed along rows, the
UNITS factors would be the row and column factors, respectively. If
UNITS are not set, the plots are assumed to be in a single line (and specified sequentially down the line). The procedure can handle missing values in the y-variate but not in the
The other parameters allow information to be saved from the analysis:
EFFECTS for the table of estimated treatment effects;
SED for the standard errors of differences between treatments effects (in either a matrix or a symmetric matrix);
WNOISE for the estimated white noise (in a variate);
TREND for trend component (in a variate); and
COMPONENTS for the two variance parameters. The first variance component is the parameter λ. For λ<0.9 the second component is the variance of the innovations in the random walk. If λ≥0.9 the second component saved is the variance of the white noise component, as the random walk component disappears in the limit as λ tends to one.
Printed output is controlled by the
data – y-values and treatments in a tabular form;
effects estimated treatment effects;
sed standard errors of differences of effects;
variance estimates of λ and the white noise variance; and
residuals trend and white noise components.
METHOD option controls the form of LV model to be fitted. By default setting of
full causes the full LV model to be fitted, with the variance parameters of the model estimated by Residual Maximum Likelihood (REML); see Gleeson & Cullis (1987). The variance parameters used, λ and κ, are those given by Baird and Mead (1991). The parameter λ is known as the tuning parameter, as it controls the degree of smoothing used in eliminating trend effects from the data. It is related to the parameter α of Besag & Kempton (1986), by the relationship
λ = α / (1 + α)
METHOD=reduced fits the reduced form of the LV model, that is the FD model. This is equivalent to putting λ = 0.
LAMBDA allows the value of the tuning parameter to be set at a fixed value, which must lie between 0 and 1. By default
LAMBDA=0, which for
METHOD=full causes the value to be estimated as described above.
VARMETHOD controls the estimator used for the estimating the variance of the residual white noise component. There are two possibilities: the normal generalized least-squares estimator
GLS, and an estimator based on the second differences of the errors
RMS2 (Besag & Kempton 1986). The simulation study of Baird & Mead (1991) showed the standard errors of treatment effects based on
RMS2 to be approximately valid under randomization for a wide range of error models. When the estimated value of λ was not close to zero, the standard errors based on
GLS were found to be approximately unbiased and more efficient than those based on
RMS2 for the LV model. However the standard errors based on
GLS could be seriously biased in some situations for the FD model or when λ was close to zero. Thus the default for
TOLERANCE option specifies the precision to which λ should be estimated.
The model is fitted in a similar manner to that outlined in Besag & Kempton (1986), but the variance components have the parameterization used by Baird & Mead (1991) and are fitted by residual maximum likelihood (Gleeson & Cullis 1987) rather than maximum likelihood; also see Baird (1987). The optimization of the likelihood is done by golden section search on the profile likelihood for λ. Residuals are constructed by creating the smoothing matrix S that corresponds to the LV model fitted (Green et al. 1985).
The procedure uses a large amount of data space and computer time when the tuning parameter is estimated by REML. The speed is proportional to the number of rows multiplied by the square of the numbers of columns.
The procedure ignores any restrictions, for example, on
Baird, D.B. (1987). A Genstat 5 procedure for a First Difference analysis. Genstat Newsletter, 19, 40-47.
Baird, D.B. & Mead, R. (1991). The empirical efficiency and validity of two neighbour models. Biometrics, 47, 1473-1487.
Besag, J.E. & Kempton R.A. (1986). Statistical analysis of field experiments using neighbouring plots. Biometrics, 42, 231-251.
Gleeson, A.C. & Cullis, B.R. (1987). Residual maximum likelihood estimation of a neighbour model for field experiments. Biometrics, 43, 277-288.
Green, P.J., Jennison, C. & Seheult. A.H. (1985). Analysis of field experiments by least squares smoothing. Journal of the Royal Statistical Society, Series B, 47, 299-315.
Williams, E.R. (1986). A neighbour model for field experiments. Biometrika, 73, 279-287.
Commands for: REML analysis of linear mixed models.
CAPTION 'LVARMODEL example',!t(\ 'The data (Jenkyn et al. 1979, Annals of Applied Biology, pp.',\ '11-28) are a series of Barley yields from a row of 38 plots',\ 'which had four fungicide spray treatments applied',\ '(0 = None, 1 = One spray, 2 = Two sprays, R = Repeated sprays).');\ STYLE=meta,plain VARIATE [NVALUES=38] Yield; VALUES=\ !( 5.77,5.73,6.08,5.26,5.89,5.37,5.95,5.95,5.59,5.16,5.89,6.14,6.01,\ 5.63,5.39,5.46,5.05,5.76,5.23,6.20,6.26,6.48,5.79,6.45,6.44,6.31,\ 6.18,6.43,5.82,6.47,5.73,6.54,5.99,5.76,5.04,4.38,5.06,5.13 ) FACTOR [Labels=!T('0','1','2','R');NVALUES=38] Treat; VALUES=\ !( 4, 3, 4, 1, 2, 1, 3, 4, 2, 1, 2, 3, 4,\ 3, 2, 4, 1, 4, 1, 3, 2, 3, 1, 2, 4, 2,\ 4, 3, 1, 3, 1, 4, 2, 3, 2, 1, 4, 3 ) CAPTION 'Fit the Reduced LV model - First Difference model.' LVARMODEL [METHOD=REDUCED] Yield; TREATMENTS=Treat CAPTION 'Fit the Full LV model with precision of estimation for lambda 0.005' LVARMODEL [TOLERANCE=0.005] Yield; TREATMENTS=Treat