Analysis of variance with an added Papadakis covariate, formed from neighbouring residuals (D.B. Baird).
|Output from the analysis of the y-variates, adjusted for covariates (
|Whether to plot the residuals against the average of neighbouring residuals (
|The neighbours whose residuals are averaged to form the residual covariate (
||Defines the treatment structure of the model; default given by the most recent
||Defines the blockings structure of the model; default given by the most recent
|Specifies any covariates in addition to the residual (Papadakis) covariate; default given by the most recent
|Limit on number of factors in a treatment term; default 3|
|Limit on the order of a contrast of a treatment term; default 4|
|Limit on the number of factors in a treatment term for the deviations from its fitted contrasts to be retained in the model; default 9|
|Standard errors to be printed with tables of means,
|Significance level (%) to use in the calculation of least significant differences; default 5|
||Variates to be analysed|
||Factor giving the row location of each plot|
||Factor giving the plot location of each plot|
||Factor giving the plot location of each unit|
||Saves the covariate formed from the mean of the neighbouring residuals|
||Title for the graph; default i.e. title created from the
||Window number for the graph; default 3|
||Pen number for the graph; default 1|
||Whether to clear the screen before plotting or to continue plotting on the old screen (
APAPADAKIS procedure analyses balanced designs with an added covariate formed from the neighbouring residuals from the initial analysis of variance (Papadakis 1937, Bartlett 1938, Wilkinson et al. 1983). This method was the first and simplest nearest-neighbour adjustment for removing the effects of spatial trends within a trial. If there is a smooth trend in the trial, the plot’s residual will be correlated with the neighbouring plots’ residuals. Fitting the average of the neighbouring residuals as a covariate can then adjust the treatment means for the trend and reduce their standard errors. This technique has been superceded by spatial
REML analyses, but may still be useful for comparison.
The model to be fitted in the analysis has three parts. The
TREATMENTSTRUCTURE specifies the treatment (or systematic, or fixed) terms for the analysis. The
BLOCKSTRUCTURE defines the “underlying structure” of the design or, equivalently, the error terms for the analysis; in the simple cases where there is only a single error term this can be omitted. The
COVARIATE option specifies any covariates to be included, in addition to the residual (Papadakis) covariate. These can be specified as options in the procedure, or defined by previous
Y parameter lists the variates to be analysed. The
COLUMNS parameters can define the 2-dimensional spatial layout of the design. Alternatively, the
UNITS parameter defines a 1-dimensional spatial layout. If
UNITS is not specified for a 1-dimensional layout,
APAPADAKIS assumes (with a warning) that the y-values are in plot order.
NEIGHBOURS option controls which neighbours are averaged to form the residual (Papadakis) covariate. The settings
all require a 2-dimensional layout. The neighbours for
rows are the two plots on either side in the same row, for
columns they are the two plots on either side in the same column, for
adjacent they are the 4 plots with an edge in common, and for
all they are the eight plots with a side or corner in common. For a 1-dimensional layout,
adjacent is the only relevant setting. This uses the plots on either side of the given plot as neighbours. Note: edge plots will have fewer neighbours.
aovtable analysis-of-variance table;
information information summary, giving details of aliasing and non-orthogonality or of any large residuals;
covariates estimates of covariate regression coefficients;
effects tables of estimated treatment parameters;
residuals tables of estimated residuals;
contrasts estimated contrasts of treatment effects;
means tables of predicted means for treatment terms;
cbeffects estimated effects of treatment terms combining information from all the strata in which each term is estimated;
cbmeans predicted means for treatment terms combining information from all the strata in which each term is estimated;
stratumvariances estimated variances of the units in each stratum and stratum variance components;
%cv coefficients of variation and standard errors of individual units; and
missingvalues estimates of missing values.
The default is intended to give the output that you will require most often from a full analysis:
aovtable, information, covariates, means and
missingvalues. However, as with
ANOVA, the settings
missingvalues will not produce any output unless there is something definite to report.
PSE option controls the standard errors printed with the tables of means. The default setting is
differences, which gives standard errors of differences of means. The setting
means produces standard errors of means,
LSD produces least significant differences, and you can suppress the standard errors altogether by setting
PSE=*. The significance level to use for calculating the least significant differences can be changed from the default of 5% with the
The treatment terms to be included in the model are controlled by the
FACTORIAL option. This sets a limit (by default 3) on the number of factors in a treatment term. Terms containing more than that number are deleted.
CONTRASTS option places a limit (by default 4) on the order of contrast to be fitted. (Contrasts are defined by using the functions
REGND in the treatment formula.) For a term involving a single factor, the orders of the successive contrasts run from one upwards, with the deviations term (if any) numbered highest. In interactions between contrasts, the order is the sum of the orders of the component parts.
If your design has few or no degrees of freedom for the residual, you may wish to regard the deviations from some of the fitted contrasts as error components, and assign them to the residual of the stratum where they occur. You can do this by the
DEVIATIONS option; its value sets a limit on the number of factors in the terms whose deviations are to be retained in the model. For example, by putting
DEVIATIONS=1, the deviations from the contrasts fitted to all terms except main effects will be assigned to error. When deviations have been assigned to error, they will not be included in the calculation of tables of means, which will then be labelled “smoothed”. However the associated standard errors of the means are not adjusted for the smoothing.
You can set option
PLOT=residuals to plot the residuals against the average of neighbouring residuals. The
TITLE parameter gives the title for the graph; if this is this not set, an automatic title will be created from the
Y variate name and the neighbouring plots that are used. The
WINDOW parameter defines the window in which the graph is drawn (default 3). The
PEN parameter specifies the pen to use (default 1). Finally, the
SCREEN parameter controls whether the graphical display is cleared before the graph is plotted.
PRINT, PLOT, NEIGHBOURS, TREATMENTSTRUCTURE, BLOCKSTRUCTURE, COVARIATE, FACTORIAL, CONTRASTS, DEVIATIONS, PSE, LSDLEVEL.
Y, ROWS, COLUMNS, UNITS, RCOVARIATE, TITLE, WINDOW, PEN, SCREEN.
You can restrict the set of units used for the analysis by applying a restriction to any of the y-variates. Only these units are included in the analysis of each y-variate.
Bartlett, M.S. (1938). The approximate recovery of information from replicated field experiments with large blocks. Journal of Agricultural Science, 28, 418-427.
Papadakis, J.S. (1937). Méthode statistique pour les expériences en champ. Bulletin Institute de L’Ameloration Des Plantes à Salonique, 23.
Wilkinson, G.N., Eckert, S.R., Hancock, T.W. and Mayo O. (1983). Nearest neighbour (NN) analysis of field experiments. Journal of the Royal Statistical Society B, 45, 151-178.
Commands for: Analysis of variance, Design of experiments, REML analysis of linear mixed models.
CAPTION 'APAPADAKIS example',\ !t('Study of 25 wheat cultivars at Slatehall farm',\ 'in a 5x5 lattice square'); STYLE=meta,plain SPLOAD [PRINT=*] '%Data%/Slatehall.gsh' "Simple randomized block analysis" TREATMENTS variety BLOCKS replicates ANOVA [FPROBABILITY=yes; PSE=lsd] yield PEN 5...8; SYMBOL='circle','square','triangle','star'; \ COLOUR='red','darkgreen','blue','black'; CFILL='match' FOR NN='Row','Column','Adjacent','All'; W=5...8; screen='clear',3('keep') TXCONSTRUCT [TEXT=Title] NN,' neighbours' APAPADAKIS [PRINT=*; NEIGHBOURS=#NN; PLOT=residual] \ yield; ROWS=fieldrow; COLUMNS=fieldcolumn; \ TITLE=Title; WINDOW=W; PEN=W; SCREEN=#screen ENDFOR APAPADAKIS [NEIGHBOURS=rows; PSE=lsd] yield; \ ROWS=fieldrow; COLUMNS=fieldcolumn "Lattice square analysis - does better than Padadakis covariate" POINTER [NVALUES=NLEVELS(replicates)] PFrow,PFcol FACTOR [LEVELS=6; NVALUES=NVALUES(replicates)] PFrow,PFcol GENERATE [TREATMENTS=variety; REPLICATES=replicates; BLOCKS=rows] PFrow & [BLOCKS=columns] PFcol FDISTINCTFACTORS PFrow; SET2=PFcol; DISTINCTSET=PF POINTER [RENAME=yes] PF; VALUES=PF BLOCKS replicates/(rows*columns) TREATMENTS variety//PF ANOVA [FPROBABILITY=yes; PSE=lsd; LSDLEVEL=5] yield