Analysis of variance with an added Papadakis covariate, formed from neighbouring residuals (D.B. Baird).

### Options

`PRINT` = string tokens |
Output from the analysis of the y-variates, adjusted for covariates (`aovtable, information, covariates, effects, residuals, contrasts, means, cbeffects, cbmeans, stratumvariances, %cv, missingvalues);` default `aovt, info, cova, mean, miss` |

`PLOT` = string token |
Whether to plot the residuals against the average of neighbouring residuals (`residuals` ); default * i.e. no plot |

`NEIGHBOURS` = string token |
The neighbours whose residuals are averaged to form the residual covariate (`adjacent, rows, columns, all` ); default `adja` |

`TREATMENT STRUCTURE` = formula |
Defines the treatment structure of the model; default given by the most recent `TREATMENTSTRUCTURE` directive |

`BLOCK STRUCTURE` = formula |
Defines the blockings structure of the model; default given by the most recent `BLOCKSTRUCTURE` directive |

`COVARIATE` = variates |
Specifies any covariates in addition to the residual (Papadakis) covariate; default given by the most recent `COVARIATE` directive |

`FACTORIAL` = scalar |
Limit on number of factors in a treatment term; default 3 |

`CONTRASTS` = scalar |
Limit on the order of a contrast of a treatment term; default 4 |

`DEVIATIONS` = scalar |
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 |

`PSE` = string token |
Standard errors to be printed with tables of means, `PSE` =* requests s.e.’s to be omitted (`differences, lsd, means` ); default `diff` |

`LSDLEVEL` = scalar |
Significance level (%) to use in the calculation of least significant differences; default 5 |

### Parameters

`Y` = variates |
Variates to be analysed |

`ROWS` = factors or variates |
Factor giving the row location of each plot |

`COLUMNS` = factors or variates |
Factor giving the plot location of each plot |

`UNITS` = factors or variates |
Factor giving the plot location of each unit |

`RCOVARIATE` = variates |
Saves the covariate formed from the mean of the neighbouring residuals |

`TITLE` = texts |
Title for the graph; default i.e. title created from the `Y` variate name and the neighbouring plots that are used |

`WINDOW` = scalars |
Window number for the graph; default 3 |

`PEN` = scalars, variates or factors |
Pen number for the graph; default 1 |

`SCREEN` = string token |
Whether to clear the screen before plotting or to continue plotting on the old screen (`clear, keep` ); default `clea` |

### Description

The `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 `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

and `COVARIATE`

directives.

The `Y`

parameter lists the variates to be analysed. The `ROWS`

and `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.

The `NEIGHBOURS`

option controls which neighbours are averaged to form the residual (Papadakis) covariate. The settings `rows`

, `columns`

and `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.

The `PRINT`

option selects which components of output are to be displayed:

`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 `information`

and `missingvalues`

will not produce any output unless there is something definite to report.

The `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 `LSDLEVEL`

option.

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.

The `CONTRASTS`

option places a limit (by default 4) on the order of contrast to be fitted. (Contrasts are defined by using the functions `POL`

, `REG`

, `COMPARISON`

, `POLND`

or `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.

The `RCOVARIATE`

parameter saves the covariate formed from the neighbouring residuals. Other results from the analysis can be saved with the `AKEEP`

directive, as for the `ANOVA`

directive.

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.

Options: `PRINT, PLOT, NEIGHBOURS, TREATMENTSTRUCTURE, BLOCKSTRUCTURE, COVARIATE, FACTORIAL, CONTRASTS, DEVIATIONS, PSE, LSDLEVEL`

.

Parameters: `Y, ROWS, COLUMNS, UNITS, RCOVARIATE, TITLE, WINDOW, PEN, SCREEN`

.

### Action with `RESTRICT`

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.

### References

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.

### See also

Directives: `ANOVA`

, `BLOCKSTRUCTURE`

, `TREATMENTSTRUCTURE`

, `ADISPLAY`

, `AKEEP`

.

Procedures: `ACHECK`

, `AGRAPH`

, `APLOT`

, `AFIELDRESIDUALS`

, `APERMTEST`

, `AMCOMPARISON`

,

`ARESULTSUMMARY`

, `ASPREADSHEET`

.

Commands for: Analysis of variance, Design of experiments, REML analysis of linear mixed models.

### Example

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