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AMTIER procedure

Analyses a multitiered design by an analysis of variance specified by up to three model formulae (C.J. Brien & R.W. Payne).


PRINT = string tokens Controls printed output from the analysis (aovtable, aovpseudotable, design, effects, fittedvalues); default aovt
F1 = formula First model formula
F2 = formula Second model formula
F3 = formula Third model formula
FACTORIAL = scalar Limit on the number of factors in a model term
F2BALANCETYPE = string token Type of balance required for F2 (orthogonal, firstorder); default orth
F3BALANCETYPE = string token Type of balance required for F3 (orthogonal, firstorder); default orth
PSEUDOTERMS = formula structures Specifies pseudo-terms for terms in the F1, F2 or F3 formulae
DESIGN = tree Saves or specifies details of the design and analysis
SEED = scalar Seed for random numbers to generate dummy variate for determining the design; default 13579
TOLERANCE = variate Tolerance for zero sweeps in dummy and y-variate analyses
DPRINT = string tokens Controls debug output (setup, analysis, dummyanalysis); default * i.e. none


Y = variates Each of these contains the data values for an analysis
RESIDUALS = variates Saves the residuals from each analysis
FITTEDVALUES = variates Saves the fitted values from each analysis
SAVE = pointers Save structure for each analysis (to use in AMTDISPLAY)


Genstat users are accustomed to the idea that more than one model formula may be required to specify an analysis of variance. For the ANOVA directive, the underlying structure of the data (which indicates the error terms for the analysis) is defined by a model formula specified by the BLOCKSTRUCTURE directive, while the treatment terms to be fitted in the analysis are defined in a model formula specified by the TREATMENTSTRUCTURE directive. However, experiments that involve multiple randomizations (Brien & Bailey 2006), such as two-phase experiments, may require more than two model formulae to define their analysis correctly (Brien & Payne 1999).

For example, Brien (1983) considers a two-phase experiment set up to evaluate a set of wines. These are evaluated at a tasting where several tasters are given the wines over a number of sittings. One wine is presented to each taster at a sitting, and each wine is evaluated only once by each taster. The order of presentation of the wines is randomized for each taster. The basic observational unit is a glass of wine presented to a particular taster in the tasting phase. These have a structure of tasters/sittings. If this phase represented the whole experiment, tasters/sittings would be the block formula, and the treatment formula would be the factor wines. So we would have

BLOCKSTRUCTURE tasters/sittings


Now suppose that the wines were produced from a field experiment and, in fact, that each one was produced from one of the plots of a randomized-block design. The second model formula would then be blocks/plots, and the final formula would be treatments (the factor identifying the treatments applied in the field).

AMTIER can analyse designs requiring up to three model formulae. It can thus analyse any design with three tiers, and also some with more than three; see, for example, the corn experiment in Brien & Bailey (2006, example 12). The formulae are specified by the options F1, F2 and F3, which must not contain the pseudofactorial operator. For the example in Brien (1983), the statement would be

AMTIER [F1=tasters/sittings; F2=blocks/plots;\

       F3=treatments] Y

The FACTORIAL option sets a limit on the number of factor in the model terms generated from the formulae.

The Y parameter specifies the response variate. Residuals and fitted values can be saved by the RESIDUALS and FITTEDVALUES parameters, respectively. The SAVE parameter can save a pointer containing the full details of the analysis. This can be used as input to the AMTDISPLAY procedure to obtain further output.

The F2BALANCETYPE and F3BALANCETYPE options control whether the terms from the second and third model formulae are allowed to be first-order balanced rather than orthogonal (Brien & Bailey 2007). The default is that the terms are required to be orthogonal. It is emphasized that this applies only to terms from the same model formula. Even if the terms from a model formula are required to be orthogonal, they may still only be structure balanced in relation to terms from other formulae. However, if terms from any model formula are non-orthogonal, then the experiment is not structure balanced (Brien & Bailey 2007) and so sums of squares for sources differ depending on their order in the model formula.

The PSEUDOTERMS option allows you to specify a list of formula structures defining pseudo-terms for some of the terms in the formulae. Each pseudoterm formula is of the form

group_term // pseudoterms_formula

All pseudo-terms must be defined explicitly as none are generated, for example from relations between the group term and other factors. Furthermore, all marginal terms to a pseudoterm need to be included in its formula, irrespective of whether they themselves are pseudoterms. Those that are not pseudo-terms need to occur in one of the three main model formulae and will not be included in the analysis sequence again as a result of their appearance in the pseudo-term formula. The pseudo-terms are placed immediately before the group term in the analysis sequence. Any repetitions of pseudo-terms are removed.

The DESIGN option can save a tree structure representing the design and analysis. You can then specify this as the design in a subsequent AMTIER statement, to avoid having to go through the process of determining the design structure with another response variate from the same experiment. The design structure is determined by a similar dummy analysis process as in the standard ANOVA directive. The TOLERANCE option specifies a variate with two values. The first defines the tolerance multiplier for zero sweeps in the dummy analysis and the second defines the multiplier for use in the analysis of the y-variates. The SEED option sets the starting value for the random generator that is used to generate variates to be used in the dummy analysis.

Printed output is controlled by the PRINT option with settings:

    aovtable to print the analysis-of-variance table,
    aovpseudotable to print the analysis-of-variance table with lines for all the pseudo-terms (generated by pseudo-factors) given explicitly,
    design to display the structure of the design,
    effects to print tables of effects and residuals, and
    fittedvalues to print a table with the y-variate, fitted valued and residuals.

The DPRINT option controls debug output, with settings:

    setup for information from the set-up stage,
    analysis for information from the analysis of the y-variates, and
    dummyanalysis for information from the dummy analysis.




Multitiered experiments are defined by Brien (1983), their design is discussed by Brien & Bailey (2006) and Brien et al. (2011), and their analysis of variance is described by Brien & Payne (1999), Brien & Bailey (2009) and Bailey & Brien (2013).

Action with RESTRICT

There must not be any restrictions.


Bailey, R.A. & Brien C.J. (2013). Randomization-based models for multitiered experiments. I. A chain of randomizations. arXiv preprint arXiv:1310.4132: 30.

Brien, C.J. (1983). Analysis of variance tables based on experimental structure. Biometrics, 39, 53-59.

Brien, C.J. & Bailey, R.A. (2006). Multiple randomizations. Journal of the Royal Statistical Society, Series B, 68, 571-609.

Brien, C.J. & Bailey, R.A. (2009). Decomposition tables for multitiered experiments. I. A chain of randomizations. The Annals of Statistics, 36, 4184-4213.

Brien, C.J., Harch, B.D., Correll, R.L. & Bailey, R.A. (2011). Multiphase experiments with at least one later laboratory phase. I. Orthogonal designs. Journal of Agricultural,

  Biological and Environmental Statistics, 16, 422-450.

Brien, C.J. & Payne, R.W. (1999). Tiers, structure formulae and the analysis of complicated experiments. The Statistician, 48, 41-52.

See also


Commands for: Analysis of variance.


CAPTION 'AMTIER example','Example from Brien & Payne (1999).';\ 
SPLOAD  [PRINT=*] '%gendir%/examples/Amtier.gsh'
        F2=(Rows*(Squares/Columns))/Halfplots; F3=Trellis*Method] Score
Updated on June 20, 2019

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