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

### Options

`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 |

### Parameters

`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` ) |

### Description

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`

`TREATMENTSTRUCTURE wines`

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. |

Options: `PRINT`

, `F1`

, `F2`

, `F3`

, `FACTORIAL`

, `F2BALANCETYPE`

, `F3BALANCETYPE`

, `PSEUDOTERMS`

, `DESIGN`

, `SEED`

, `TOLERANCE`

, `DPRINT`

.

Parameters: `Y`

, `RESIDUALS`

, `FITTEDVALUES`

, `SAVE`

.

### Method

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.

### References

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

Procedures: `AMTDISPLAY`

, `AMTKEEP`

.

Commands for: Analysis of variance.

### Example

CAPTION 'AMTIER example','Example from Brien & Payne (1999).';\ STYLE=meta,plain SPLOAD [PRINT=*] '%gendir%/examples/Amtier.gsh' AMTIER [PRINT=aovtable; FACTORIAL=5;\ F1=((Occasions/Intervals/Sittings)*Judges)/Positions;\ F2=(Rows*(Squares/Columns))/Halfplots; F3=Trellis*Method] Score