Estimates the power λ in a Box-Cox transformation, that maximizes the partial log-likelihood in `ANOVA`

(W. van den Berg).

### Options

`PRINT` = string tokens |
Controls printed output (`aovtable` , `lambda` , `monitoring` ); default `aovt` , `lamb` |
---|---|

`TREATMENTSTRUCTURE` = formula |
Defines the treatment model; if this is not set, the default is taken from any existing setting defined by the `TREATMENTSTRUCTURE` directive |

`BLOCKSTRUCTURE` = formula |
Defines any block model; if this is not set, the default is taken from any existing setting defined by the `BLOCKSTRUCTURE` directive |

`COVARIATE` = variates |
Specifies any covariates; if this is not set, the default is taken from any existing setting defined by the `COVARIATE` directive |

`FACTORIAL` = scalar |
Limit in the number of factors in the terms generated from the `TREATMENTSTRUCTURE` formula; 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 |

`PLOT` = string token |
Whether to plot the partial log-likelihood (`partialloglikelihood` ); default `part` |

`CIPROBABILITY` = scalar |
Probability level for the confidence interval for lambda; default 0.95, i.e. a 95% confidence interval |

`TRIALVALUES` = variate |
Values of λ for which the partial log-likelihood is to be calculated; default !(`-4,` `-3.75` `...` `4)` |

`TRANSFORM` = string token |
How to transform the y-variate (`estimate` , `trialvalue` ); default `tria` |

`STEPLENGTH` = scalar |
Steplength for estimating λ; default 0.01 |

`MAXCYCLE` = scalar |
Maximum number of iterations; default 100 |

`TOLERANCE` = scalar |
Tolerance for convergence; default 0.00001 |

`ASAVE` = identifier |
Saves the `ANOVA` save structure from the analysis of variance |

### Parameters

`Y` = variates |
Response variate |
---|---|

`NEWY` = variates |
Saves the transformed response variate |

`LAMBDA` = scalars |
Saves the estimated value of λ |

`LOWER` = scalars |
Saves the lower confidence limit for λ |

`UPPER` = scalars |
Saves the upper confidence limit for λ |

### Description

`ABOXCOX`

uses profile likelihood to estimate the parameter λ for a Box-Cox transformation (Box & Cox 1964) in a an `ANOVA`

analysis. The transformation is defined as

*y ^{λ}* = (

*y*– 1) / λ λ ≠ 0

^{λ}*y ^{λ}* = log

_{e}(

*y*) λ = 0

The `TRIALVALUES`

option supplies trial values of λ (in a variate) at which the partial log-likelihood is evaluated. If the maximum value is within the range of the supplied values, `ABOXCOX`

then finds the value of λ that maximizes the partial log-likelihood, using a Newton-Raphson algorithm. It also estimates confidence limits for λ. The probability for the interval is specified by the option `CIPROBABILITY`

; default 0.95 (i.e. 95%). Note: if the confidence region includes the value one, there is no evidence (at the specified probability level) to support taking a transformation.

The response variate is supplied by the `Y`

parameter, and must contain only positive values. The transformed variate can be saved by the `NEWY`

parameter. The `TRANSFORM`

option controls whether the transformation uses the estimated value of λ or the best of the trial values (default). Using the trial value will usually provide results that are easier to interpret. For example, if the estimated value is close to zero, it may be clearer to use a logarithmic transformation than the power transformation. The estimated value of λ can be saved by the `LAMBDA`

parameter, and its confidence limits can be saved by the `LOWER`

and `UPPER`

parameters.

The treatment model can be specified using the `TREATMENTSTRUCTURE`

option, the block structure (if any) on the subjects can be specified by the `BLOCKSTRUCTURE`

option, and the `COVARIATE`

option can be used to list any covariates. If any of these options is unset, the default is taken from any existing setting defined by the directives `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

or `COVARIATE`

, respectively. The `FACTORIAL`

option can be used to set a limit on the number of factors in the terms generated from the `TREATMENTSTRUCTURE`

option.

Contrasts can be specified by using the functions `POL`

, `REG`

, `COMPARISON`

, `POLND`

or `REGND`

in the `TREATMENTSTRUCTURE`

formula, as in `ANOVA`

. The `CONTRASTS`

option places a limit on the order of contrasts that are fitted. The `DEVIATIONS`

option sets a limit on the number of factors in the terms whose deviations from the fitted contrasts are to be retained in the model. See `ANOVA`

for more details.

Printed output is controlled by the `PRINT`

option, with settings:

`aovtable` |
prints the analysis-of-variance table of the transformed variate; |
---|---|

`lambda` |
prints the estimated value of λ, and its confidence limits; and |

`monitoring` |
reports the progress of the estimation. |

The default is to print the analysis-of-variance table and the estimate of λ with its confidence limits.

The `ASAVE`

option can be used to save the `ANOVA`

save structure from the analysis of the transformed variate. This can then be used to produce further output, by the usual commands `ADISPLAY`

, `APLOT`

and so on.

By default, a plot of the partial log-likelihood is produced. This can be suppressed by setting option `PLOT=*`

.

The `STEPLENGTH`

option specifies the steplength for the estimation process (default 0.00001), the `MAXCYCLE`

option specifies the maximum number of iterations (default 100), and the `TOLERANCE`

option specifies the tolerance for convergence (default 0.00001).

Options: `PRINT`

, `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

, `COVARIATE`

, `FACTORIAL`

, `CONTRASTS`

, `DEVIATIONS`

, `PLOT`

, `CIPROBABILITY`

, `TRIALVALUES`

, `STEPLENGTH`

, `MAXCYCLE`

, `TOLERANCE`

, `ASAVE`

.

Parameters: `Y`

, `NEWY`

, `LAMBDA`

, `LOWER`

, `UPPER`

.

### Method

The partial log-likelihood for λ can be found on pages 178-180 of Pawitan (2001). The confidence limits are estimated by cubic interpolation, using the `INTERPOLATE`

directive. This is feasible only if at least two values have been evaluated on either side of the maximum. The `TRIALVALUES`

option can be used to include additional values if this fails.

### Action with `RESTRICT`

The y-variate may be restricted.

### References

Box, G.E.P. & Cox, D.R. (1964). An analysis of transformations. *Journal of the Royal Statistical Society Series B*, 26, 211–252.

Pawitan, Y. (2001). *In All Likelihood: Statistical Modelling And Inference Using Likelihood.* Oxford: Clarendon Press.

### See also

Directive: `ANOVA`

.

Procedure: `YTRANSFORM`

.

Commands for: Analysis of variance.

### Example

CAPTION 'ABOXCOX example'; STYLE=meta " Data from Box, G.E.P. & Cox, D.R. (1964). An analysis of transformations. Journal of the Royal Statistical Society Series B, 26, 211-252." FACTOR [LEVELS=3; VALUES=16(1...3)] Poison & [LEVELS=4; VALUES=(1...4)12] Treatment VARIATE [NVALUES=48] Time READ Time 0.31 0.82 0.43 0.45 0.45 1.10 0.45 0.71 0.46 0.88 0.63 0.66 0.43 0.72 0.76 0.62 0.36 0.92 0.44 0.56 0.29 0.61 0.35 1.02 0.40 0.49 0.31 0.71 0.23 1.24 0.40 0.38 0.22 0.30 0.23 0.30 0.21 0.37 0.25 0.36 0.18 0.38 0.24 0.31 0.23 0.29 0.22 0.33 : TREATMENTS Poison * Treatment ABOXCOX Time; NEWY=TimeTran " TimeTran is transformation of Time using trial value closest to optimum." ANOVA TimeTran