Performs multivariate analysis of variance and covariance (R.W. Payne & G.M. Arnold).

### Options

`PRINT` = string tokens |
Printed output required from the multivariate analysis of covariance (`ssp` , `tests` , `permutationtest` ); default `test` |
---|---|

`APRINT` = string tokens |
Printed output from the univariate analyses of variance of the y-variates (as for the `ANOVA` `PRINT` option); default `*` |

`UPRINT` = string tokens |
Printed output from the univariate unadjusted analyses of variance of the y-variates (as for the `ANOVA` `UPRINT` option); default `*` |

`CPRINT` = string tokens |
Printed output from the univariate analyses of variance of the covariates (as for the `ANOVA` `CPRINT` option); default `*` |

`TREATMENTSTRUCTURE` = formula |
Treatment formula for the analysis; if this is not set, the default is taken from the setting (which must already have been defined) by the `TREATMENTSTRUCTURE` directive |

`BLOCKSTRUCTURE` = formula |
Block formula for the analysis; if this is not set, the default is taken from any existing setting specified by the `BLOCKSTRUCTURE` directive and if neither has been set the design is assumed to be unstratified (i.e. to have a single error term) |

`COVARIATES` = variates |
Covariates for the analysis; by default `MANOVA` uses those listed by a previous `COVARIATE` directive (if any) |

`FACTORIAL` = scalar |
Limit on the number of factors in a treatment term |

`LRV` = pointer |
Contains elements first for the treatment terms and then the covariate term (if any), allowing the LRV’s to be saved from one of the analyses; if a term is estimated in more than one stratum, the LRV is taken from the lowest stratum in which it is estimated |

`FPROBABILITY` = string token |
Printing of probabilities for F statistics (`no` , `yes` ); default `no` |

`SELECTION` = string tokens |
Which test statistics to print when `PRINT=test` (`lawleyhotellingtrace` , `pillaibartletttrace` , `roysmaximumroot` , `wilkslambda}` ; default `lawl` , `pill` , `roys` , `wilk` |

`NTIMES` = scalar |
Number of permutations to make when `PRINT=perm` ; default 999 |

`EXCLUDE` = factors |
Factors in the block model of the design whose levels are not to be randomized |

`SEED` = scalar |
Seed for the random number generator used to make the permutations; default 0 continues from the previous generation or (if none) initializes the seed automatically |

### Parameter

`Y` = variates |
Y-variates for an analysis |
---|

### Description

Procedure `MANOVA`

performs multivariate analysis of variance or covariance. The data variates are specified by the `Y`

parameter.

The model for the design is specified by options of the procedure. `TREATMENTSTRUCTURE`

specifies a model formula to define the treatment terms in the analysis; if this is unset, `MANOVA`

will use the model already defined by the `TREATMENTSTRUCTURE`

directive, or will fail if that too has not been set. `BLOCKSTRUCTURE`

defines the underlying structure of the design, and `MANOVA`

will use the model (if any) previously defined by the `BLOCKSTRUCTURE`

directive if this is not set; these can both be omitted if there is only one error term (i.e. if the design is unstratified). The `COVARIATES`

option specifies any covariates; by default `MANOVA`

will take those already listed (if any) by the `COVARIATE`

directive. The `FACTORIAL`

option can be used to set a limit on the number of factors in the terms generated from the treatment formula.

The `LRV`

option allows a pointer to be saved containing an LRV structure for each treatment term. When covariates have been specified, the pointer will also contain a final LRV structure for the covariate term. If a term is estimated in more than one stratum, the LRV is taken from the stratum that occurs last in the `BLOCKTERMS`

pointer. The structures in the LRV hold the canonical variate loadings, roots and trace for the respective term.

The `PRINT`

option indicates the output required from the multivariate analysis of covariance, with settings `ssp`

to print the sums of squares and products matrices, `tests`

to print the various test statistics, and `permutationtest`

to calculate probabilities for the test statistics using a permutation test.

The `SELECTION`

option controls which test statistics are given when `PRINT=tests`

. The available statistics are Wilks’ Lambda (with approximate F test), the Pillai-Bartlett trace, Roy’s maximum root test and the Lawley-Hotelling trace. The default is to print them all.

By default, when `PRINT=perm`

, `MANOVA`

makes 999 random permutations and determines the probability of each test statistic from its distribution over these randomly generated datasets. The `NTIMES`

option allows you to request another number of allocations, and the `SEED`

option allows you to specify the seed to use for the random numbers used to make the permutations. The permutations are done by the `RANDOMIZE`

directive, using the block model defined by the `BLOCKSTRUCTURE`

option. The `EXCLUDE`

option allows you to restrict the randomization so that one or more of the factors in the block model is not randomized. The most common situation where this is required is when one of the treatment factors involves time-order, which cannot be randomized.

The `APRINT`

, `UPRINT`

and `CPRINT`

control output from the univariate analyses of each of the y-variates, corresponding to `ANOVA`

options `PRINT`

, `UPRINT`

and `CPRINT`

, respectively. `FPROBABILITY`

controls whether or not probabilities are produced for F-ratios and for Chi-square variables in the analysis; by default these are omitted.

Options: `PRINT`

, `APRINT`

, `UPRINT`

, `CPRINT`

, `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

, `COVARIATES`

, `FACTORIAL`

, `LRV`

, `FPROBABILITY`

, `SELECTION`

, `NTIMES`

, `EXCLUDE`

, `SEED`

.

Parameter: `Y`

.

### Method

The relevant theory, with formulae and references for the test statistics, can be found in Chatfield & Collins (1986, Chapter 9). The procedure analyses the data variates by `ANOVA`

first as y-variates, and then as covariates in order to obtain the SSP matrices. The SSP matrices are then adjusted for the covariates, using matrix manipulation in `CALCULATE`

, and LRV decompositions are done, before the test statistics are calculated (again using `CALCULATE`

).

### Action with `RESTRICT`

If any of the y-variates is restricted, the analysis will involve only the units not excluded by the restriction.

### Reference

Chatfield, C. & Collins, A.J. (1986). *Introduction to Multivariate Analysis (revised edition)*. Chapman & Hall, London.

### See also

Procedures: `RMULTIVARIATE`

, `MVAOD`

.

Commands for: Multivariate and cluster analysis, Repeated measurements.

### Example

CAPTION 'MANOVA example',\ !t('Data from Chatfield & Collins, Introduction to Multivariate',\ 'Analysis, 1986 edition, page 143 (analysis on pages 176-178).');\ STYLE=meta,plain FACTOR [LEVELS=!(4,20,34); VALUES=6(4,20,34)] Temp FACTOR [LABELS=!T(Male,Female); VALUES=3(1,2)3] Sex VARIATE [NVALUES=18] InitWt,FinalWt,TumourWt READ InitWt,FinalWt,TumourWt 18.15 16.51 0.24 18.68 19.50 0.32 19.54 19.84 0.20 19.15 19.49 0.16 18.35 19.81 0.17 20.68 19.44 0.22 21.27 23.30 0.33 19.57 22.30 0.45 20.15 18.95 0.35 18.87 22.00 0.25 20.66 21.08 0.20 21.56 20.34 0.20 20.74 16.69 0.31 20.02 19.26 0.41 17.20 15.90 0.28 20.22 19.00 0.18 18.38 17.92 0.30 20.85 19.90 0.17 : MANOVA [PRINT=ssp,tests; TREATMENTSTRUCTURE=Temp*Sex;\ COVARIATES=InitWt; FPROBABILITY=yes] FinalWt,TumourWt