Produces experimental designs efficient under analysis of covariance (D.B. Baird).

### Options

`PRINT` = string tokens |
Controls printed output (`design` , `cefficiency` , `means` , `histogram` , `cutoff` ); default `desi` , `ceff` , `cuto` |
---|---|

`TREATMENTSTRUCTURE` = formula |
Treatment terms to be fitted |

`BLOCKSTRUCTURE` = formula |
Block model for the design |

`COVARIATES` = variates |
Covariates for the design |

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

`GRBLOCKSTRUCTURE` = formula |
Formula use for randomization; default uses `BLOCKSTRUCTURE` |

`EXCLUDE` = factors |
(Block) factors whose levels are not to be randomized |

`UNITS` = text, variate or factor |
Labels for the units of the design |

### Parameters

`PROPORTION` = scalars |
Upper proportion of the combined cov. ef. distribution from which the design is to be chosen (or zero to take the best design found); default 0.5 |
---|---|

`NSIMULATIONS` = scalars |
Number of designs to simulate for the empirical distribution of combined cov. ef.’s; default 100 |

`WEIGHTS` = variates |
Weighting for the treatment terms to use when calculating the combined cov. ef.; default 1 (i.e. all equal) |

`CEFLIMIT` = scalars |
Minimum value of the cov. ef. for each or variates treatment term for a design to be included in the set of acceptable designs; default 0 (i.e. all designs acceptable). |

`ORDER` = scalars |
Order of polynomial to fit for each covariate; or variates default 1 (i.e. only linear covariates) |

`SEED` = scalars |
Seed for random number generator for randomizing the simulated designs; default 0 |

`SAVE` = pointers |
Saves the treatment factor allocations for the selected design; if unset, these overwrite the values of the treatment factors themselves |

`CUTOFF` = scalars |
Critical value of the combined cov. ef. from the simulated distribution |

`CEFFICIENCY` = variates |
Covariate efficiencies for the treatment terms from the selected design |

`SIMULATIONS` = variates |
Simulated combined cov. ef.’s |

### Description

When a covariate is fitted in an analysis of variance, there can be a loss of efficiency in the estimation of the treatment effects. A measure of this loss of efficiency is printed in a column of the analysis of variance table headed. “cov. ef.” (an abbreviation for the covariance efficiency factor). A value of the cov. ef. close to 1 represents very little loss in efficiency through fitting the covariate. In good designs, the treatment means for a covariate will be similar, so that only small adjustments will be required in estimating the response-variate treatment means in the analysis of covariance.

Where the covariates are available before the allocation of treatments to units, the randomization of the design may be restricted to ensure high covariate efficiency for all treatment factors. Cox (1957, 1982) suggests the approach of restricting the randomization, so that only designs with values of cov. ef. in the top ranked set of the full randomization set are chosen. The proportion of acceptable designs is set via the `PROPORTION`

option. If this is greater than zero, `COVDESIGN`

randomly generates the number of designs specified by the `NSIMULATIONS`

parameter to obtain an empirical distribution for the covariance efficiencies. It then generates further designs until it finds one within the acceptable proportion. Alternatively, Harville (1974, 1975) suggests ignoring randomization, and only taking the design with the optimal cov. ef. value. If `PROPORTION`

is set to zero, `COVDESIGN`

instead takes the best design out of the `NSIMULATIONS`

randomly generated designs. You can provide a seed, using the `SEED`

parameter, for the randomizations used to generate the designs. The default, `SEED=0`

, sets the seed automatically.

The treatment and block models for the design are specified by the `TREATMENTSTRUCTURE`

and `BLOCKSTUCTURE`

options, respectively, and the `COVARIATES`

option lists the covariates. The `FACTORIAL`

option specifies the maximum order of treatment term to fit; default 3. Usually the design is randomized according to the `BLOCKSTRUCTURE`

, but you can specify an alternative model for randomization using the `GRBLOCKSTRUCTURE`

option. The `EXCLUDE`

option can supply a list of blocking factors that are not to be randomized, similarly to the `EXCLUDE`

option of the `RANDOMIZE`

directive. `COVDESIGN`

usually uses `RANDOMIZE`

for the randomization, and so there is the constraint that the block-factor combinations must all have replication one, and the block model must contain only the operators `*`

and `/`

. However, if `EXCLUDE`

is unset and the randomization structure consists of a single factor, `COVDESIGN`

uses the `URAND`

function of `CALCULATE`

and the `SORT`

directive. Under these circumstances, for example, the blocks need not be of equal sizes.

When there are several treatment terms, each one may have a different cov. ef. `COVDESIGN`

combines these into a combined cov. ef. over all the terms, calculated as the geometric mean. You can provide a variate of weights for the terms, using the `WEIGHTS`

parameter.

The `CEFLIMIT`

parameter can be set to restrict the designs from which the resulting design is selected. A design is acceptable if the values of cov. ef. for each treatment term (main effects and interactions, if fitted) are greater than the corresponding value in `CEFLIMIT`

. The values in `CEFLIMIT`

must be between 0 and 1. If `CEFLIMIT`

is a scalar, a common minimum value is applied to each treatment term.

Higher order covariate balance on the treatments can be obtained by including polynomial covariate terms. The `ORDER`

parameter specifies the degree of the polynomial to be included for each covariate. For example, setting `ORDER`

to 2 would force both the mean and variance of the covariates in each treatment group to be balanced. The default `ORDER=1`

includes only the usual linear covariates.

The `PRINT`

option controls prointed output, with settings:

`cutoff` |
the critical value for the combined cov. ef. defining the acceptable `PROPORTION` of designs; |
---|---|

`design` |
the treatment allocations in the resulting design; |

`efficiency` |
the cov. ef.’s in the resulting design; |

`histogram` |
histogram of the combined cov. ef.’s in the simulations; |

`means` |
covariate means by treatments |

By default `PRINT=design,cefficiency,cutoff`

. The `UNITS`

option allows you to specify a text, variate or factor to label the units in the design output.

The `SAVE`

parameter allows you to supply a pointer to save the values of the treatment factors for the best design. If this is not set, `COVDESIGN`

saves them by redefining the values of the original treatment factors. The cov. ef.’s for the best design can be saved using the `CEFFICIENCY`

parameter, in a scalar if there is only one treatment term, or in a variate if there are several. The combined cov. ef.’s from the simulations can be saved using the `SIMULATIONS`

parameter. The `CUTOFF`

parameter can save the critical value for the combined cov. ef. defining the acceptable `PROPORTION`

of designs.

Options: `PRINT`

, `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

, `COVARIATES`

, `FACTORIAL`

, `GRBLOCKSTRUCTURE`

, `EXCLUDE`

, `UNITS`

.

Parameters: `PROPORTION`

, `NSIMULATIONS`

, `WEIGHTS`

, `CEFLIMIT`

, `ORDER`

, `SEED`

, `SAVE`

, `CUTOFF`

, `CEFFICIENCY`

, `SIMULATIONS`

.

### Action with `RESTRICT`

Any restrictions on covariate, treatment factors, block factors or units structures are cancelled.

### References

Cox, D.R. (1957). The use of a concomitant variable in selecting an experimental design. *Biometrics*, 13, 150-158.

Cox, D.R. (1982). Randomization and concomitant variables in the design of experiments. In: *Statistics and Probability* (ed. G. Kallianpur, R. Krishnaiah & J.K. Ghosh), 777-790. North Holland, New York.

Harville, D.A. (1974). Nearly optimal allocation of experimental units using observed covariate values. *Technometrics*, 16, 589-599.

Harville, D.A. (1975). Computing optimum designs for covariance models. In: *A Survey of Statistical Design and Linear Models* (ed. J.N. Srivastava). North-Holland. Amsterdam.

### See also

Commands for: Design of experiments, Analysis of variance.

### Example

CAPTION 'COVDESIGN example',!t(\ 'Animal feeding experiment with two prior',\ 'covariates, live weight and fleece weight (kg),',\ 'and two treatment factors, four types of diet',\ 'at two levels of intake.') VARIATE [VALUES=108,109,111,112,115,122,123,124,\ 130,132,135,137,139,140,142,143] Tag VARIATE [VALUES= 45, 56, 48, 63, 58, 49, 46, 53,\ 67, 47, 50, 56, 52, 50, 58, 51] LiveWt VARIATE [VALUES=3.1,4.2,3.2,3.5,3.8,3.6,3.9,4.1,\ 4.3,3.3,3.9,4.5,3.6,3.3,3.7,3.9] FleeceWt FACTOR [Levels=4;VALUES=4(1...4)] Diet FACTOR [Levels=2;VALUES=2(1,2)4] Intake COVDESIGN [COVARIATES=LiveWt,FleeceWt; TREATMENTS=Diet+Intake; UNITS=Tag]\ NSIMULATIONS=200; PROPORTION=0.1; SAVE=Des; SEED=347893