Calculates efficiency of estimating effects in cross-over designs (B. Jones & P.W. Lane).

### Options

`PRINT` = string tokens |
What reports to produce (`summary` , `efficiency` , `variance` , `carryover` , `contrasts` , `dummyanalysis` , `incidence` ); default `summ` , `effi` , `cont` |
---|---|

`NPERIODS` = scalar |
Number of periods in the design; no default |

`CARRYOVER` = string token |
Whether to included effects of carryover (`yes` , `no` ); default `no` |

`CONTRASTTYPE` = string token |
Type of treatment contrasts if `POLYNOMIAL` and `OWN` parameters are unset (`pairwise` , `control` ); default `pair` |

`INCIDENCE` = pointer |
Saves incidence matrices; default `*` |

### Parameters

`SEQUENCES` = formula |
Text, variate or factor with sequence of levels of a single treatment; no default |
---|---|

`POLYNOMIAL` = scalars |
Order of polynomials to represent each term in the `SEQUENCES` parameter; default `*` , i.e. represent effects according to `OWN` parameter or `CONTRASTTYPE` option |

`OWN` = matrices |
Specific contrasts for each term in the sequences parameter; default `*` , i.e. represent effects according to `POLYNOMIAL` parameter or `CONTRASTTYPE` option |

`EFFICIENCY` = symmetric matrices, variates or diagonal matrices |
Saves efficiencies; default `*` |

`VARIANCE` = symmetric matrices, variates or diagonal matrices |
Saves variances; default `*` |

### Description

The simplest use of procedure `XOEFFICIENCY`

is for a cross-over design with a single treatment factor. The `SEQUENCES`

parameter should then be set to a factor indicating the treatment level to be applied at each period for each patient: the ordering must be such that the sequence of levels for the first patient come first, then the sequence for the second patient, and so on. The `NPERIODS`

option must be set to specify the number of periods. If preferred, the sequences can be input just as a text or variate structure containing the textual or numeric codes of the treatment, leaving the procedure to form the factor internally.

The procedure calculates the efficiency of estimating the treatment effects. By default, it reports the efficiency of the design for each estimated pairwise difference of treatment levels, together with the mean of these differences. Alternatively, the `CONTRAST`

option can be set to `control`

to request efficiencies of the differences of each level with the reference level of the treatment factor (the first level, by default). Another possibility is to set the `POLYNOMIAL`

parameter to the order of polynomial effects to be estimated for the treatment levels; orthogonal polynomials will be used, based on the marginal replication of the treatment levels. Finally, the `OWN`

parameter can be set to a matrix that specifies comparisons between the treatment levels: the matrix must have one column for each treatment level and one row for each desired contrast. If either of `POLYNOMIAL`

or `OWN`

is set, the `CONTRASTTYPE`

option is ignored.

The `PRINT`

option controls which reports are displayed. By default, a summary of the design is given, and then a symmetric matrix of the efficiencies of each difference between pairs of treatment levels together with the mean pairwise efficiency. In addition, the chosen contrasts are displayed, unless the default pairwise contrasts are required. The `variance`

setting displays the variance of each contrast. The `incidence`

setting displays two tables of the numbers of observations in the design: the first is classified by Subject and Treatment, and the second by Treatment and Period. (There is no point displaying the classification by Subject and Period, since this always consists of a 1 in each cell for the designs dealt with by this procedure.) The `aov`

setting produces a skeleton analysis of variance of the specified design, if the design is generally balanced.

By default, carry-over effects are ignored. If the `CARRYOVER`

option is set to `yes`

, first-order carry-over effects are included in the model, and efficiencies for treatments will be adjusted accordingly. If the `carryover`

setting is included in the `PRINT`

option, the efficiencies and variances of the carry-over contrasts are displayed in the same way as for the treatment contrasts (that is, with regard to the setting of the `CONTRASTTYPE`

option, `POLYNOMIAL`

and `OWN`

parameters, and the `efficiency`

and `variance`

settings of the `PRINT`

option). If the `incidence`

option is included, a further three incidence tables will be displayed: Treatment by Carry-over, Subject by Carry-over, and Carry-over by Period.

The `INCIDENCE`

option allows the incidence information, as printed by `PRINT=incidence`

, to be stored. It should be set to the identifier of a pointer, which will be set up by the procedure with elements labelled to identify the matrices concerned. If there is no carry-over, the pointer will point to two matrices, ordered as for the `PRINT`

option; if there is carry-over, there will be five matrices.

The `EFFICIENCY`

and `VARIANCE`

parameters allow the variances and efficiencies of the treatment effects to be stored in symmetric matrices (for pairwise differences), variates (for differences with control), or diagonal matrices (for polynomial or own contrasts). If the option `CARRYOVER`

is set to `yes`

, the stored results will be for the carry-over effects; to get the results for the treatment effects, the procedure must be invoked again with the `CARRYOVER`

option set to `no`

.

Options: `PRINT`

, `NPERIODS`

, `CARRYOVER`

, `CONTRASTTYPE`

, `INCIDENCE`

.

Parameters: `SEQUENCES`

, `POLYNOMIAL`

, `OWN`

, `EFFICIENCY`

, `VARIANCE`

.

### Method

The efficiency of a contrast is calculated as the ratio of its theoretically optimal variance to its variance in the supplied design, expressed as a percentage. The optimal variance may not actually be attainable. It is calculated as the variance for the contrast in a design with the same marginal replication of treatment levels, but where the treatment factor is orthogonal to all other factors in the design. For example, the optimal variance for a contrast between two treatment levels (omitting any estimate of dispersion) is calculated as (1/*n*_{1} + 1/*n*_{2}), where *n*_{1} and *n*_{2} are the replications of the two levels. The actual variance of the supplied design is calculated by fitting a linear model by linear regression, including terms as specified in the options. The inverse matrix then provides the variance, omitting the estimate of dispersion which would cancel out in the ratio anyway.

### Action with `RESTRICT`

No structures should be restricted.

### See also

Procedures: `AFCARRYOVER`

, `AGCROSSOVERLATIN`

, `XOCATEGORIES`

, `XOPOWER`

.

Commands for: Design of experiments.

### Example

CAPTION 'XOEFFICIENCY example'; STYLE=meta " (1) Design with eight patients, four periods, five treatments; treatment input as text codes; no carryover effects" XOEFFICIENCY [NPERIODS=4] !t(A,E,B,D, B,A,C,E, D,C,E,B, E,D,A,C,\ C,D,B,E, D,E,C,A, E,A,D,B, A,B,E,D) " (2) Same design with treatment input as a factor with ordinal levels; carryover estimated and reported; own contrasts specified" FACTOR [LEVELS=5; VALUES=1,5,2,4, 2,1,3,5, 4,3,5,2, 5,4,1,3,\ 3,4,2,5, 4,5,3,1, 5,1,4,2, 1,2,5,4] sequence MATRIX [ROWS=4; COLUMNS=5; VALUES=-1,1,0,0,0, -1,0,1,0,0,\ -1,0,0,1,0, -1,0,0,0,1] contrasts XOEFFICIENCY [PRINT=summary,efficiency,variance,carryover,contrasts;\ NPERIODS=4; CARRYOVER=yes] sequence; own=contrasts