Produces an analysis of variance for repeated measurements (R.W. Payne).

### Options

`PRINT` = string tokens |
Controls output about the covariance structure (`vcovariance` , `correlation` , `epsilon` , `test` ); default `epsi` , `test` |
---|---|

`APRINT` = string tokens |
Printed output from the analysis of variance (as for the `ANOVA` `PRINT` option); default `*` |

`TREATMENTSTRUCTURE` = formula |
Defines the treatments given to the subjects; if this is not set, the default is taken from any existing setting defined by the `TREATMENTSTRUCTURE` directive |

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

`COVARIATE` = variates |
Specifies any covariates on the subjects 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 |

`TIMEPOINTS` = variate, text or factor |
When the `DATA` parameter supplies a separate variate of observations for each time this can specify numbers or labels for the time points, when there is a single `DATA` variate this must supply a factor to indicate the time of each observation |

`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 |

`FPROBABILITY` = string token |
Printing of probabilities for variance ratios in the aov table (`no` , `yes` ); default `no` |

`PSE` = string tokens |
Standard errors to be printed with tables of means (`differences` , `lsd` , `means` ); default `diff` |

`MAXCYCLE` = scalar |
Maximum number of iterations for estimating missing values; default 20 |

`LSDLEVEL` = scalar |
Significance level (%) to use in the calculation of least significant differences; default 5 |

`EPSILON` = scalar |
Saves the correction factor epsilon |

`SAVEFACTORS` = pointer |
Saves the factors used in the analysis of variance |

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

### Parameter

`DATA` = variates |
Data observations either in a list of variates (one for each time), or a single variate (with `TIMEPOINTS` set to a factor indicating the time of each observation) |
---|

### Description

A repeated-measures design is one in which subjects (animals, people, plots, etc) are observed several times. Each subject receives a randomly allocated treatment, either at the outset, or repeatedly through the experiment. The subjects are observed at successive occasions to see how the treatment effects develop.

The design might thus seem analogous to a split-plot design, with subjects corresponding to whole plots, and the occasions of observation to the sub-plots. There are, however, some important differences between the two situations. With repeated measurements, there is likely to be a greater correlation between observations that are made at adjacent time points than between those that are more greatly spaced. Furthermore, the `Times`

factor cannot, by its very nature, be allocated at random to the occasions within subjects. In the customary split-plot situation we can usually assume that there is an equal correlation between the sub-plots of each whole plot and, even if this were not so, the sub-plot treatment should have been allocated at random to the sub-plots within each whole plot. The formal conditions for the validity of the split-plot analysis will be discussed in more detail below, together with advice on how to proceed if they do not hold.

It is worth pointing out first, though, that this problem affects only the `Subjects.Times`

stratum. The `Subjects`

stratum contains an analysis of variance of the measurements totalled over the subjects, and this part of the analysis will be valid whatever the within-subject correlation structure. A further point is that, when measurements are taken on only two occasions, the analysis in the `Subjects.Times`

stratum will also be valid; there can then be only one within-subject correlation, and the analysis in the `Subjects.Times`

stratum is of the difference between the observations at time 2 and time 1 on each subject.

Another potential problem arising from the systematic nature of the `Times`

factor is that effects arising from the “length of treatment time” will be confounded with any effects arising from the duration of the experiment, such as age of subject (which may be important with short-lived material such as aphids), season of year, time of day, and so on. This does not affect the validity of the analysis, and some of the confusion may be capable of being unravelled by running the experiment during more than one period. Nevertheless, care needs to be taken in drawing conclusions about time-effects.

The `Subjects.Times`

information, describing the way in which the treatment effects change differentially with time, is often the aspect of most interest in the study. The formal requirement for the validity of the analysis in the sub-plot stratum of a split-plot design is that all the normalised contrasts in that stratum have an equal variance. The only practical arrangement of covariances between times that satisfies this condition would have a single variance down the diagonal and a single covariance off-diagonal. This pattern is known as a uniform covariance structure or, equivalently, the matrix is said to show compound symmetry; Box (1950) describes how this can be tested. In the usual split-plot analysis, the `Subjects.Times`

sum of squares is assumed to be distributed as σ^{2} × χ^{2}_{r} where σ^{2} is a constant and χ^{2}_{r} has a chi-square distribution on *r* degrees of freedom. Similarly, under the assumption that there is no `Treatments.Times`

interaction, the `Treatments.Times`

sum of squares is assumed to be distributed as σ^{2} × χ^{2}_{t} where χ^{2}_{t} has a chi-square distribution on *t* degrees of freedom. If the variance-covariance structure does not exhibit compound symmetry, it is possible to show that the distributions can still be approximated by chi-square distributions, but the degrees of freedom are instead epsilon × *r* and epsilon × *t*. The correction factor epsilon lies between one, which would give the ordinary split-plot analysis, and 1/(number of times minus one), which would leave just one degree of freedom within each subject (remember that when there are only two observation on each subject, and thus just one within-subject degree of freedom, the analysis is valid). Epsilon can be estimated by maximum likelihood, as described by Greenhouse & Geisser (1959), and the estimated value can be saved by the `EPSILON`

option. A further point is that this correction applies to the calculation of least significant differences as well as to the F ratios in the analysis of variance table. So, instead of a t distribution on *r* degrees of freedom, these must use the square root of an F distribution on epsilon and epsilon × *r* degrees of freedom.

The printing of information about the covariances is controlled by the strings listed for the `PRINT`

option: `vcovariance`

variance-covariance matrix, `correlation`

correlation matrix, `epsilon`

Greenhouse-Geisser epsilon, `test`

test for compound symmetry.

The output from the analysis of variance is controlled by the `APRINT`

option, with settings identical to those in the `PRINT`

option of the `ANOVA`

directive. The `FPROBABILITY`

, `PSE`

, `MAXCYCLE`

and `LSDLEVEL`

options also operate exactly as in `ANOVA`

.

The treatments applied to the subjects can be specified (as a model formula) 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.

The observed data are specified by the `DATA`

parameter in one of two ways. The first is to supply a list of variates, each one containing the measurements made on the subjects at one of the successive occasions on which they were observed. The `TIMEPOINTS`

option can then supply a variate or text to define numbers or labels to use in output to identify the time point corresponding to each `DATA`

variate; if `TIMEPOINTS`

is unset, the labels are formed automatically from the identifiers of the `DATA`

variates themselves. The `DATA`

variates are appended into a single variate for the analysis, and the block and treatment factors are expanded to match. You can specify a pointer using the `SAVEFACTORS`

option to save the expanded factors. The elements of the pointer are labelled by the factor names, and the time factor is also included, with the label `'Time`

`factor'`

. You would need to use these, for example, if you wanted to plot the means using `AGRAPH`

.

The second possibility is to supply a single `DATA`

variate containing the data from all the times. The `TIMEPOINTS`

option must then be set to a factor indicating the time of each observation. The block and treatment factors must already have been expanded to match the `DATA`

variate, and each subject should be represented by a unique combination of the block factors. If not, Genstat prints a warning and assumes that the subjects occur in the same order within each time. To simplify the use of `AREPMEASURES`

in general programs, the `SAVEFACTORS`

pointer is also formed when the data are in a single variate. (However, it then contains the original factors.)

The `ASAVE`

option lets you save the save structure from the `ANOVA`

analysis.

Options: `PRINT`

, `APRINT`

, `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

, `COVARIATE`

, `FACTORIAL`

, `TIMEPOINTS`

, `CONTRASTS`

, `DEVIATIONS`

, `FPROBABILITY`

, `PSE`

, `MAXCYCLE`

, `LSDLEVEL`

, `EPSILON`

, `SAVEFACTORS`

, `ASAVE`

.

Parameter: `DATA`

.

### Method

The procedure uses the standard Genstat directives for calculations and manipulation to obtain the various matrices and tests. Formulae for these are given by Box (1950), Greenhouse & Geisser (1959) and Winer (1962) pages 523 and 594-599, although note that equation (1) on page 595 should contain *N*′ & *n*′_{i}, not *N* & *n _{i}*.

### Action with `RESTRICT`

The procedure does not allow for restrictions, and will cancel any that have been applied.

### References

Box, G.E.P. (1950). Problems in the analysis of growth and wear curves. *Biometrics*, 6, 362-389.

Greenhouse, S.W. & Geisser, S. (1959). On methods in the analysis of profile data. *Psychometrika*, 24, 95-112.

Winer, B.J. (1962). *Statistical Principals in Experimental Design (second edition)*. McGraw-Hill, New York.

### See also

Directive: `ANOVA`

.

Commands for: Repeated measurements.

### Example

CAPTION 'AREPMEASURES example',\ !t('Data from Winer (1962), Statistical Principals in Experimental',\ 'Design (2nd edition), McGraw-Hill, New York, p. 597.');\ STYLE=meta,plain FACTOR [LEVELS=2; VALUES=5(1,2)] Groups POINTER [NVALUES=3] Data VARIATE [NVALUES=10] Data[1,2,3] READ Data[1,2,3] 4 7 2 3 5 1 7 9 6 6 6 2 5 5 1 8 2 5 4 1 1 6 3 4 9 5 2 7 1 1 : AREPMEASURES [PRINT=vcovariance,correlation,epsilon,test; APRINT=aov,means;\ FPROB=yes; PSE=diff,lsd; TREATMENTS=Groups] Data[]