M ultivariate D ata

T he analysis of the 2x2 design from a m ultivariate perspective was first presented by Zim m erm ann and Rahlfs (see [93]). The authors argue th a t the m ultivari­ ate approach has certain advantages over the univariate one, since sim ultaneous testing of hypothesis of interest are possible, while restrictive assum ptions on th e w ithin-subject covariance stru ctu re can be avoided. The authors assume a simple carry-over model w ith a general w ithin-subject covariance stru ctu re. A sim ultaneous hypothesis concerning tre a tm e n t and carry-over effect is first p er­ formed. This hypothesis is usually rejected a t conventional significance levels and this leads to a test for exam ining th e im portance of carry-over difference. If carry-over effect is shown to be different from zero, then only first period d a ta are used for testing treatm en t effect, while in the case where residual effect is statistically un im p ortan t then all four cell m eans are used for draw ing infer­ ence for treatm en t effect. This work is extended in th e case of the cyclic design

(ABC,BCA ,CA B) where carry-over effect in th e th ird period represents residual effect from the second and th e first period. A sim ilar m ulti-stage procedure to th e 2x2 case for testing tre a tm e n t effect is proposed. The authors conclude th a t for th e 2x2 design the univariate approach gives identical results to th e m ultivari­

ate one. For m ulti-period designs different hypothesis can be tested under the two approaches, though the m ultivariate procedures has the advantage th a t less restrictive assum ptions are imposed. A hybrid procedure is finally recom mended which uses th e advantages of bo th approaches.

In m ulti-period multi-sequence cross-over trials a single outcom e variable is usu­ ally of interest. There are occasions however, where two or more outcome vari­ ables m ay be observed w ithin a treatm en t period. This is simply a cross-over design w ith m ultivariate observations and can be analyzed using stan d ard m ulti­ v ariate techniques, see M ardia et al [60] or Kraznowski [46]. G render and Johnson

(see [24]) discuss an example of a cross-over trial w ith a bivariate response, where the effect of caffeine on stress reactions was studied by m easuring systolic and di­ astolic blood pressures before perform ing a task and after adm inistering caffeine or placebo. An adequate wash-out period was allowed in this study.

Let yijk be th e response vector of the subject w ithin the period who has been random ized in the sequence. The model can be expressed in m atrix n o tatio n as follows;

E iVijk) = /i + 7Tj - l - T- t - A ( 2 . 2 1 )

where fi, ttj, r, A are vectors corresponding to the overall m ean, period, treatm en t and carry-over effects. Note th a t G render and Johnson do not include a sequence effect, since even in the m ultivariate 2x2 case th a t term is confounded w ith the

carry-over effect. The above model can be presented in a concise form as follows:

E { Y ) = A(j) (2.2 2)

where each row of Y corresponds to responses of each individual. All m ultivariate cross-over hypothesis can be w ritten in the form:

C(f)M = 0 (2.23)

and an appropriately constructed F -test can be used to test the hypothesis above. Surprisingly enough G render and Johnson propose a m ulti-variate analogue of th e two stage procedure to test the hypothesis of treatm en t effect. Obviously the def- ficiencies of th a t scheme are well known, when one outcome variable is m easured in each period, bu t it is my view the same defficiences will be evident in the

m ultivariate case as well. A common covariance m atrix E is assumed for the observations taken on a subject across responses.

G render and Johnson extend the above work to accom m odate analysis of two or more responses taken repeatedly across tim e. For example, in the previous trial, diastolic and systolic blood pressure can be m easured more th a n once w ithin a period. In such circum stances, the interaction of tim e w ith period, treatm en t and carry-over should be tested and if not im p o rtan t then one can average responses over tim e points and use the analysis outlined above. More specifically, G render and Johnson propose a three stage procedure, where th e tim e by carry-over in­ teractio n is tested first, followed by a test of no carry-over differences. Based on th e outcom e of the test for carry-over, either d a ta from b o th periods are used or only the first period d a ta considered for analysis purposes. The W ilks likelihood ratio criterion, which transform s to a F -statistic for th e 2x2 cross-over case, is

used for the hypothesis testing of various effects.

In a subsequent paper (see [25]), G render and Johnson fit polynom ial models for a 2x2 cross-over design where several responses are m easured w ithin a period,

repeatedly over time. W ith such d ata, a m ulti-variate test of equality of means a t tim e points w ithin sequence by period cells, is first perform ed. This hypothesis is usually rejected at conventional levels of significance and the next step is try to claim parallelism of mean profiles across groups defined by the sequence by pe­ riod cells. If th a t hypothesis accepted, then averaging response(s) across tim e is th e way forward. However, if the parallelism hypothesis is rejected, then the aim m ight be to discover how mean profiles across sequence by period groups differ. To th a t purpose, a polynom ial model can be fitted to subject specific data. The estim ated param eters of the polynomial model are subsequently analyzed using ap p ro priate techniques for cross-over plans. G render and Johnson illustrate the technique by fitting second order polynom ials in a study which investigates the effect of eating onions on triglyceride levels of p atients w ith h eart disease.