Classical and m odern non-param etric approaches For many years the use of classical non-param etric procedures, like W ilcoxon rank

sum test, have dom inated th e analysis of cross-over d a ta where th e d istrib u tio n al assum ptions (e.g. norm ality) have been violated. This approach is illu strated in Koch (see [45]) for the 2x2 design. Koch assumes the sim ple carry-over m odel with a random subject effect. For estim ating treatm en t effect, th e w ithin subject differences are calculated and the Wilcoxon rank sum te st is applied to these differences. Sim ilarly for testing the hypothesis of no residual effects, th e w ithin subject sums are first evaluated and then the W ilcoxon te st is applied to these sums.

McHugh and Gomez-M arin (see [67]) examine and com pare a random ization model for analyzing the 2x2 cross-over design w ith the sim ple carry-over model.

An additivity assum ption is then introduced in the random ization m odel and a new com parison w ith the simple carry-over model is perform ed. T he random iza­ tion model assumes th a t the test and reference products can be tried only on a finite population of size N . C onceptually each of the N experim ental subjects can

be allocated to any one of th e four possible sequence by period com binations, gen­ erating a hypothesized AN responses, which can be used to describe th e observed responses. It tu rn s out th a t th e tre a tm e n t estim ato r based on th e random ization argum ent alone, has a d istin ct different variance from th e tre a tm e n t estim ato r under the simple carry-over model. T h e add itiv ity assum ption, introduced next, simply assumes th a t th e 4A^ conceptually responses are com posed of a subject, a treatm en t and a carry-over effect. T h e results of th e random ization m odel w ith the additivity condition are com parable to th e results of th e sim ple-carry-over model, as far as the precision of the tre a tm e n t effect estim ate is concerned. Tsai and P atel (see [8 8]) were one of th e first to apply m odern non-param etric

approaches to th e analysis of th e 2x2 tria l w ith baselines. T sai and P a te l im ple­ m ent these m ethods to a 2x2 design th a t includes a placebo run-in period and a

wash-out interval of adequate length between the two active tre a tm e n t periods. Baseline m easurem ents are taken b o th during th e run-in and w ash-out periods. Tsai and Patel, consider a sim ilar approach to th a t of Jones and K enward for the m anagem ent of carry-over effects. A te st for th e significance of th e residual effect from the first tre a tm e n t period to th e w ash-out interval is first perform ed, by taking the differences between th e two baseline m easurem ents and th en applying a Wilcoxon rank sum test to the derived d a ta from the two sequence groups. W here Tsai and P a te l’s work differs from conventional approaches, is the way they test for residual effect from th e first to the second tre a tm e n t period, and the way they test for tre a tm e n t effect. Before testin g for tre a tm e n t and carry-over effects, Tsai and P atel remove th e effect of baselines. D ata from first and second treatm en t period are m odeled separately. A linear regression is perform ed, w ith treatm en t period d a ta as response and corresponding baseline m easurem ent as covariate. These models are n o t fitted by m inim izing the sum of th e squares of the difference between th e response and its expected value, b u t ra th e r a slightly com plicated function of th a t difference is optim ized. R obust linear fit minimizes for each period j , where j = 1,2, the following function:

(f> {{Vijk — Oij — PjXijk) /Oj) (2.26)

i,k

Note th a t a common regression coefficient is assum ed for b o th sequences w ithin a period. The function 4>{x) is H u b er’s function (see [34]), and p aram eter estim ates

are derived by solving a system of nonlinear equations sim ultaneously. T he above equation also implies a different variance p a ra m ete r for th e two period groups. An alternative way to identify a relationship betw een two continuous variables, is by fitting a locally weighted robust regression curve. Cleveland (see [8]) was the first to introduce th e idea, which allows us to use neighborhood points of a given point {x,y) to ob tain a fitted value for y. W ith these points, a weighted least squares fit is perform ed, where th e w eighted function is sym m etric ab o u t x

and decreases to zero as th e distance from x increases. As before, th is m eth o d is applied separately to the d a ta from th e two periods.

Using either of the above approaches, a p air of residuals {rnk,ri2k) can be cal­

culated for each subject, an d th e hypothesis of no carry-over effect from first to second treatm en t period or of no tre a tm e n t difference is based on these residual pairs. A Wilcoxon rank sum te st is applied to th e sets { r m + r u i, . . . , rn„^4-ri2ni) and (r2ii + f22i, • • • , 7'2in2+ ^22712) for carry-over testing, where rii and ri2 are num ­

ber of subjects random ized to the two sequence groups. For th e com parison of treatm en ts a Wilcoxon rank sum test is perform ed on th e differences rnk — Vi^k-