T h e Bayesian A pproach

T he power of this approach lies on the ability of the analyst to report not only an estim ate of treatm en t (or carry-over) difference accompanied w ith its stan d ard error, bu t the whole distribution of it, making easier the task to answer fu rth er queries of interest about these param eters. The first to present a Bayesian an al­ ysis of the cross-over experim ent was Grieve (see [26]), who was able to derive explicitly th e jo in t posterior d istrib u tio n of treatm en t and carry-over effect, as well as, the m arginal posterior distrib u tio n of the carry-over effect. M arginal in­ ference for th e treatm en t effect, which is the m ain purpose for running the clinical trial, was not possible to be evaluated analytically, b u t Grieve (see [27]) was able to provide a very good approxim ation to it, based on P a til’s approxim ation to a B ehrens-Fisher type distribution. Also the constrain cr^ < was considered in th e analysis, b u t it turned out to make very little difference to the final conclu­ sions.

O ur approach will be based on graphical modeling theory for expressing q u alita­ tive relationships between d a ta and unknown param eters, and on Gibbs sam pling for perform ing the necessary com putations to derive the posterior quantities of

interest. T he presentation of a statistic a l problem using graphs, where nodes rep­ resent random quantities and missing links represent conditional independence assum ptions, has th e m ain advantage of breaking a com plex m odel to sim pler ones. This implies th a t th e stru c tu re of the problem is easier to com m unicate and furtherm ore th e graph provides the basis for the co m p u tatio n task (Gibbs sam pling, see Gilks et al [2 2]).

In th e 2x2 cross-over trial exam ined here, recall th a t responses [yij) on a specific patien t are independent conditional on th eir m ean jiij and th e w ithin-subject com ponent of variance Each p a tie n t’s m ean is a linear function of four pa­ ram eters ; p atien t, period, tre a tm e n t and carry-over effect. Each one of these param eters is considered as a random variable and a prior d istrib u tio n is assigned to it. Note th a t in th e frequentist approach only th e ’’subject effect” is taken as random , w ith the rest of th e param eters regarded as fixed quantities. T his model is known in the frequentist lite ra tu re as the random intercept model.

Schem atically the situ atio n is presented in Figure (3.5). In th a t diagram logical links (dashed arrows) have been used for represented determ inistic relationships, while solid arrows represent stochastic dependencies. T he G ibbs sam pler now generates a M arkov chain for each variable. T he chain is produced by using the conditional d istrib u tio n of each unobserved node in the graph given th e rest. In the long run th e generated draws compose a sam ple from th e posterior d istri­ bution of th a t variable. T he diagram indicates the way in which a sam ple of a random variable is linked w ith random draws of oth er variables, so th a t the statistical restrictions of the model are satisfied (see Spiegelhalter [87]).

The likelihood function can be expressed as th e pro d u ct of th e following term s:

l / i j k ~ ^

fJ'ijk = + 7Tj H- T d { i j ) + A d ( i j _ i )

Sik - N (0, ( j |)

Fully Bayesian analysis requires th e specification of prior d istrib u tio n s for all u n ­ known param eters appearing in th e above equations. If inform ation regarding those param eters was available from previous cross-over trials this could be in­ corporated a t th a t stage. In th e absence of any prior knowledge th e influence of the prior distrib u tio n s in th e final conclusions should be m inim al. In our case the

y V

A,

following least-inform ative priors were chosen:

~ G am m a (lO~®, 1 0"®)

T he sta rtin g values chosen to in itiate th e Gibbs sam pler set th e location p aram ­ eters a t zero, while th e variance com ponents a t one. A long chain was run, so th a t conclusions are insensitive to in itial values and most im p o rtan tly to ensure th a t th e chain has converged to its lim ited distribution. In this exam ple, conver­ gence m onitoring was also perform ed by generating five sim ulated sequences w ith different sta rtin g points and using CODA software to evaluate G elm an-R u b in ’s R -statistic for tre a tm e n t and carry-over effect. The R-values were alm ost identi­ cally equal to 1, re-assuring th a t convergence occurred. For each variable 15000

values were generated and only the last 5000 values used for draw ing inference. T he sam pled values used for draw ing inference for the various p aram eters are displayed graphically in Figure (3.6). All calculations were perform ed using th e BUGS software. BUGS code is provided a t the end of this chapter.

T he posterior distribu tio n of carry-over has mean 13.30 with variance 85.70. We conclude th a t carry-over m ust be negligible, although the 95% equal-tailed confi­ dence interval for th a t effect is (-141.00,201.00) indicating a wide range of possible values for th e carry-over difference. This is expected since carry-over is estim ated using betw een-patient inform ation, which implies th a t no m a tte r if eith er a Fre­ quentist 95% confidence interval is formed or a 95% Bayesian H PD region is calculated th e interval looks always wide. Note here th a t because of the sym ­ m etry of th e posterior distributions for all location param eters, 95% equal-tailed intervals or 95% H PD regions lead to sim ilar inferential conclusions.

O th er posterior quantities of interest for bo th models, not only for tre a tm e n t and carry-over difference, b u t also for the w ithin and between p atien t variability are sum m arized in Table (3.4).

T he advantage of the Bayesian approach is th a t we can form an idea of th e m ost likely values of tre a tm e n t (carry-over) effect. In Figure (3.7), the posterior d istri­ butio n of 2A indicates th a t th e probability of th a t param eter lying in a sym m etric interval around zero is really high. In th e same figure the posterior d istrib u tio n for th e tre a tm e n t difference suggests th a t under the simple carry-over model it is

Sampled values for the treatment effect under the no-carryover model o o CM o o o CM

§

Sampled values for the treatment effect under the simple carryover model

§

Sampled values for the carryover effect under the simple carryover model

1000 2000 3000 4000 5000

Iteration

Figure 3.6: Sam pled values for tre a tm e n t and carry-over effect under various assum ptions concerning the carry-over term

Posterior distribution of residual effect for ttie simple carryover model