2.5.2 2x2 case w ith m issing data
2.8.4 P oisson D a ta
There is an extensive lite ra tu re covering generalized linear m odel approaches for modeling purposes in repeated m easures settings (see [56]). One of th e few p a pers paying special atten tio n to the analysis of count cross-over d a ta is th e one by Layard and Arvesen (see [54]). T h e au th o rs suggest th a t cross-over experi ment should be avoided if it is tho u g h t th a t carry-over effects could occur. So, a Poisson d istribution is assum ed for th e count d a ta , while a log-link relates the mean of th a t distribution to th e linear predictor. T he linear predictor contains term s for subject, period and tre a tm e n t only. Layard-A versen’s analysis condi tions upon subject totals. In this way testin g for drug by period interactions is not feasible, though for tackling th is problem they recom m end an alternative procedure based on a t-test for appropriately transform ed p a tie n t d ata. They illustrate their approach using two examples, where a 2x2 design was used to run the trials.
square for illustrative purposes. T he approach based on conditioning on th e sub ject to ta l suffers from the fact th a t pairwise com parisons among tre a tm e n ts are
not easily perform ed. An altern ative route where d a ta are first appropriately transform ed and then a weighted linear regression is perform ed on th e tra n s formed values, w ith weights determ ined beforehand, is recommended.
2.9
V ariance C om ponents E stim ation
Laird et al (see [50]) propose an interesting m ethod of estim ating the variance com ponents, when com pound sym m etry stru ctu re (i.e. random subject effects) is assum ed for the responses on a subject, in two period cross-over studies. More specifically, if denotes the difference and the sum of the two responses on the subject, then the following two models are fitted,
d = (2^!7)
s =■ -A5/3 -}- 6s (2.28)
where Xd and Xg denote th e design m atrices for the sum and difference vector. From these models two m ean square errors, M S Ed and M S Eg, are derived and the covariance/ correlation param eters are estim ated as follows:
= { M S E s + M S E i ) / 4 : (2.29)
p = { M S E , - M S E i ) ! { M S E s + M S E i ) (2.30)
Laird et al combine the estim ates of j3 derived from equations (2.27) and (2.28) to derive th e GLS estim ate. Obviously th is m ethod generalizes in a straightforw ard way, when baseline m easurem ents are included as covariates.
M atthew s (see [63]) considers the estim ation of th e dispersion param eters in the general case of a p-period cross-over tria l w ith a continuous outcom e, where n
subjects are recruited. T he model assum ed, includes subject, period and tre a t ment effects (all fixed), while carry-over term is not considered. The linear model can be sum m arized in the following equation
where y is an T i p - d i m e n s i o n a l vector, 5 , tt, t is th e subject, period and treatm en t
effect respectively. The variance m atrix of th e error-vector e has a block diagonal form
W = ® V (2.32)
where th e m a trix Vpxp describes th e in tra-su b ject correlation structure. This stru c tu re takes th e form of a statio n ary first-order autoregressive process, w ith its elem ent equal to (1 — M atthew s removes the nuisance param eters, su b ject and period term s, by pre-m ultiplying b o th sides of th e above equation w ith an appropriate m atrix. T he m odel for th e transform ed response looks as follows:
z = A t + €* (2.33)
T his m odel contains only the param eters we are interested in, r, p and The au th o r th en applies ordinary m axim um likelihood and derives an analytic ex pression for the correlation coefficient p. T he above approach, called restricted m axim um likelihood, is equivalent to integrating out the nuisance param eters from the full likelihood function. M atthew s com pares th e above m ethod w ith a conditional profile likelihood approach, where a likelihood function containing only th e p aram eter of interest, p, can be w ritten down explicitly. Sim ulation studies are used to com pare the two inference m ethods plus the stan d ard m axi m um likelihood approach. T he designs used are a four-sequence three-period one
(ABB, AAB, duals) and a four-sequence four-period one (ABBA, AABB, duals), where 1 2 subjects are allocated in each sequence. M atthew s concludes th a t b o th
conditional an d restricted likelihood approaches perform b e tte r th an the stan d ard m axim um likelihood in term s of bias, though th e restricted likelihood approach is to be preferred because it can easily be generalized to the case where in tra su b ject covariance stru ctu re is described by more th a n one param eter. S tan d ard w eighted least squares can be used to estim ate r , the tre a tm e n t effect, w ith p replaced by its estim ate. U ncertainty concerning the estim ation of p can safely be ignored in our inferences for r , since p and r are orthogonaly estim ated. G uilbaud (see [31]) estim ates variance com ponents in the 2x2 case, assum ing
on draw ing inference for the ratio 6 = which m easures th e relative vari ability w ithin subjects under the two treatm en ts. G uilbaud derives initially the exact d istrib u tio n of the following quantity 7 = (cr^ — c r |) / (cr^ -f <7| ) , from which
inferences ab o u t Q can be made. As before, th e key statistic is based on the w ithin-subject sum and difference pair {sik,dik), where k indexes subject and i
sequence group. The au th or proves th a t (7* — j ) / s * follows a t-d istrib u tio n on
n — 3 degrees of freedom, where n is the to ta l num ber of p articip an ts recruited in the study. The value of 7* equals the common slope of two parallel lines fitted to
the two sequence groups by ordinary least squares, w ith th e dik tre a ted as fixed predictor, while the Sik treated as the response. T he s* is sim ply th e stan d ard error of th a t slope.