A hybrid procedure for detecting global treatment effects in multivariate clinical trials: theory and applications to fMRI studies

Abstract

In multivariate clinical trials, a key research endpoint is ascertaining whether a candidate treatment is more efficacious than an established alternative. This global endpoint is clearly of high practical value for studies, such as those arising from neuroimaging, where the outcome dimensions are not only numerous but they are also highly correlated and the available sample sizes are typically small. In this paper, we develop a two-stage procedure testing the null hypothesis of global equivalence between treatments effects and demonstrate its application to analysing phase II neuroimaging trials. Prior information such as suitable statistics of historical data or suitably elicited expert clinical opinions are combined with data collected from the first stage of the trial to learn a set of optimal weights. We apply these weights to the outcome dimensions of the second-stage responses to form the linear combination z and t tests statistics while controlling the test's false positive rate. We show that the proposed tests hold desirable asymptotic properties and characterise their power functions under wide conditions. In particular, by comparing the power of the proposed tests with that of Hotelling's T(2), we demonstrate their advantages when sample sizes are close to the dimension of the multivariate outcome. We apply our methods to fMRI studies, where we find that, for sufficiently precise first stage estimates of the treatment effect, standard single-stage testing procedures are outperformed.

Global Rank Tests for Multiple, Possibly Censored, Outcomes

“…Alternatively, one can use Bayesian methods by setting a prior on the weights, and updating the weights with additional data. Minas et al () use a type of Bayesian method to estimate weights in the case of multivariate normal data, basing the priors on previous studies, and computing the posterior with a subset of pilot data taken from the main study data. Something similar to the above procedure can potentially fit within a group‐sequential design framework as well.…”

Adaptive Multivariate Global Testing

“…We consider both single stage and sequential J -stage designs for all these tests. Finally, the two-step, single-stage linear combination z + and t + tests proposed in Minas et al (2012) are also considered. Note that the latter tests can be derived as special cases of the z * and t *-tests for J = 2, (α 1,1 , α 0,1 ) = (0, 1) and C ( p 2 ) = p 2 .…”

“…Furthermore, it might be useful to emphasize that for fixed design parameters, the power of the linear combination test with weighting vector (either fixed or initial) set equal to the optimal weighting vector ω * attains the maximum power and provides an upper bound to all the other presented procedures, including Hotelling's T 2 -test as proved in Minas et al (2012) (Corollary 1). Compared to the z -tests with fixed weighting vectors w , as we can see in Figure 3, the adaptive z * lose some power for w̃ (= w̃ z 1 ) close to optimal but gains substantial amounts of power for ω̃ far from optimal, importantly avoiding the problem of z -tests having zero power for w̃ orthogonal to optimal.…”

“…Similarly to the tests developed by O'Brien (1984), Läuter, Glimm, and Kropf (1998), and Minas et al (2012), the proposed tests are based on linear combinations of the observation vectors. The crucial element in this approach is the weighting vector reducing the observation vectors to the scalar linear combinations.…”

“…O'Brien proposed deriving the weighting vectors under the assumption of uniform mean structure, while Läuter et al showed that if the weighting vector is derived from the observation sums of products matrix, the Type I error is controlled and high power is attained under certain factorial structures. On the other hand, the tests in Minas et al (2012) can attain high power levels independently of the mean and covariance structure but a part of the sample is used in a separate pilot study to learn the weighting vector.…”