Validación del Procedimiento

AD

and

t

tests

As inz+ _{and t}+ _{tests, we use the standard Gaussian-inverse-Wishart priors for the} observation mean and covariance matrix,

(µ_|Σ,_I0)∼NK(m0,Σ/n0), (Σ| I0)∼IWK×K ν0,S0−1

, (6.22)

wherem0 represents a prior estimate of the value of µand n0 corresponds to the number of observations on which this prior estimate is based on, while ν0 and S0 respectively represent the degrees of freedom and the scale matrix of the inverse- Wishart prior.

Under this standard Bayesian model (see Gelman et al. [2004]), the posterior distribution of µ and Σ given the information set _Ij = {I0,X(j)}, consisting of the prior information _I0 and the data collected up to the j−th interim analysis

X_(j)= [X1X2. . .Xj] is (µ_|Σ,_Ij)∼NK mj,Σ/n(j) , (Σ_{| I}j)∼IWK×K νj,Sj−1 . (6.23)

Here, m_j = n0m0+n(j)x¯(j) n0+n(j) , (6.24) and S_j =S₀+ν(j)Sx(j)+ n0n(j) n0+n(j) ( ¯x_(j)−m₀)( ¯x_(j)−m₀)T, (6.25) whereν_(j)=n_(j)−1, νj =ν0+n(j) withn(j)=n1+n2+· · ·+nj and

¯ x_(j)= j X l=1 nl X i=1 x_il/n_(j), Sx(j) = 1 nj−1 j X l=1 nl X i=1 x_il−x¯_(j) x_il−x¯_(j)T (6.26)

are the sample mean and sample covariance matrix of X_(j). Note that, due to the positive definiteness of the prior estimates S0, the posterior estimates Sj, j =

1,2, . . . , J, are also positive definite.

We wish to use this information to select the weighting vectorswj optimally.

Optimality here is expressed in terms of predictive power of the test. The predictive power for the first stage given the prior information set_I0 isb1 =P r(p1 ≤α1,1 | I0) and for thej₋th stage,j = 2,3, . . . , J, given the information set _Ij−1 is

bj =                 

1, _Ij−1 such thatC(p_l)≤αl,1 for l∈ {1,2, . . . , j−1}, 0, _Ij−1 such thatC(pl)≥αl,0 for l∈ {1,2, . . . , j−1},

J P l=j P r C(pl′)∈(α_l,1′, α_l,0′), l′ < l; C(p_l)≤α_l,1| I_j−1 , otherwise. (6.27)

The next result presents the weighting vectors that we suggest to use for the stage- wise linear combination z andt tests.

Theorem 6.4.1. Under (6.1) and (6.22), the j₋th stage predictive power, bzj, j=

1,2, . . . , J, of theJ₋stageztest in (6.27) is maximized with respect to the weighting vector w_j if and only if w_j is proportional to

Under (6.1) and (6.22) and for n_{(j−1) → ∞}, the j₋th stage predictive power, btj,

j = 1,2, . . . , J, of the J₋stage t test in (6.27) is maximized with respect to the weighting vectorwj if and only if wj is proportional to

w_t+

j =S

−1

j−1mj−1, (6.29)

where mj, Sj as in (6.24) and(6.25), respectively.

Proof. The zstatistic of the j₋th stage, j= 1,2, . . . , J, can be written as

zj = ¯θj+e, e∼N(0,1).

Under (6.1), (6.22), (θj | Ij−1) is normally distributed with mean ˆ

θj =wjTmj−1/σj

and variance (n0+n(j−1))−1. Thus, (zj | Ij−1)∼N

ˆ

θj√nj,1 +nj/ n0+n(j−1)

. The result is then proved using theorem 6.3.1 and following the same steps as in 4.3.1 whereθj is replaced by ˆθj.

For thet₋test, we wish to compute the asymptotic,n_{(j−1) → ∞}, distribution oftj | Ij−1. By Bayes’ rule, under (6.1) and (6.22), we have that,

(xij | Ij−1)∼tK ν0+n(j−1)−K+ 1,mj−1, cSj−1, i= 1,2, . . . , nj where c = (n0 + n(j−1) + 1) (n0+n(j−1))(ν0+n(j−1)−K+ 1) −1 . Hence for n(j−1) → ∞, (xij | Ij−1)∼NK(mj−1,Sj−1).

Using the last result and corollary 7.2.3 in Anderson [1984] we have that

From the same result we have that

Sxj|Ij−1

∼WK×K(nj−1,Sj−1/(nj−1)),

and hence by proposition 3.4.2 in Mardia et al. [1979]

s2 j

w_jTS_j−1w_j/(nj−1)|Ij−1

!

∼χ2_nj−1, independent of ¯yj. (6.31)

From (6.30) and (6.31), it follows that thet statistic in (6.4) can be written as

tj | Ij−1= Z+ ˆϑj−1√nj p X2_/(n j−1) , whereZ_∼N(0,1) and X2_∼χ2

nj−1. That is,tj | Ij−1 is approximately non-central

tdistributed with non-centrality parameter ˆϑj−1√nj−1, where ˆ

ϑj−1=

w_jT₋₁m_j−1

q

w_jT₋₁S_j−1wj−1

and nj −1 degrees of freedom, as n(j−1) → ∞. By replacing θj with ˆϑj in the

proof of proposition 6.4.1, it follows that, for n_{(j−1) → ∞}, the predictive power functionbtj in (6.27) is maximized with respect to the weighting vector, wj, if wj

is proportional tow_t+

j in (6.29).

We refer to the J₋stage linear combination z and t tests with weighting vectorsw_z+

j andwt

+

j as the adaptivez

+

AD andt+AD tests, respectively.

We can easily prove that these tests satisfy the conditional invariance prin- ciple and they control the type I error. We next prove type I error control for the two-stage adaptive z+_AD and t+_AD tests, while results for J₋stage tests follow com- pletely analogously. The main argument is that the weighting vectors w_z+

j | Ij−1

and w_t+

j | Ij−1 are fixed and thus, under H0, the conditional distributions of the

distributed and the correspondingp₋values uniformly distributed,U(0,1).

Type I error control

For the two-stagez+_AD,t+_ADtests, it is sufficient to show that, underH0, thepvalues of the first and second stage, respectively, are

p1, p2 | I0 ∼U(0,1), independent,

which implies that, if the critical valuesα1,0, α1,1, α2,1 satisfy the type I error rate equation in (6.10), the type I error rate is controlled. For the rest of this section, all the distributions are computer underH0.

First see that, if the weighting vectors of the z or t statistics in (6.4) are fixed, theirpvalues are uniformly distributedU(0,1) [George and Mudholkar, 1990]. Conditional on, respectively,_I0 andI1={I0,X1}the weighting vectors of the first and second stage of the adaptivez_AD+ and t+_AD tests are fixed and thus,

(p_{1 | I}0)∼U(0,1), (p2 | I1)∼U(0,1). (6.32) It is then sufficient to show that (p2| I0)∼U(0,1) since this implies also that (p2 |

I0) andX1 are stochastically independent and thusp1, p2| I0 are also independent. Letg(_·), f(_·), and ˜g(_·;X1) be the density functions of (p2 | I0), X1 and (p2 | I1), respectively. Note that, by (6.32), ˜g(p2;X1) = 1, p2 ∈ [0,1] and thus the result follows from g(p₂) = Z ˜ g(p₂;X1)f(X1)dX1= Z f(X1)dX1= 1, p2 ∈[0,1], which implies that (p2 | I0)∼U(0,1).

Connections to other linear combination tests

Thez+andt+tests developed in chapter 4 are special cases of the adaptivez_AD+ and

t+_AD tests. For this, the pilot study is considered as the first stage of the study. The

z+ and t+ tests are thus two-stage tests with (α1,1, α1,0) = (0,1), that is, no early stopping is permitted, and C(p2) = p2. The weighting vector used in z+ and t+ tests are, under this representation, equal tow_z+

2 andwt+2, respectively, constructed based on prior information and first stage (pilot) data.

Furthermore, linear combination z and ttests with fixed weighting vectors, such as O’Brien’s OLS and GLS (for Σ unknown) tests [O’Brien, 1984], can be implemented under the adaptive design by setting n0 ≫ nT which effectively sets

the weighting vector equal to the first stage weighting vector w_z+

1. Alternatively, for group sequential linear combination z and t tests with fixed weighting vectors, one may consider the methodology described in section 5.3.1 (see pp. 72-73).

Two-sided and one-sided p-values

In the proposedz_AD+ , t+_AD as well asz+,t+ tests, the weighting vectors are allowed to be in any direction the prior information and observed data suggests. Therefore, they are not necessarily restricted to be positive although this can be attained by ma- nipulation. This approach is motivated by neuroimaging in which, as we explained earlier, contrasting effects are often exhibited and are of interest to investigators.

Due to this approach, it is natural to consider two-sidedp-values rejecting the null hypothesisH0for large absolute values of thezortstatistics regardless of their sign. This suggests that only the direction of the weighting vector or, equivalently, the effect structure is of interest and not the sign.

However, since in our methods the sign of the weighting vector in addition to the direction is chosen, one may consider one-sidedp-values in order to improve power. Specifically, the z_AD+ and t+_AD tests described earlier can be implemented

will be attained if the sign of the weighting vector is correctly chosen.

In adaptive testing methodology, it is often preferable to use one-sided p- values since for univariate stage-wise tests this prevents rejections based on effects with contrasting signs. The issue is more complicated in multivariate testing and if one wish to prevent such situations should consider one-sided tests (see section 3.3.2). Primarily due to our motivating application, in this thesis, we mainly consider tests with two-sided p-values.

6.5

Conclusions

The methodology developed in this chapter provides an important generalisation of the testing procedures described in chapter 4. The issues arising with external pilot studies are overcome with a more efficient use of the pilot data. The latter are here used not only for selecting the weighting vector, but also for testing. This potentially leads to a considerable improvement to the power performance of the test, particularly in the case where the first stage weighting vector is close to optimal. In addition, the tests are further generalised to allow for more than two stages, with the weighting vector being adapted to observed data at every interim analysis. Moreover, early stopping of the study, which is particularly important in medical settings, is permitted in these designs.

The tests control type I error under general conditions and they are optimal with respect to predictive power. They can be used in situations where a pre- specification of the adaptation rules is required. In the next chapter, we derive a power characterisation of linear combination tests, which is then used in chapter 8 to perform an extensive power analysis of these tests, including comparisons to alternative global tests. The test statistic adaptation is expected to improve power performance in cases where the initial statistic is far from optimal, while it may lead to lose of power if the latter is close to optimal.