Computational details

Let:

Y_ijkt be the response of subject j in sequence i at time t with treatment k.

be a vector of responses for subject j in sequence i. The components of Y are arranged in the ascending order of time.

.be the design vector of treatment T in sequence i.

be a design vector for treatment R in sequence i.

n_i > 1 be the number of subjects in sequence i.

Assume the mean responses of , follow a linear model:

where m_T, m_R are the population mean responses of Y with treatments T and R respectively, and X_i are design/model matrices.

Let:

be the number of occasions T is assigned to the subjects in sequence i.

be the number of occasions R is assigned to the subjects in sequence i.

Then:

Assume that the covariances:

follow the model:

where the parameters:

are defined as follows:

intra-subject correlation coefficients are:

intra-subject covariances are:

intra-subject variances are:

For PBE and IBE investigations, it is useful to define additional parameters:

Except for s_D^*, all the quantities above are non-negative when they exist. It satisfies the equation:

In general, this s_D^* may be negative. This method for PBE/IBE is based on the multivariate model. This method is applicable to a variety of higher-order two-treatment crossover designs including TR/RT/TT/RR (the Balaam Design), TRT/RTR, or TxRR/xRTT/RTxx/TRxx/xTRR/RxTT (Table 5.7 of Jones and Kenward, page 205).

Given the i^th sequence, let:

where:

is the sample mean of the i^th sequence.

V_i is the within-sequence sample covariance matrix.

It can be shown that:

Furthermore, it can be shown that {S_iT, S_iR} (for PBE) are statistically independent from {D} and {S_iWT, S_iWR}, and that the four statistics D, S_iI, S_iWT, S_iWR (for IBE) are statistically independent.

Let a_i be sets of normalized weights, chosen to yield the method of moments estimates of the h. Then define the estimators of the components of the linearized criterion by:

Using the above notation, one may define unbiased moment estimators for the PBE criteria:

and for the IBE criteria:

Construct a 95% upper bound for h based on the TRTR/RTRT design using Howe’s approximation I and a modification proposed by Hyslop, Hsuan and Holder (2000). This can be generalized to compute the following sets of n_q, H, and U statistics:

where the degrees of freedom n_q are computed using Satterthwaite’s approximation. Then, the 95% upper bound for each h is:

If H_q < 0, that indicates bioequivalence; H_q > 0 fails to show bioequivalence.

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