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Modeling Approach

The Joint Model

Let X be the gene expression matrix where Xj is the jth gene expression of the ith compound, i = 1,... ,n and j = 1,... ,m. Let Y denote the measurement for the bioassay data. Both gene expression and bioassay read-outs are assumed to be normally distributed. Let Z be the binary chemical structure or fingerprint feature matrix in which the kith element takes a value of one, zki = 1, or zero, zki = 0, if the kth fingerprint feature is respectively present/absent in the ith compound.

For a given fingerprint feature, the gene-specific joint model that allows testing for which gene is also differentially expressed and which gene is predictive of the response irrespective of the effect of the fingerprint feature is given by

where the error terms have a joint zero-mean normal distribution with gene- specific covariance matrix, Xj:

The parameters aj and в represent the fingerprint feature effects for the jth gene and the response, respectively, and p,j and are gene-specific and the response-related intercepts, respectively. Note that this model is identical to the joint model for a single-trial setting discussed in Chapter 3. Thus, the gene-specific association with the response can be obtained using the adjusted association (Buyse and Molenberghs, 1998), a coefficient that is derived from the covariance matrix, Xj, of gene-specific joint model (16.2):

Indeed, pj = 1 indicates a deterministic relationship between the gene expression and the response after accounting for the effect of a fingerprint feature.

 
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