The joint effect of several variables is a prevailing statistical concept in biology. The public health importance of developing methods to better assess joint effects is evident when studying gene combinations that function together to produce a disease phenotype, or biomarker pairs that jointly affect prognosis or treatment response.
The “weakest-link” paradigm, introduced earlier by Richards and Day, constructs derived covariates accounting for the joint effect of multiple variables. The weakest-link method posits a one-dimensional locus in covariate space, called the curve of optimal use (COU). For a data set with two predictors and an associated outcome, the COU separates the two-dimensional covariate space into two subsets. The subset of an observation determines its weakest-link covariate, which alone locally affects the corresponding outcome. With a modest generalization, one can extend weakest-link methods to assess interactions between more than two variables.
Current methods for detecting interesting variable combinations have shortcomings. Some methods, such as logic regression, require dichotomization, and lose information. Other methods such as support vector machines, are too computationally intensive, especially with large data sets.
With these issues in mind, the primary objectives in expanding the practical applications of weakest-link methodology are: (1) to develop a semi-parametric method to screen hundreds or thousands of variables for combinations associated with an outcome, (2) to adapt the method for a more complicated data structure found in a multi-parameter cell based cytometry study, where data sets typically consist of thousands of cell observations per outcome.
In a high-throughput microarray data set of breast cancer patients, conventional additive linear models and weakest-link models identified multiple combinations of biomarkers associated with lymph node positivity. Simulations of high-throughput data sets found that weakest-link models had better success than additive models in detecting covariate pairs used to generate outcomes; weakest-link models were preferable even in some situations when the additive model was the true outcome-generating model.
The weakest-link approach showed promising results in modeling recurrence-free survival in a cytometry data set of lung cancer samples. Weakest-link models, compared to logic regression and linear regression, provided the best results according to cross-validation assessments.