Disciplines have their favorite or conventional basis set. For example, polynomials are common in psychology, and Fourier transforms are often used in engineering and physics. There exists a vast set of possibilities for basis sets in generalized splines, which can be used to impose expected structure on a model via piecewise sums of any functions. Many researchers do not further explore the possibilities for other basis sets in their research, often because the choice of basis set is perceived as irrelevant or conventional. However, the properties of basis set transformations
can be leveraged in powerful ways, especially when they can give data or model parameters more natural, domain-relevant interpretations.

By using problem-appropriate basis sets, we can find new approaches to almost any scientific question. Here, I will show how Regression Spline Mixed Models (RSMM) can combine the nonparametric features of splines with a hierarchical random effects framework to explore EEG data at any of the many levels that are collected and of interest to researchers. I will then show how a verbalized hypothesis can be translated into a basis set for formal testing. These methods can be valuable to researchers working with EEG and similar time-series biological data (fMRI, MEG, EKG, pupilometry, and others) because they allow for analyses at the many levels of the natural hierarchy of an EEG experiment. Having the ability to work at these levels is useful for recognizing outliers, learning about variance and statistical significance, and inspiring further analyses or future studies. In addition, the ability to convert verbalized hypotheses into elements in a basis set assigns interpretability to model parameters and their standard errors.

Future work involves an in-depth comparison of existing and proposed methods, particularly with respect to standard error calculation and testing. Creative use of basis sets may also inspire new approaches to other statistical problems - for example, using a polynomial basis set to form a general framework for higher-order interaction models. Traditional approaches to testing continuous-by-continuous interaction models involve artificially grouping one variable and exploring how the groups moderate the relationship between the continuous predictor and response; this new perspective may bring insights in addition to avoiding loss of information.