Table 1 Summary of available models based on functional regression framework.

From: Introduction of FixFRM package for gene-based association test and its innovative application to mixture exposure analysis

Model

Intercept

Covariate

Genotype

Genetic variants

Genetic effect

Beta-smooth only model

\(\:{\alpha\:}_{0}\)

\(\:{\varvec{Z}}_{i}^{T}\varvec{\alpha\:}\)

\(\:{\varvec{G}}_{i}\)

\(\:\beta\:\left(t\right)\) expanded by OLS smoother

\(\:\beta\:\left(t\right)\) expanded by linear spline basis

Fixed effect model

\(\:{\alpha\:}_{0}\)

\(\:{\varvec{Z}}_{i}^{T}\varvec{\alpha\:}\)

\(\:{X}_{i}\left(t\right)\) expanded by OLS smoother

\(\:\beta\:\left(t\right)\) expanded by OLS smoother

\(\:\beta\:\left(t\right)\) expanded by linear spline basis

FPCA model

\(\:{\alpha\:}_{0}\)

\(\:{\varvec{Z}}_{i}^{T}\varvec{\alpha\:}\)

\(\:{X}_{i}\left(t\right)\) expanded by functional principal component analysis

\(\:\beta\:\left(t\right)\) expanded by OLS smoother

\(\:\beta\:\left(t\right)\) expanded by linear spline basis

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