Table 1.

Forms of $f_r({\mathcal{X}}_r,t)$ depending on the covariates in ${\mathcal{X}}_r$ and linearity or smoothness in these covariates (rows), and on whether the effect is constant or varying over t (columns). For scalar categorical covariates, synthetic scalar covariates in effect or reference category coding are created. Note that effects can become interaction effects if ${\mathcal{X}}_r$ additionally contains such scalar categorical covariates. For example, we estimate group-specific effects of the functional covariates for MS patients and healthy controls in our DTI application.

${\mathcal{X}}_r$	$f_r({\mathcal{X}}_r,t)$ constant over t	$f_r({\mathcal{X}}_r,t)$ varying over t
∅ (no covariates)	scalar intercept α	functional intercept α(t)
functional covariate x(s)	linear functional effect ${\int }_{\mathcal{Y}}x\left(s\right)\beta \left(s\right)ds$	linear functional effect ${\int }_{\mathcal{Y}}x\left(s\right)\beta (s,t)ds$
	smooth functional effect ${\int }_{\mathcal{Y}}F(x\left(s\right),s)ds$	smooth functional effect ${\int }_{\mathcal{Y}}F(x\left(s\right),s,t)ds$
scalar covariate z	linear effect zδ	functional linear effect zδ(t)
	smooth effect γ(z)	smooth effect γ(z, t)
vector of scalar covariates z	interaction effect z1z2δ	functional interaction effect z1z2δ (t)
	varying coefficient z1δ(z2)	functional varying coefficient z1δ(z2, t)
	smooth effect γ(z)	smooth effect γ(z,t)
grouping variable g	random intercept bg	functional random intercept bg(t)
grouping variable g and scalar covariate z	random slope zbg	functional random slope zbg(t)