The linear partial spline model assumes that (Wahba, 1990)

where the first part is a linear model of covariates , and as in (). Note that an SS ANOVA model discussed in the next section can also be used for when is multivariate. Partial spline models provide a tool to model multiple covariates when the relationship is unknown for only a few variables. Note that some 's may be functions of . For example, allows a jump in the th derivative at .

Let be the design matrix of : , and . If is of full column rank, the estimate of has the same representation as in (). Furthermore, coefficients and are solutions to equations () with replaced by .

The linear model for
in ()
can be easily specified by adding these covariates to the right
hand side of `formula`. For example, supposing and
a cubic spline for , we can fit model () by

ssr(y~x1+x2+x3+t, rk=cubic(t))