Truncated power basis functions are simple semiparametric functions that approximate curves. We define a truncated power function at a given knot ${\kappa }_k$ as

where $p$ is the order of the polynomial function, and $k=\mathrm{\hspace{0.25em}1},\mathrm{\dots },K$ represents the number of knots [21]. The functions are differentiable up to $p-\mathrm{\hspace{0.25em}1}$ times [20, 21, 22, 23]. In modeling mean functions, TPBFs approximate curves based on polynomial expansions. A mixed effects model based on truncated power basis is

(2)$$Y_i=X_i\beta +{\beta }_{0i}+{\sum }_{k=1}^p{\alpha }_k{\left(x_i-{\kappa }_k\right)}_{+}^p+{ϵ}_i$$

where $Y_i$ is the n${}_i$$\times$ 1 vector of responses for the $i^{th}$ subject, $n_i$ represents the total number of responses per subject, $X_i\beta$ is the fixed part of the model, and ${\beta }_{0i}$ is the subject specific random intercept. The term ${\left(x_i-{\kappa }_k\right)}^p$ is a $p^{th}$ order truncated power basis of degree $p$, with ${\kappa }_k$ representing the $k^{th}$ knot [20]. Eqn. 2 is a polynomial piece-wise regression model with separate slopes, (${\alpha }_k$), fit to different partitions of the predictor variable. Thus, ${\left(x_i-{\kappa }_k\right)}_{+}$ is an indicator variable indicating the partition where ${\left(x_i-{\kappa }_k\right)}_{+}$ is positive. Knots are the points where adjacent partitions meet. For effective estimation, the TPBF approach requires an adequate number of knots or penalization [21].

Our cubic TPBF model of energy expenditure is

(3)$$\begin{array}{c}\hfill Y_{ij}={\beta }_0+{\beta }_1*tim{e}_i+{\beta }_2*tim{e}_i^2+{\beta }_3*{\mathrm{time}}_i^3\\ \hfill +{\beta }_4*lo{w}_i+{\beta }_5*hig{h}_i\\ \hfill +{\sum }_{i=1}^k{\mu }_k{\left(tim{e}_i-{\kappa }_k\right)}_{+}^3+b_{0i}+{\epsilon }_{ij}\end{array}$$

${\beta }_1$, ${\beta }_2$, ${\beta }_3$ are the fixed coefficients for the linear, quadratic, and cubic terms for time, respectively, and ${\beta }_4$ and ${\beta }_5$ represent the low interferon tau and high interferon tau groups’ contrasts, respectively, with the control group. The ${\left(tim{e}_i-{\kappa }_k\right)}_{+}^3$ term is the cubic spline basis. We treat the truncated cubic basis splines and the intercept, $b_{0i}$, as random and assume ${\mu }_k\sim Normal(0,{\sigma }_u^2)$ and $b_{0i}\sim Normal(0,{\sigma }_b^2)$. When ${\sigma }_u^2=0$, Eqn. 2 reduces to a mixed effects model. The random effects ${\left(tim{e}_i-{\kappa }_k\right)}_{+}^{\left(p\right)}$, which we model as normal random curves with mean zero [23], are not present in the LMEM in Eqn. 1. The smoothness of the spline regression rises with increasing degree of the polynomial [23]. The smoothing parameter, $\lambda =\sqrt{\frac{{\sigma }_u^2}{{\sigma }_{\epsilon }^2}}$, controls the smoothness of the curve, while the mean square error of the model grows with increasing $\lambda$ [22, 23]. Although easy to construct, models based on the TPBF can be numerically unstable due to correlations between the basis functions. When the range for $x_i$ in Eqn. 2 is wide, the basis functions increase rapidly as $x$ rises. To resolve this issue, the range for $x_i$ may be re-scaled to $[0,1]$. These disadvantages make the models prone to computational difficulties [21]. B-splines allow analysts to avoid these problems [21, 24, 25].

B-splines allow flexible approaches to analyzing data [21, 25]. B-splines are piece-wise polynomial functions of order $p$ connected at their inner knots [19, 21, 24, 26]. While B-splines are equivalent to TPBFs on any given interval $[{\kappa }_0,{\kappa }_m]$, they are more numerically stable [20, 21, 27]. B-splines are transformations of TPBFs [20, 21]. To illustrate their equivalence, let $X_T$ and $X_B$ be design matrices for the TPBF and the B-spline basis functions of the same degree and same knot locations, respectively. Then $X_B=X_T{L}_p$ where $L_p$ is a square invertible matrix [20].

B-spline basis functions are nonzero over the interval $[k_0,k_{m+1}]$. Next, let $\kappa =({\kappa }_0,{\kappa }_1,\mathrm{\dots },{\kappa }_m)$ be a set of $m+\mathrm{\hspace{0.25em}1}$ non-decreasing knots. The domain for B-splines is $[{\kappa }_0,{\kappa }_m]$, with $k_0=\mathrm{\hspace{0.25em}0}$ and $k_m=\mathrm{\hspace{0.25em}1}$, typically representing the two boundary knots [24]. We define the $k^{th}$ B-spline basis function of degree $p$ recursively as

${\mathrm{B}}_{\mathrm{k},\mathrm{p}}(\kappa )={\displaystyle \frac{\kappa -{\kappa }_{\mathrm{k}}}{{\kappa }_{\mathrm{k}+\mathrm{p}}-{\kappa }_{\mathrm{k}}}}{\mathrm{B}}_{\mathrm{k},\mathrm{p}-1}(\kappa )+{\displaystyle \frac{{\kappa }_{\mathrm{k}+\mathrm{p}+1}-\kappa }{{\kappa }_{\mathrm{k}+\mathrm{p}+1}-{\kappa }_{\mathrm{k}+1}}}{\mathrm{B}}_{\mathrm{k}+1,\mathrm{p}-1}(\kappa )$

In our analyses, we specified the B-spline models as

(4)$$Y_i=X_i\beta +{\beta }_{0i}+{\gamma }_{i}B+\delta B+{ϵ}_i$$

where $Y_i$ is the $n_i\times 1$ vector of responses for the $i^{th}$ subject, $X_i\beta$ and $\delta B$ are the fixed effects, and ${\beta }_{0i}$ and ${\gamma }_i$ are the subject-specific random intercepts and random slopes for the B-spline basis functions, respectively.

The TPBF models fit the data substantially better than the LMEMs (Table 4; Figs. 2d–f,3d–f). The linear spline TPBF model fit raw heat production at the scale of minutes best, while the quadratic spline model fit hourly mean heat production best. As in the LMEMs, there were no statistically significant treatment effects in any TPBF model. Also, TPBF models of hourly mean heat production fit better and had lower AIC values when compared to the AIC values for the analyses conducted at the minute-levels. For the cubic spline models, the higher order terms for time (quadratic and cubic) were not statistically significant. However, both the linear and quadratic terms for time were statistically significant quadratic spline models in the paired models that differed only in time scale.

The B-spline SMEMs (Table 5; Figs. 2g–i,3g–i) fit the data better than the TPBF models and LMEMs. As with all of the other models, there were no statistically significant treatment effects in the B-spline models. Also, B-spline models of hourly mean heat production fit better and had lower standard errors of coefficients than models of raw heat production at the time scale of minutes, although the patterns of coefficients for paired models were similar. The quadratic B-spline analyzed at the hourly level performed the best of all models we estimated for both mean hourly untransformed heat production and untransformed heat production at the time scale of minutes. Time was not statistically significant in the linear spline models for the analyses performed at the hourly and minute levels. However, the linear and quadratic terms for time were statistically significant in the quadratic spline model performed on a the hour level time scale. The quadratic and cubic terms for time were not statistically significant in the cubic spline models.