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Fitting Nonlinear Ordinary Differential Equation Models with Random Effects and Unknown Initial Conditions Using the Stochastic Approximation Expectation–Maximization (SAEM) Algorithm

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Abstract

The past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in social sciences. Clearly lacking, however, are modeling tools that allow researchers to fit dynamic models to irregularly spaced data, particularly data that show nonlinearity and heterogeneity in dynamical structures. We consider the issue of fitting multivariate nonlinear differential equation models with random effects and unknown initial conditions to irregularly spaced data. A stochastic approximation expectation–maximization algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model, namely, the Van der Pol oscillator equations. The empirical utility of the proposed technique is illustrated using a set of 24-h ambulatory cardiovascular data from 168 men and women. Pertinent methodological challenges and unresolved issues are discussed.

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Notes

  1. The local truncation errors of a numerical solver at each time point are equal to \(\varvec{c} \Delta ^{g+1}_{i, j}\), where \(g\) is the order of the ODE solver and \(\varvec{c}\) is a vector of constants that depends on elements such as the differentials of the ODEs (for further details see Press, Teukolsky, Vetterling, & Flannery, 2002; Ralston & Rabinowitz, 2001).

  2. In contrast, in cases involving SDEs, \(p(\tilde{\varvec{\mathrm {X}}}|\varvec{b}; \varvec{\theta })\) is not fixed even when \(\varvec{b}\) is known and there is considerable increase in estimation complexity.

  3. In the present context, we specify the gain constant to be \(\gamma ^{(m)} = a_2/(m^{a_1}+a_2-1), \quad m=1, \ldots , K_1+K_2,\) where the real number \(a_1\) and the integer \(a_2\) are preassigned. In stage 1, \(a_1\) and \(a_2\) are selected such that the gain constant assumes some large values to prevent the SAEM algorithm from settling into local minima too quickly. In stage 2, the gain constant is slowly tapered toward zero to allow the algorithm to stabilize toward a final set of estimates (e.g., by setting \(a_1 \in (.5, 1]\) to be close to 1, and \(a_2\) to be a small integer, say, \(a_2=2\)). The transition from stage 1 to 2 is governed by another predefined criterion function (for details see Gu & Zhu, 2001; Zhu & Gu, 2007).

  4. In the present study, we define the stopping rule to be

    $$\begin{aligned} K_2&= \inf \Bigg \{m: \tilde{\varvec{s}}^{'(m)}_{\varvec{\mathrm {Y}}} \Big [\tilde{\varvec{\mathrm {I}}}^{(m)}_{\varvec{\mathrm {Y}}}\Big ]^{-1} \tilde{\varvec{s}}^{(m)}_{\varvec{\mathrm {Y}}} + \mathop {\mathrm {tr}}\Big \{ \Big [\tilde{\varvec{\mathrm {I}}}^{(m)}_{\varvec{\mathrm {Y}}}\Big ]^{-1}\hat{\varvec{\mathrm {\Sigma }}} \Big \}/m \le \text { some small constant}\Bigg \}. \end{aligned}$$
    (13)

    \(\hat{\Sigma }\) denotes an estimate of the covariance matrix of Monte Carlo error. In practice, we used the sample covariance matrix of \(\overline{\varvec{s}}_{\varvec{\mathrm {Z}}}^{(m)}\) as a rough estimate of \(\hat{\varvec{\mathrm {\Sigma }}}\).

  5. This decision was made because there were insufficient repeated measurements to estimate this parameter accurately, especially for individuals with a diurnal cycle that is longer than 24 h, or individuals with less than 24 h worth of measurements.

  6. Because the model fitting procedures were based on the second-order Heun’s method whereas the true data were generated using a fourth-order Runge Kutta approach, the errors entailed from approximating the trajectories from the fourth-order solver by means of a second-order solver were expected to lead to some biases in the point estimates. Thus, the coverage performance of the confidence intervals as assessed, e.g., by the proportion of 95 % CIs covering each true population parameter value, can be expected to deviate from the nominal coverage rate of 0.95.

    Table 2 Parameter estimates for the Van der Pol oscillator model with \(T = 150\), true initial condition = fixed, fitted initial condition = fixed.
    Full size table
    Table 3 Parameter estimates for the Van der Pol oscillator model with \(T = 300\), true initial condition = fixed, fitted initial condition = fixed.
    Full size table
    Table 4 Parameter estimates for the Van der Pol oscillator model with \(T = 150\), true initial condition = fixed, fitted initial condition = random.
    Full size table
    Table 5 Parameter estimates for the Van der Pol oscillator model with \(T = 300\), true initial condition = fixed, fitted initial condition = random.
    Full size table
    Table 6 Parameter estimates for the Van der Pol oscillator model with \(T = 150\), true initial condition = random, fitted initial condition = random.
    Full size table
    Table 7 Parameter estimates for the Van der Pol oscillator model with \(T = 300\), true initial condition = random, fitted initial condition = random.
    Full size table
    Table 8 Parameter estimates for the Van der Pol oscillator model with \(T = 150\), true initial condition = random, fitted initial condition = fixed.
    Full size table
    Table 9 Parameter estimates for the Van der Pol oscillator model with \(T = 300\), true initial condition = random, fitted initial condition = fixed.
    Full size table

  7. To ensure the positive definiteness of \(\varvec{\mathrm {\Sigma }}_b\), we chose to estimate the lower triangular entries of \(\varvec{\mathrm {L}}\) and the diagonal entries of \(\varvec{\mathrm {D}}\) in the \(\varvec{\mathrm {\Sigma }}_b = LDL^\mathrm{T}\) decomposition, with the constraint that the diagonal elements of \(\varvec{\mathrm {D}}\) were positive (Anderson, 2003). These constraints might have affected the accuracy of the point and SE estimates for the initial condition variance–covariance parameters as well.

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Acknowledgments

Funding for this study was provided by NSF Grant BCS-0826844, NIH Grants RR025747-01, P01CA142538-01, MH086633, EB005149-01, AG033387, and R01GM105004.

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Authors and Affiliations

  1. The Pennsylvania State University, 413 Biobehavioral Health Building, University Park, PA, 16802 , USA

    Sy-Miin Chow

  2. University of North Carolina at Chapel Hill, Chapel Hill, USA

    Zhaohua Lu & Hongtu Zhu

  3. Duke University, Durham, USA

    Andrew Sherwood

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Appendices

Appendix 1: Score Vector and Information Matrix of the Complete-Data Loglikelihood Function

The elements in \(\varvec{s}_{\varvec{\mathrm {Z}}}(\varvec{\theta }; \varvec{\mathrm {Z}})\) and \(\varvec{\mathrm {I}}_{\varvec{\mathrm {Z}}}(\varvec{\theta }\varvec{\mathrm {Z}})\), namely, the score vector and information matrix of the complete-data loglikelihood function, are computed as

$$\begin{aligned} \varvec{s}_{\varvec{\mathrm {Z}}}(\varvec{\theta }; \varvec{\mathrm {Z}})&= \frac{\partial L(\varvec{\mathrm {Z}};\varvec{\theta })}{\partial \varvec{\theta }} = \displaystyle {\sum _{i=1}^{n}} \begin{bmatrix} \Big (\sum _{j=1}^\mathrm{T}\frac{\partial L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta }) }{\partial \varvec{\beta }}\Big )\\ \Big (\sum _{j=1}^\mathrm{T}\frac{\partial L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mu }}}\Big )\\ \Big (\sum _{j=1}^\mathrm{T}\frac{\partial L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}} \Big )\\ \Big (\sum _{j=1}^\mathrm{T}\frac{\partial L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta })}{\partial \varvec{\theta }_{\varvec{\epsilon }}}\Big )\\ \Big (\frac{\partial L_{i}(\varvec{b}; \varvec{\theta })}{\partial \varvec{\theta }_{\varvec{b}}}\Big ) \end{bmatrix}, \end{aligned}$$
(17)
$$\begin{aligned} \varvec{\mathrm {I}}_{\varvec{\mathrm {Z}}}(\varvec{\theta };\varvec{\mathrm {Z}})&= -\frac{\partial ^2 L(\varvec{\mathrm {Z}};\varvec{\theta })}{\partial \varvec{\theta } \partial \varvec{\theta }^\mathrm{T}}=- \displaystyle {\sum _{i=1}^{n}} \mathop {\mathrm {Diag}}\begin{bmatrix} \sum _{j=1}^\mathrm{T}\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b};\varvec{\theta }) }{\partial \varvec{\beta } \partial \varvec{\beta }^\mathrm{T}}\\ \sum _{j=1}^\mathrm{T}\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mu }} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\mu }}} \\ \sum _{j=1}^\mathrm{T}\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\Lambda }}}}\\ \sum _{j=1}^\mathrm{T}\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}| \varvec{b}; \varvec{\theta })}{\partial \varvec{\theta }_{\varvec{\epsilon }} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\epsilon }}}\\ \frac{\partial ^2 L_{i}(\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{b}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}} \end{bmatrix}, \end{aligned}$$
(18)

where \(\mathop {\mathrm {Diag}}(.)\) denotes a block diagonal matrix formed by stacking the appropriate second partial derivative matrices in its diagonal section and zero matrices in its off-diagonal sections.

Using Heun’s method, with \(\tilde{\varvec{x}}_i(t_{i,j})\) as defined in Eq. (4) and \(\varvec{z}_i(t_{i,j}) = [\varvec{y}_i(t_{i, j})-\varvec{\mu }-\varvec{\mathrm {\Lambda }}\tilde{\varvec{x}}_i(t_{i, j})]\), first-order partial derivative elements of the complete-data loglikelihood are given by

$$\begin{aligned} \frac{\partial L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\beta }}&= -\,\frac{1}{2} \Bigg \{ \frac{\partial \varvec{\theta }_{f,i}}{\partial \varvec{\beta }} \frac{\partial \tilde{\varvec{x}}_i(t_{i,j})}{\partial \varvec{\theta }_{f,i}} \frac{\partial \varvec{z}_i(t_{i,j})}{\partial \tilde{\varvec{x}}_i(t_{i,j})} \frac{\partial \varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \varvec{z}_i(t_{i,j})}{\partial \varvec{z}_i(t_{i,j})} \Bigg \} \nonumber \\&= \varvec{\mathrm {H}}^\mathrm{T}_i \frac{\partial \tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}}{\partial \varvec{\theta }_{f,i}}\varvec{\mathrm {\Lambda }}^\mathrm{T} \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \varvec{z}_i(t_{i,j}), \nonumber \\ \frac{\partial L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mu }}}&= -\,\frac{1}{2} \Bigg \{\frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}} \frac{\partial \varvec{z}_i(t_{i,j})}{\partial \varvec{\mu }}\frac{\partial \varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \varvec{z}_i(t_{i,j})}{\varvec{z}_i(t_{i,j})} \Bigg \} \nonumber \\&= \frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}} \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{z}_i(t_{i,j}),\nonumber \\ \frac{\partial L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}}&= -\,\frac{1}{2} \Bigg \{\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}} \frac{\partial \varvec{z}_i(t_{i,j})}{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}\frac{\partial \varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \varvec{z}_i(t_{i,j})}{\partial \varvec{z}_i(t_{i,j})} \Bigg \} \nonumber \\&= \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}} \mathop {\mathrm {vec}}\Big [ \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{z}_i(t_{i,j})\tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}\Big ],\nonumber \\ \frac{\partial L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\epsilon }}}&= -\,\frac{1}{2} \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{\partial \varvec{\theta }_{\varvec{\epsilon }}} \Bigg \{\frac{\partial \varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \varvec{z}_i(t_{i,j})}{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma }}_{\varvec{\epsilon }})} + \frac{\partial \log |\varvec{\mathrm {\Sigma }}_{\varvec{\epsilon }}|}{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma }}_{\varvec{\epsilon }})} \Bigg \} \nonumber \\&= \frac{1}{2} \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{\partial \varvec{\theta }_{\varvec{\epsilon }}}\Bigg \{ [\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \otimes \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}] \mathop {\mathrm {vec}}[\varvec{z}_i(t_{i,j}) \varvec{z}_i(t_{i,j})^\mathrm{T}-\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}] \Bigg \}, \nonumber \\ \frac{\partial L_{i}(\varvec{b}; \varvec{\theta })}{\partial \varvec{\theta }_{\varvec{b}}}&= -\frac{1}{2} \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{\partial \varvec{\theta }_{\varvec{b}}} \Bigg \{ \frac{\partial \varvec{b}^\mathrm{T}_i \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1} \partial \varvec{b}_i}{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma }}_{\varvec{b}})} + \frac{\partial \log |\varvec{\mathrm {\Sigma }}_{\varvec{b}}|}{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma }}_{\varvec{b}})} \Bigg \} \nonumber \\&= \frac{1}{2} \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{\partial \varvec{\theta }_{\varvec{b}}} \Bigg \{ [\varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1} \otimes \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1}] \mathop {\mathrm {vec}}(\varvec{b}_i \varvec{b}_i^\mathrm{T}-\varvec{\mathrm {\Sigma _{\varvec{b}}}}) \Bigg \}, \end{aligned}$$
(19)

where the \(\mathop {\mathrm {vec}}(\varvec{\mathrm {W}})\) operator stacks the columns of the \(m \times n\) matrix \(\varvec{\mathrm {W}}\) into an \(mn\)-dimensional column vector and \(\frac{\partial \tilde{\varvec{x}}_i(t_{i, j})^\mathrm{T}}{\partial \varvec{\theta }_{f,i}}\) is dictated by the dynamic model under consideration. Terms such as \(\frac{\partial \varvec{\mu } }{\partial \varvec{\theta }_{\varvec{\mu }}}, \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}}, \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{\partial \varvec{\theta }_{\varvec{\epsilon }}}, \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\eta }}}})}{\partial \varvec{\theta }_{\varvec{\eta }}}\), and \(\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{\partial \varvec{\theta }_{\varvec{b}}}\) also depend on the model specification adopted in a particular application. Cases where some elements of \(\varvec{\mathrm {\Lambda }}, \varvec{\mathrm {\Sigma }}_{\varvec{\epsilon }}, \varvec{\mathrm {\Sigma }}_{\varvec{\eta }}\), and \(\varvec{\mathrm {\Sigma }}_{\varvec{b}}\) are fixed at known values can be readily accommodated through appropriate specification of these matrices of partial derivatives.

Second-order partial derivative elements of the complete-data loglikelihood function are computed as

$$\begin{aligned} \frac{\partial ^2 L_{i,j}({\varvec{\mathrm {Y}}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\beta } \partial {\varvec{\beta }}^\mathrm{T}}&= \Big (\varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{\mathrm {\Lambda }}^\mathrm{T}\otimes \varvec{\mathrm {I}}_{p_{\beta }} \Big ) \frac{\partial \mathop {\mathrm {vec}}}{\partial {\varvec{\beta }}^\mathrm{T}}\Bigg \{\varvec{\mathrm {H}}^\mathrm{T}_i\frac{\partial \tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}}{\partial \varvec{\theta }_{f,i}} \Bigg \}\\&\quad +\; \Bigg (\varvec{\mathrm {H}}^\mathrm{T}_i \frac{\partial {\tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}}}{\partial \varvec{\theta }_{f,i}}\Bigg ) \frac{\partial \varvec{\mathrm {\Lambda }}^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \varvec{z}_i(t_{i,j})}{\partial \varvec{z}_i(t_{i, j})^\mathrm{T}}\frac{\partial \varvec{z}_i(t_{i,j})}{\partial \tilde{\varvec{x}}_i(t_{i,j})^\mathrm{T}} \frac{\partial \tilde{\varvec{x}}_i(t_{i, j})}{\partial {\varvec{\theta }}^\mathrm{T}_{f,i}} \frac{\partial \varvec{\theta }_{f,i}}{\partial {\varvec{\beta }}^\mathrm{T}} \\&= \Big (\varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{\mathrm {\Lambda }}\otimes \varvec{\mathrm {I}}_{p_{\beta }} \Big ) (\varvec{\mathrm {I}}_{n_x} \otimes \varvec{\mathrm {H}}^\mathrm{T}_i)\frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta _{f,i}}^\mathrm{T}}\Bigg \{\frac{\partial \tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}}{\partial {\varvec{\theta }}_{f,i}} \Bigg \}\varvec{\mathrm {H}}_i\\&\quad -\; \varvec{\mathrm {H}}^\mathrm{T}_i \frac{\partial \tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}}{\partial \varvec{\theta }_{f,i}}\varvec{\mathrm {\Lambda }}^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{\mathrm {\Lambda }}\frac{\partial \tilde{\varvec{x}}_i(t_{i,j})}{\partial \varvec{\theta }^\mathrm{T}_{f,i}}\varvec{\mathrm {H}}_i,\\ \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mu }} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\mu }}}&=\Big (\varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \otimes \varvec{\mathrm {I}}_{p_{\mu }}\Big )\frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{\mu }}}\Bigg \{\frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}}\Bigg \} - \frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}} \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}^\mathrm{T}}, \\ \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\Lambda }}}}&= \Bigg \{\mathop {\mathrm {vec}}[\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{z}_i(t_{i,j})\tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}]^\mathrm{T}\otimes \varvec{\mathrm {I}}_{p_{\Lambda }}\Bigg \}\frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\Lambda }}}}\Bigg \{\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}} \Bigg \} \\&\quad -\,\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}} [\tilde{\varvec{x}}_i(t_{i,j})\tilde{\varvec{x}_i}(t_{i,j})^\mathrm{T}\otimes \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}] \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\Lambda }}}}, \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\epsilon }} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\epsilon }}}&= \frac{1}{2}\Big [\mathop {\mathrm {vec}}\Big (\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{z}_i(t_{i,j})\varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \Big )^\mathrm{T} \otimes \varvec{\mathrm {I}}_{p_{\epsilon }} \Big ] \frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\epsilon }}}} \Bigg \{\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{ \partial \varvec{\theta }_{\varvec{\mathrm {\epsilon }}} } \Bigg \} \\&\quad -\;\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{ \partial \varvec{\theta }_{\varvec{\mathrm {\epsilon }}} } [\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \otimes \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{z}_i(t_{i,j})\varvec{z}_i(t_{i,j})^\mathrm{T}\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} ]\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{ \partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\epsilon }}} }\\&\quad -\;\frac{1}{2}[\mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma ^{-1}_{\varvec{\epsilon }}}})^\mathrm{T} \otimes \varvec{\mathrm {I}}_{p_{\epsilon }}]\frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\epsilon }}}}\Bigg \{\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{ \partial \varvec{\theta }_{\varvec{\mathrm {\epsilon }}} } \Bigg \}\\&\quad +\;\frac{1}{2}\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{ \partial \varvec{\theta }_{\varvec{\mathrm {\epsilon }}} } (\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} \otimes \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1})\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}})}{ \partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\epsilon }}} },\\ \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 L_{i}(\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{b}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}&=\frac{1}{2}\Big [\mathop {\mathrm {vec}}\Big (\varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1}\varvec{b}_i\varvec{b}^\mathrm{T}_i \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1} \Big )^\mathrm{T} \otimes \varvec{\mathrm {I}}_{p_{b}} \Big ] \frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}} \Bigg \{\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{ \partial \varvec{\theta }_{\varvec{b}} } \Bigg \} \nonumber \\&\quad -\; \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{ \partial \varvec{\theta }_{\varvec{b}} } \left[ \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1} \otimes \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1}\varvec{b}_i\varvec{b}^\mathrm{T}_i \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1} \right] \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{ \partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}\nonumber \\&\quad -\;\frac{1}{2} [\mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma ^{-1}_{\varvec{b}}}})^\mathrm{T} \otimes \varvec{\mathrm {I}}_{p_{b}}]\frac{\partial \mathop {\mathrm {vec}}}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}\Bigg \{\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{ \partial \varvec{\theta }_{\varvec{b}} } \Bigg \}\nonumber \\&\quad +\; \frac{1}{2}\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{ \partial \varvec{\theta }_{\varvec{b}} } (\varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1} \otimes \varvec{\mathrm {\Sigma _{\varvec{b}}}}^{-1})\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Sigma _{\varvec{b}}}})}{ \partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}, \nonumber \\ \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta _{\varvec{\mu }}} \partial \varvec{\theta _{\varvec{\mathrm {\Lambda }}}}^\mathrm{T}}&= -\, \frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}} \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1} [\tilde{\varvec{x}}_i(t_{i,j})^\mathrm{T} \otimes \varvec{\mathrm {I}}_{n_y} ] \frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }^\mathrm{T}_{\varvec{\mathrm {\Lambda }}}},\nonumber \\ \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mu }}\partial \varvec{\beta }^\mathrm{T}}&= -\,\frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}} \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\frac{\partial \varvec{z}_i(t_{i,j})}{\partial \tilde{\varvec{x}}_i(t_{i,j})^\mathrm{T}} \frac{\partial \tilde{\varvec{x}}_i(t_{i, j})}{\partial \varvec{\theta }^\mathrm{T}_{f,i}} \frac{\partial \varvec{\theta }_{f,i}}{\partial \varvec{\beta }^\mathrm{T}},\nonumber \\&=-\,\frac{\partial \varvec{\mu }}{\partial \varvec{\theta }_{\varvec{\mu }}} \varvec{\mathrm {\Sigma _{\varvec{\epsilon }}}}^{-1}\varvec{\mathrm {\Lambda }}\frac{\partial \tilde{\varvec{x}}_i(t_{i,j})}{\partial \varvec{\theta }^\mathrm{T}_{f,i}}\varvec{\mathrm {H}}_i,\nonumber \\ \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}\partial \varvec{\mathrm {\beta }}^\mathrm{T}}&=\frac{\partial \mathop {\mathrm {vec}}(\varvec{\mathrm {\Lambda }})}{\partial \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}} \Bigg \{ (\varvec{\mathrm {I}}_{n_x}\otimes \varvec{\mathrm {\Sigma }}_{\epsilon }^{-1}) \Big [ -(\tilde{\varvec{x}}_i(t_{i,j})^\mathrm{T} \otimes \varvec{\mathrm {I}}_{n_y})\varvec{\mathrm {\Lambda }}\frac{\partial \tilde{\varvec{x}}_i(t_{i,j})}{\partial \varvec{\theta }^\mathrm{T}_{f,i}}\varvec{\mathrm {H}}_i \nonumber \\&\quad +\; (\varvec{\mathrm {I}}_{n_x} \otimes \varvec{z}_i(t_{i,j})) \frac{\partial \tilde{\varvec{x}}_i(t_{i,j})}{\partial \varvec{\theta }^\mathrm{T}_{f,i}} \varvec{\mathrm {H}}_i\Big ]\Bigg \}. \end{aligned}$$
(20)

Other second-order derivative elements are equal to null matrices, including \(\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\theta }^\mathrm{T}_{\varvec{\theta }_{\varvec{\mu }}}},\) \( \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\theta }^\mathrm{T}_{\varvec{\theta }_{\varvec{\mathrm {\Lambda }}}}}, \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\theta }^\mathrm{T}_{\varvec{\theta }_{\varvec{\epsilon }}}}, \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}, \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\theta _{\varvec{\mu }}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}, \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta })}{\partial \varvec{\theta _{\varvec{\mathrm {\Lambda }}}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}\) and \(\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b};\varvec{\theta })}{\partial \varvec{\theta _{\varvec{\mathrm {\epsilon }}}}\partial \varvec{\theta }^\mathrm{T}_{\varvec{b}}}\). Under the assumption that the model is correctly specified, the elements in \(\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta _{\varvec{\mathrm {\Lambda }}}} \partial \varvec{\theta }^\mathrm{T}_{\varvec{\epsilon }}}, \frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta _{\varvec{\mu }}} \partial \varvec{\theta _{\varvec{\epsilon }}}^\mathrm{T}}\) and \(\frac{\partial ^2 L_{i,j}(\varvec{\mathrm {Y}}|\varvec{b}; \varvec{\theta }) }{\partial \varvec{\theta _{\varvec{\beta }}} \partial \varvec{\theta _{\varvec{\epsilon }}}^\mathrm{T}}\) are close to zeros at the MLEs of the modeling parameters. These elements are thus set to null matrices in the proposed estimation algorithm to stabilize the algorithm when initial parameter estimates are far from the MLEs and are not shown here. In addition, the off-diagonal elements shown in the last three equations in (20) are non-zero even near the MLEs. However, setting all the off-diagonal blocks of the information matrix of the complete-data loglikelihood function to null matrices helps stabilize the algorithm in case this information matrix is not positive definite in the optimization process. In our preliminary simulations, we verified that setting these three matrices to null matrices, as opposed to the forms as shown in Eq. (20), actually helped reduce numerical problems in the optimization process while having negligible effects on the final point and SE estimates because we are not using this matrix directly as the Fisher information matrix to derive the final SE estimates. We thus proceeded to setting all the off-diagonal elements, including the last three matrices shown in Eq. (20), to null matrices.

Appendix 2: Sampling From \(p(\varvec{b}| \varvec{\mathrm {Y}};\varvec{\theta }^{(m-1)})\)

The superscript \((m-1)\) of \(\varvec{\theta }\) is temporarily suppressed for notational simplicity. It can be shown that \(p(\varvec{b}| \varvec{\mathrm {Y}};\varvec{\theta })=\prod _{i=1}^{n} p(\varvec{b}_i| \varvec{\mathrm {Y}};\varvec{\theta })\), where \(p(\varvec{b}_i| \varvec{\mathrm {Y}};\varvec{\theta })\) is non-standard and cannot be sampled directly. Specifically, \(p(\varvec{b}_i| \varvec{\mathrm {Y}};\varvec{\theta }) \propto p(\varvec{b}_i;\varvec{\theta }_{\varvec{b}})p(\varvec{\mathrm {Y}}_i | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta })\), in which \(p(\varvec{\mathrm {Y}}_i | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta })\) is given by

$$\begin{aligned} p(\varvec{\mathrm {Y}}_i | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta }) = \prod _{j=1}^\mathrm{T} p(\varvec{y}_i(t_{i, j}) | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta }), \end{aligned}$$
(21)

where \(p(\varvec{y}_i(t_{i, j}) | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta })\) is a multivariate normal density function with mean \(\varvec{\mu }+\varvec{\mathrm {\Lambda }}\tilde{\varvec{x}}_i(t_{i, j})\) and covariance matrix \(\varvec{\mathrm {\Sigma }}_{\epsilon }\). As \(\varvec{b}_i\) is involved in the nonlinear \(f(.)\) in \(p(\varvec{y}_i(t_{i, j}) | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta }), p(\varvec{b}_i| \varvec{\mathrm {Y}};\varvec{\theta })\) is usually non-standard. To sample from \(p(\varvec{b}_i| \varvec{\mathrm {Y}};\varvec{\theta })\), we adopt a Metropolis-Hastings (MH) algorithm as follows. At the \(m\)th iteration with current values in \(\varvec{b}_i^{(m)}\), a new candidate \(\varvec{b}_i\) is generated from a proposal distribution, chosen to be the normal distribution \(\text{ N }(\varvec{b}_i^{(m)},\sigma _{b}^{2} \varvec{\mathrm {\Omega }}_{bi})\), where \(\sigma _{b}^{2}\) is a scaling constant, \(\varvec{\mathrm {\Omega }}_{bi} = (\varvec{\mathrm {\Sigma }}_{\varvec{b}}^{-1}+\sum _{j=1}^\mathrm{T}\varvec{\mathrm {D}}_{bit}^\mathrm{T}\varvec{\mathrm {\Sigma }}_{\epsilon }^{-1} \varvec{\mathrm {D}}_{bit})^{-1}, \varvec{\mathrm {D}}_{bit}=\partial \tilde{\varvec{x}}_i(t_{i, j})/\partial \varvec{b}_{i}^\mathrm{T} |_{\varvec{b}_i=\varvec{b}_i^{*}}\), and \(\varvec{b}_i^{*}\) is a fixed value with high \(p(\varvec{b}_i^{*}| \varvec{\mathrm {Y}};\varvec{\theta })\). One possibility is to use the mean of \(p(\varvec{b}_i;\varvec{\theta }_{\varvec{b}})\) as \(\varvec{b}_i^{*}\), which we have found to lead to good performance. The new \(\varvec{b}_i\) is accepted with probability

$$\begin{aligned} \text{ min }\left\{ 1,\frac{p(\varvec{b}_i;\varvec{\theta }_{\varvec{b}}) \prod _{j=1}^\mathrm{T} p(\varvec{y}_i(t_{i, j}) | \varvec{b}_i;\varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta })}{p(\varvec{b}^{(m)}_i;\varvec{\theta }_{\varvec{b}}) \prod _{j=1}^\mathrm{T} p(\varvec{y}_i(t_{i, j}) | \varvec{b}_i^{(m)}; \varvec{\theta }_{\varvec{\mu }}, \varvec{\theta }_{\varvec{\mathrm {\Lambda }}}, \varvec{\theta }_{\varvec{\epsilon }},\varvec{\beta })} \right\} , \end{aligned}$$
(22)

The scaling constant, \(\sigma _{b}^2\), can be chosen such that the average acceptance rate is approximately 0.4.

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Chow, SM., Lu, Z., Sherwood, A. et al. Fitting Nonlinear Ordinary Differential Equation Models with Random Effects and Unknown Initial Conditions Using the Stochastic Approximation Expectation–Maximization (SAEM) Algorithm. Psychometrika 81, 102–134 (2016). https://doi.org/10.1007/s11336-014-9431-z

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