Web-based Supplementary Materials for Modeling of Hormone Secretion-Generating Mechanisms With Splines: A Pseudo-Likelihood Approach by Anna Liu and Yuedong Wang Web Appendix A This appendix computes mean and covariance for the response variable y(t. Since α i is independent of N i, we have E(y(t γ + αe p(t xdn i (x γ + α p(t xh(xdx. Let A(t be the unique compensator for N i (t in the extended Doob-Meyer Decomposition theorem (Theorem 2.2.3 of Fleming & Harrington (99. Then E(A(t E(N i (t h(xdx and M(t N i(t A(t is a martingale with predictable quadratic variation < M, M > A. From Theorem 2.4.4 of Fleming & Harrington (99, E E E p(s xdm(x p(t xdm(x p(s xp(t xd < M, M > (x p(s xp(t xda(x p(s xp(t xh(xdx.
Also, Cov (α i p(s xdn i (x, α i p(t xdn i (x E (α i p(s xdn i (x α p(s xh(xdx (α i p(t xdn i (x α p(t xh(xdx E (α i p(s xdm(x + (α i α p(s xh(xdx (α i p(t xdm(x + (α i α p(t xh(xdx (α 2 + σ 2 α Therefore, ( ( p(s xp(t xh(xdx + σα 2 p(s xh(xdx p(t xh(xdx. Cov(y(s, y(t σγ 2 + (α 2 + σα 2 p(s xp(t xh(xdx ( ( + σα 2 p(s xh(xdx p(t xh(xdx + σ 2 I(s t. 2
Web Appendix B This appendix proves equation (4. We show the derivation for the mean only. The derivation for the covariance is similar. The mean formula is a direct result of the following equation. k k k p(t xh(xdx exp( β 2 (t xh(xdx + exp( β 2 yh(t ydy + k+ t exp( β 2 yh(t ydy + exp( β (x th(xdx exp( β yh(t + ydy k k k+ exp( β 2 (y + kh(t y kdy + exp( β 2 k k exp( β 2 yh(t ydy + exp( β yh(t + ydy k exp( β 2 yh(t ydy/( exp( β 2 + exp( β (y + kh(t + y + kdy exp( β k k exp( β yh(t + ydy exp( β yh(t + ydy/( exp( β, where we used the periodic property of h. Web Appendix C This appendix provides details to the estimation procedure. The minimization problem (6 in the paper is solved iteratively using the extended Gauss-Newton method in Ke & Wang (24b. Note that θ is fixed for the problem (6 and W i s are fixed at each iteration. Let N t η ξ(t γ + α p(β, t x exp(η(xdx, which is a non-linear functional in η. At each iteration, we approximate N t η by its first-order Taylor expansion at a previous estimate η (Lusternik & Sobolev 974: N t η N t η + D t (η η, 3
where D t N t / η ηη is the Fréchet differential. It is easy to check that (Debnath & Mikusinski 999 D t η α p(β, t x exp(η (xη(xdx. Let N ti η (N ti η,, N tini η T, D ti η (D ti η,, D tini η T, D ti η (D ti η,, D tini η T, and ỹ i y i N ti η + D ti η. We update η by solving { m min η W 2 (per i (ỹ i D ti η T W i (ỹ i D ti η + Nλ (η (t 2 dt. ( Since D tij s are linear bounded functionals, the solution to ( has the form (Wahba 99, Wang 998 η(t d + m n i c ij D tij R (, t, (2 i j where R (s, t B 4 ([s t]/24, [s t] is the fractional part of s t, and B 4 (x x 4 2x 3 +x 2 /3. R is the reproducing kernel for the space W 2 (per {. The minimization problem ( reduces to where ỹ (ỹ T,, ỹ T m T, z ij z (z T,, z T m T, Σ ii { min (ỹ dz Σc T W (ỹ dz Σc + Nλc T Σc, (3 d,c α p(β, t ij x exp(η (xdx, z i (z i,, z ini T, (D tij D ti j R (, n i n j j j, Σ (Σ ii m i,i, c i (c i,, c ini T, c (c T,, c T m T, and W diag( W,, W m. Furthermore, let ỹ W /2 ỹ, z W /2 z, Σ W /2 Σ W /2 and c W /2 c. The minimization problem (3 reduces to min { ỹ d z Σ c 2 + Nλ c T Σ c, (4 d, c which is computed by the R function ssr in the ASSIST package (Ke & Wang 24a. We estimate the smoothing parameter λ at each iteration using the generalized cross-validation (GCV method which minimizes V (λ /N (I A(λỹ 2 [(/Ntr(I A(λ] 2 4
or the generalized maximum likelihood (GML method which minimizes M(λ ỹ T (I A(λỹ [det + (I A(λ] where A(λ is the hat matrix (Wahba 99, Gu 22. N We now discuss how to compute N tij η, D tij η, z ij, W and D tij D ti j R (, α 2, p(β, t ij xp(β, t i j y exp(η (x + η (yr (x, ydxdy. A naive application of Gaussian quadrature to approximate involved integrals directly is computational intensive since the integrands depend on observation time points and, consequently, the approximations have to be done for every unique time point. In addition, a relatively large number of points are needed to approximate these integrals over large intervals. Let t (j, j,, N, be the sequence of the ordered unique observation time points from all subjects, where N denotes the total number of unique observation time points. After some tedious algebra (one example shown below, when the pulse shape function p is a double exponential function, these integrals can be computed from integrals (j + t (j (j + t(j2 + t (j t (j2 INT (xdx and INT 2 (x, ydxdy for some integrands INT and INT 2. For example, to calculate the integral in N t η, we use the following equation p(β, t x exp(η (xdx exp( β exp( β 2 2xh (t xdx + exp( β exp( β 2 + exp( β exp( β 2 + exp( β exp( β 2t exp( β 2 + exp(β t exp( β exp( β 2xh (t xdx + exp( β 2 t t exp( β xh (t + xdx + exp( β exp( β 2(t yh (ydy + exp( β 2 t exp( β t (y th (ydy + exp( β exp(β 2yh (ydy + exp( β 2(+t exp( β 2 t exp( β t yh (ydy + exp(β (t exp( β 5 exp( β xh (t + xdx exp( β 2xh (t xdx exp( β t xh (t + xdx exp( β 2( + t yh (y dy exp( β (y t + h (y + dy exp(β 2yh (ydy exp( β yh (ydy.
The above derivation used results in Web Appendix B and the periodicity of the function h. Note that the integrands in the last step are independent of observation time points. Therefore, the integrals can be computed by cumulating integrals (j+ t (j and (j+ t (j exp(β 2 yh (ydy exp( β yh (ydy, j,, N, which are approximated numerically by a three-point Gaussian quadrature. Integrals in D tij η, z ij and W are approximated similarly. The term D tij D ti j R (, involves double integrals (j + t (j approximated by a nine-point Gaussian quadrature. (j2 + t (j2 INT 2 (x, ydxdy which are The total number of observations N is very large in our real example and simulations. Therefore, solving the minimization problem (6 is computational intensive. Note that observation time points are the same for all subjects in our real example and simulations. Therefore, n i, t ij, ξ i and W i are all independent of i. To save computational time, we used average observations across subjects as the response vector. Specifically, let ȳ ( m i y i/m,, m i y in/m. Then the minimization problem (6 in the paper reduces to { min (ȳ ξ T W (ȳ ξ + nλ η W 2 (per (η (t 2 dt, where the subscripts i in n i, ξ i and ˇW i are dropped. The extended Gauss-Newton procedure discussed above was applied with m. An alternative approach for saving computational time is to use a subset of basis functions as in Kim & Gu (24. References Debnath, L. and Mikusinski, P. (999. Introduction to Hilbert Spaces with Applications, Academic Press, London. Fleming, T. R. and Harrington, D. P. (99. Counting processes and survival analysis, Wiley, New York. 6
Gu, C. (22. Smoothing Spline ANOVA Models, Springer-Verlag, New York. Ke, C. and Wang, Y. (24a. ASSIST: A suite of S functions implementing spline smoothing techniques. Available at http://www.pstat.ucsb.edu/faculty/yuedong/software.html. Ke, C. and Wang, Y. (24b. Smoothing spline nonlinear nonparametric regression models, Journal of the American Statistical Association 99: 66 75. Kim, Y. J. and Gu, C. (24. Smoothing spline Gaussian regression: more scalable computation via efficient approximation, Journal of the Royal Statistical Society Series B 66: 337 356. Lusternik, L. and Sobolev, V. J. (974. Elements of Functional Analysis, Hindustan Publishing Co., New Delhi. Wahba, G. (99. Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM. Wang, Y. (998. Smoothing spline models with correlated random errors, Journal of the American Statistical Association 93: 34 348. 7