Appendix A: An Alternative Estimation Procedure Dual Penalized Expansion

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1 Supplemental Materials for Functional Linear Models for Zero-Inflated Count Data with Application to Modeling Hospitalizations in Patients on Dialysis by Şentürk, D., Dalrymple, L. S. and Nguyen, D. V. Appendix A: An Alternative Estimation Procedure Dual Penalized Expansion We also explore an alternative estimation procedure called dual penalized expansion (DPE), inspired by the work of Goldsmith et al. [20], for generalized functional linear, hurdle and ZIP models, where penalized dual (separate) basis expansions for the predictor process and the coefficient functions of interest are considered. In contrast to PR, the dual penalized expansion (DPE) approach expands the original longitudinal predictor directly on its functional principal component basis, without reconstruction of X(t). In addition, similar to PR, the regression function, β(t), is expanded on spline basis functions. Step 1: Dimension Reduction via Basis Expansion The DPE approach considers dual expansions of the predictor process, X(t), and a coefficient function, β(t), using two separate sets of basis functions: K x X i (t) µ X (t) + ξ iv ψ v (t), v=1 K b β(t) b u φ u (t), u=1 where ξ iv = {X i (t) µ X (t)}ψ v (t)dt. The basis functions ψ v (t) and φ u (t) are taken to be the functional principal components (PCs) basis and truncated power spline basis, respectively. K x and K b are chosen large and they satisfy the identifiability constraint K x K b. We take K x = K b = 10, which provides adequate approximation as was found in Goldsmith et al. [20] for sparse designs. The expansion of γ(t) in a functional hurdle or a ZIP model follows similarly to the above expansion for β(t). More precisely, γ(t) K a ϑ=1 a ϑθ ϑ (t), where θ ϑ (t) is truncated power spline basis. Using the double expansion above, K b [ Kx X i (t)β(t)dt b u {ξ iv u=1 v=1 } ψ v (t)φ u (t)dt + 1 ] µ X (t)φ u (t)dt = (ξi T J ψφ + µ)b

2 where b = (b 1,..., b Kb ) T, ξ i = (ξ i1,..., ξ ikx ) T, J ψφ is a K x K b matrix with (v, u)th entry equal to ψ v (t)φ u (t)dt, and µ is a 1 K b vector with the uth entry equal to µ X (t)φ u (t)dt. When ψ(t) and φ(t) are taken to be the same set of orthonormal basis functions, such as the truncated power series spline basis defined above for t [0, 1], J ψφ reduces to an identity matrix. For sparse longitudinal data, direct expansion of X i (t) on a spline basis is not feasible since ξ iv = {X i (t) µ X (t)}ψ v (t)dt cannot be well approximated. Thus, X i (t) is expanded by functional PCs basis functions, an approach that is suitable for sparse longitudinal data. Hence, estimation of the functional PCs basis functions, eigenscores and the mean function of X(t), as described in Section 3.1 Step 0, allows estimation of the terms in (ξi T J ψφ + µ). Similar expansions lead to X i (t)γ(t)dt (ξi T J ψφ + µ)a for the functional hurdle and ZIP models. Thus, substituting the above derived equality for X i (t)β(t)dt from above into the generalized functional linear model (1) gives g(µ i ) = β 0 + X i (t)β(t)dt + α r Z ri β 0 + (ξi T J ψφ + µ)b + α r Z ri. The reduced generalized linear model has a n (1+K b +p) design matrix, (1 n, ξj ψφ + µ, Z) and parameter vector (β 0, b, α) T, where b = (b 1,..., b Kb ) T, α = (α 1,..., α p ) T, 1 n is a n 1 vector of ones, n K x matrix ξ = (ξ 1,..., ξ n ) T and Z is the n p matrix equal to Z = (Z 1,..., Z p ) for Z r = (Z r1,..., Z rn ) T. The functional hurdle model in (3) reduces to a classical hurdle model g 1 (p i ) β 0 + (ξi T J ψφ + µ)b + g 2 (λ i ) γ 0 + (ξi T J ψφ + µ)a + α r Z ri, ζ r Z ri, and with analogously defined design matrices and parameter vectors, and with design matrix (1 n, ξj ψφ + µ, Z) and parameter vector (γ 0, a, ζ) T where a = (a 1,..., a Ka ) T, ζ = (ζ 1,..., ζ p ) T. See Section 3.1 for details. Step 2: Penalized Maximum Likelihood 2

3 The penalized maximum likelihood estimation procedure for DPE proceeds as proposed for the PR approach, described in details in step 2 of Section 3.1, with the estimated design matrix Ŵ and vector Ŵi replaced by (ˆξĴψφ + ˆ µ) and (ˆξ T i Ĵ ψφ + ˆ µ), respectively, in the linear predictor and likelihood equations. Difference between PR and DPE PR and DPE both use FPCA to expand the functional predictor and the penalized truncated power series for the regression coefficient. However, since PR uses the FPCA decomposition in the reconstruction step, it effectively uses a smaller number of functional principal components (e.g. rarely selects more than 3 components in applications) compared to the DPE expansion. The main reason for DPE using a larger number of PCs in the FPCA expansion of the predictor process is the identifiability condition involved in the dual expansion procedure. In order for the spline coefficients to be identifiable in the expansion of the regression function, the FPCA decomposition of the predictor process needs to contain at least the same number of principal components (i.e. K x K b ). We take K x = K b = 10, which provides adequate approximation as was found in Goldsmith et al. [20] for sparse designs. The number of PCs used is the main difference between PR and DPE. In the following sections we study the finite sample properties of DPE in comparison to PR and PCR via simulations. Appendix B: Simulation Results with DPE Finite sample properties of DPE were evaluated in the same simulation settings as described in Section 4.1. Independent preliminary simulation studies were carried out to select the optimal regularization parameters for DPE. Median δ values minimizing CV error across multiple runs for the generalized functional linear model fits are.0055 for both sample sizes in the sparse design and they are.0005 and.001 for the denser design at n = 200 and n = 400, respectively. The selected (δ 1, δ 2 ) regularization pairs for the functional hurdle model for the 3

4 sparse design are (.01,.01) and (.05,.03); they are (.01,.05) and (.01,.01) for the denser design at n = 200 and n = 400, respectively. For the functional ZIP model, they are (.05,.05) and (.075,.05) in the sparse and denser designs for n = 200 and are both (.05,.05) for n = 400. For convenience of comparisons, we present the Tables 1, 2 and 3 from Section 4.2 of the manuscript once more, but this time with DPE results included along with PR and PCR results. Overall, both PR and DPE perform similarly and lead to significant efficiency gains in regression function estimation in generalized functional linear and functional linear mixture models over the more standard approach of PCR. More specifically, PR and DPE perform similarly in all model parameters and simulation set-ups for the generalized functional linear model. For the regression function estimation, PR trends towards some modest efficiency gains over DPE in the functional hurdle model for both binary and zero-truncated Poisson parts for the sparse design setting at n = 200. For the regression function estimation in the functional ZIP model, PR leads to efficiency gains in the Poisson part for sparse designs for both sample sizes. 4

5 Table 1: Simulation results for generalized functional linear model. Median, mean and squared deviation error (ME) reported for all model parameters for different estimation techniques in fitting a generalized functional linear model over different sparsity levels of the longitudinal predictor and at different sample sizes in 200 Monte Carlo runs. PR: penalized reconstruction; DPE: dual penalized expansion; PCR: principal components regression. Design n Median 25% 75% Median 25% 75% Median 25% 75% ME β ME β0 ME α PR DPE PCR PR DPE PCR PR DPE PCR PR DPE PCR

6 Table 2: Simulation results for functional hurdle model. Median, mean, squared deviation error (ME) reported for all model parameters for different estimation techniques in fitting a functional hurdle model over different sparsity levels of the longitudinal predictor and at different sample sizes in 200 Monte Carlo runs. PR: penalized reconstruction; DPE: dual penalized expansion; PCR: principal components regression. Design n Median 25% 75% Median 25% 75% Median 25% 75% ME β ME β0 ME α Binary PR DPE PCR PR DPE PCR PR DPE PCR PR DPE PCR ME γ ME γ0 ME ζ Zero-truncated Poisson PR < DPE < PCR PR < DPE < PCR < PR < DPE < PCR < PR < DPE < PCR <

7 Table 3: Simulation results for functional ZIP model. Median, mean, squared deviation error (ME) reported for all model parameters for different estimation techniques in fitting a functional ZIP model over different sparsity levels of the longitudinal predictor and at different sample sizes in 200 Monte Carlo runs. PR: penalized reconstruction; DPE: dual penalized expansion; PCR: principal components regression. Design n Median 25% 75% Median 25% 75% Median 25% 75% ME β ME β0 ME α Binary PR DPE PCR PR DPE PCR PR DPE PCR PR DPE PCR ME γ ME γ0 ME ζ Poisson PR < DPE < PCR PR < DPE < PCR PR < DPE < PCR PR < <.001 < DPE <.001 < PCR