Functional mixed-effects model for periodic data

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Biostatistics (2006), 7, 2, pp. 225–234 doi:10.1093/biostatistics/kxj003 Advance Access publication on October 5, 2005

Functional mixed-effects model for periodic data LI QIN∗ Statistical Center for HIV/AIDS Research and Prevention, Fred Hutchinson Cancer Research Center, Seattle, WA 98109, USA [email protected] WENSHENG GUO Department of Biostatistics and Epidemiology University of Pennsylvania School of Medicine, Philadelphia, PA 19104, USA

S UMMARY Periodic data are frequently collected in biomedical experiments. We consider the underlying periodic curves giving rise to these data, and account for the periodicity in their functional model to improve estimation and inference. We propose to incorporate the periodic constraint in the functional mixed-effects model setting. Both the fixed functional effects and random functional effects are modeled in the same periodic functional space, hence the population-average estimates and subject-specific predictions are all periodic. An efficient algorithm is given to estimate the proposed model by an O(N) modified Kalman filtering and smoothing algorithm. The proposed method is evaluated in different scenarios through simulations. Treatments to none-full period data and missing observations along the period are also given. Analysis of a cortisol data set obtained from a study on fibromyalgia is conducted as illustration. Keywords: Functional data analysis; Kalman filter; Periodic constraint; Periodic spline; Smoothing spline; State space model.

1. I NTRODUCTION Data in many biomedical experiments arise as curves, so it is natural to consider a curve as the basic observational unit in the data analysis, which is termed functional data analysis (Ramsay and Silverman, 1997). In many situations, measurements taken at different time points (or indices) display a periodic pattern, or the underlying curves are periodic. This information is the intrinsic nature of the functional space residing the curves. In this paper, we propose a new class of functional mixed-effects models that can incorporate the periodicity in both the fixed effects and random effects. The periodicity is imposed by constraining the underlying functional space of the model. The functional effects are modeled through a unified approach and are estimated by an efficient algorithm. An important periodic pattern of experimental data is the circadian rhythm, which is the periodic behavior of the data as a function of 24 h. It is well known that many hormones including cortisol have circadian rhythms. Figure 1 shows the 24 h cortisol profiles of 11 fibromyalgia (FM) patients and 11 healthy subjects, collected in a study conducted at the University of Michigan Medical Center ∗ To whom correspondence should be addressed.

c The Author 2005. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. 

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Fig. 1. Hourly cortisol concentration data profiles from 11 FM patients and 11 normal subjects for 24 h.

(Guo, 2002). FM is a stress-related syndrome that may relate to cortisol through a complex feedback system. The objective of the study is to investigate the relationship between the circadian patterns of cortisol, in normal subjects and patients with FM. As an initial attempt, we are interested in estimating the cortisol profiles of FM patients and normal individuals, and identifying the time course when the two groups are different in terms of their cortisol concentration. This may help unveil the mystery behind the unknown mechanism of FM. A key step of the analysis is to incorporate the circadian patterns in the model. We propose to incorporate it by modeling both functional fixed effects and functional random effects as periodic splines; consequently, both the group-average profiles and subject-specific predictions are of 24-h cycle. More details about this example will be given in Section 5. Substantial developments have been accomplished in functional models. Reviews of the early work can be found in Ramsay and Silverman (1997). A review of the recent developments using smoothing splines was given by Guo (2004). For example, Hastie and Tibshirani (1993) proposed varying-coefficient models and used cubic splines to model the functional coefficients. Zhang et al. (1998) considered semiparametric mixed models with nonparametric fixed effects and parametric random effects. Brumback and Rice (1998) modeled nested curves by cubic splines. Verbyla et al. (1999) used cubic splines to model fixed effects. Guo (2002) proposed functional mixed-effects models with nonparametric fixed effects and nonparametric random effects. However, all these papers use cubic splines that do not incorporate the periodicity of the functions. Models for periodic curves can be viewed as a special case of constrained functional models, which can be estimated by different smoothing techniques. Mammen et al. (2001) introduced a general framework of constrained smoothing via projection. A periodic curve can be modeled through projecting the data onto the smoothed periodic space. Readers are referred to Mammen et al. (2001) and the references therein for other types of constrained smoothing, and to Wahba (1990, Chapter 2) for periodic smoothing splines as projections. Wang and Brown (1996) considered a periodic spline model for hormone

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circadian rhythms without random effects and Zhang et al. (2000) used periodic cubic splines to model the fixed-effects functions and modeled the serial correlation by a nonhomogeneous Ornstein Uhlenbeck process. In general, these approaches require the inversion of large dimensional matrices and thus are very computationally demanding. An alternative low-rank smoother for the curves is P-spline (Ruppert et al., 2003), while it is only an approximation to smoothing splines. In this paper, we take a different approach than projection, in which numerical constraints are imposed on the usual smoothing splines, resulting in a significant computational saving. We extend the numerical constrained single-curve model of Ansley et al. (1993) to a mixed-effects functional model setting, enabling unified estimation and inference for the functional effects. We represent the whole proposed functional model as a multivariate state space model and adopt the fast algorithm of Koopman and Durbin (2000). This approach can estimate the exact smoothing splines in order O(N). The rest of the paper is structured as follows. In Section 2, we propose the model specification for the functional effects with periodic constraint. In Section 3, we present an efficient estimation procedure. A simulation study can be found in Section 4. We illustrate the proposed method through the application to the cortisol data in Section 5. 2. T HE MODELS 2.1

The general model

Suppose yi j ( j = 1, . . . , Ti , i = 1, . . . , n) is the observation on the ith curve at index ti j . For simplicity, we assume t ∈ [0, 1] (or can be re-scaled to [0, 1]). We consider the following general functional mixedeffects model (2.1) yi j = X i j β (ti j ) + Z i j α i (ti j ) + ei j , ei j ∼ N (0, σe2 ), where β (t) = {β1 (t), . . . , β p (t)}T is a p×1 vector of functional fixed effects, α i (t) = {α1i (t), . . . , αqi (t)}T is a q × 1 vector of functional random effects, which are modeled as realizations of the q × 1 vector of zero mean Gaussian process A(t) = {a1 (t), . . . , aq (t)}T , X i j = {X i j [1], . . . , X i j [ p]} is the design matrix for the fixed effects, Z i j = {Z i j [1], . . . , Z i j [q]} is the design matrix for the random effects, and ei j is the measurement error. 2.2

Model specification for β (t) and α i (t) with periodic constraint

Suppose that βk (t) and αil (t) are periodic functions on [0, 1] with period 1 (or can be mapped to period 1), with continuous up to the (m − 1)th derivatives. Similar to Ansley et al. (1993), we can model β (t) and α i (t) by periodic polynomial smoothing splines of order 2m − 1: βk (t) =

m−1
ν=0

αil (t) =

tν 1/2 bkν + λbk σe ν!

m−1
ν=0

ailν

t



tν 1/2 + λal σe ν!

0



0

t

(t − u)m−1 dWbk (u), (m − 1)!

k = 1, . . . , p,

(t − u)m−1 dWal (u), (m − 1)!

l = 1, . . . , q,

βk(v−1) (0) = βk(v−1) (1), (v−1)

αil (ν)

(ν)

(0) = αil(v−1) (1),

(2.2)

v = 1, . . . , m, v = 1, . . . , m,

where βk (t) and αil (t) are the νth derivatives of βk (t) and αil (t), {bk0 , . . . , bk,m−1 }T ∼ N (0, κ I ) with κ → ∞ and {ail0 , . . . , ail,m−1 }T ∼ N (0, σl2 D).

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Remarks: (1) We use polynomial smoothing splines with numerical constraints to account for the periodicity of the functional effects. This is to reduce the computational load through the connection with state space models. Other smoothing splines (Wahba, 1990; Gu, 2002) such as ‘splines on the circle’ can also be used, while with a more intensive computation procedure. (2) The constraints put on the functional effects essentially constrain the curves and their variances to be periodic functions. 3. E STIMATION Due to the connection of smoothing splines and linear mixed-effects models (see a recent review in Guo, 2004), our proposed model can be rewritten as linear mixed-effects type of model and estimated using standard software. However, this approach is very computationally intensive because of the need to invert large dimensional matrices. Therefore, we represent the proposed model in a state space form and adopt the O(N) algorithm of Koopman and Durbin (2000) for estimation. To simplify the presentation, we first assume that observations on different curves were collected at the same design points (i.e. ti j = tk j = t j , for k = i and Ti = T for all i). This assumption is relaxed later. 3.1 β ∗k (t)

State space representation for periodic splines

(1) (m−1) {βk (t), βk (t), . . . , βk (t)}T

(1)

(m−1)

= and α il∗ (t) = {αil (t), αil (t), . . . , αil Define the following state equations corresponding to models (2.2): β ∗k (t j ) = Hk j β ∗k (t j−1 ) + ω k j ,

ω k j ∼ N (0, λbk σe2 Wk j ),

α il∗ (t j ) = Hil j α il∗ (t j−1 ) + ω il j , ω il j ∼ N (0, λal σe2 Wil j ), ⎧ v−ν ⎨ (δt j ) , ν
v ∈ {1, . . . , m}, δt j = t j − t j−1 , (v−ν)! where Hk j (ν, v) = Hil j (ν, v) = ⎩0, ν > v, Wk j = Wil j = {wνv }m,m ν=1,v=1 , and wνv =

(t)}T . We have

(3.3)

(δt j )2m−ν−v+1 (2m−ν−v+1)(m−ν)!(m−v)! .

We adopt the idea of Ansley et al. (1993) to numerically constrain β v−1 (0) = β v−1 (1) and α iv−1 (0) = α iv−1 (1). The state space model without constraint is given by y j = F j γ (t j ) + e j ,

e j ∼ N (0, σe2 In ), j = 1, . . . , T,

γ (t j ) = H j γ (t j−1 ) + ω j ,

(3.4)

ω j ∼ N (0, W j ),

β ∗T (t), α ∗T (t)}T , where y j = {y1 j , . . . , yn j }T , e j = {e1 j , . . . , en j }T , γ (t) = {β ∗ ∗T ∗T ∗T ∗T (t), . . . , α ∗T (t)}T . The T ∗ ∗T α 1 (t), . . . , α n (t)}T , α i∗ (t) = {α β 1 (t), . . . , β p (t)} , α (t) = {α α i1 β (t) = {β iq n × m( p + qn) matrix F j = [{X 1∗Tj , . . . , X n∗Tj }T , diag{Z 1∗ j , . . . , Z n∗j }], where X i∗j [r ] = X i j [k], if r = m(k − 1) + 1, k = 1, . . . , p, and 0 otherwise; Z i∗j [r ] = Z i j [l], if r = m(l − 1) + 1, l = 1, . . . , q, and 0 T , . . . , ω T }T , H = diag{H , . . . , H , H ω 1T j , . . . , ω Tpj , ω 11, otherwise; ω j = {ω j 1j pj 11, j , . . . , Hnq, j }, and nq, j j 2 2 W j = diag{λb1 σe W1 j , . . . , λbp σe W pj , λa1 σe2 W11, j , . . . , λaq σe2 Wnq, j }. The periodic constraint γ (0) = γ (1) is then added to the above unconstrained model by augmenting γ (t) by γ (0): γ˜ (t) = {γγ (t)T , γ (0)T }T and adding m( p + qn) pseudo observations zero at t = 1 to numerically enforce γ (0) = γ (1). The pseudo observations form [m( p + qn)/n] + 1 columns of pseudo data vectors, where [x] denotes the largest integer that is no greater than x. Note that the last column does

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not need to be a full column because we adopt the univariate algorithm of Koopman and Durbin (2000). The augmented model is y˜ j = F˜ j γ˜ (t j ) + e˜ j ,

j = 1, . . . , T ∗ , (3.5)

γ˜ (t j ) = H˜ j γ˜ (t j−1 ) + ω˜ j ,

ω˜ j ∼ N (0, W˜ j ),

where T ∗ = T + mq + [mp/n] + 1, y˜ j = y j , e˜ j = e j when j = 1, . . . , T , y˜ j = 0, and e˜ j = 0 for the n,2m( p+qn) added pseudo observations, F˜ j = {F˜ j (i, k)} , with i=1,k=1

F˜ j (i, k) =

⎧ Fj , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1,

k = 1, . . . , m( p + qn),

j = 1, . . . , T ,

k = i + ( j − T − 1)n,

j > T,

⎪ ⎪ −1, k = i + ( j − T − 1)n + m( p + qn), ⎪ ⎪ ⎪ ⎩ 0, otherwise.

j > T,

T T ω Tj , 0m( The state transition matrix H˜ j and the variance–covariance matrix W˜ j of ω˜ j = {ω p+qn)×1 } are defined as     Hj 0m( p+qn)×m( p+qn) Wj 0m( p+qn)×m( p+qn) , W˜ j = . H˜ j = 0m( p+qn)×m( p+qn) Im( p+qn) 0m( p+qn)×m( p+qn) 0m( p+qn)×m( p+qn)

P P The initial state γ˜ (0) ∼ N (0, P˜0 ), with P˜0 = P0 P0 , where P0 denotes the variance–covariance matrix 0 0 for γ (0).

3.2 Parameter estimation and algorithm The parameters in the state space model (3.5) include smoothing parameters and variance parameters. The unknown parameters can be estimated by the restricted maximum likelihood, which is equivalent to the generalized maximum likelihood (Wahba, 1985). The generalized likelihood can be sequentially calculated by the Kalman filter and maximized through numerical optimization routines. Given the estimates of the unknown parameters, a smoothing algorithm (Koopman and Durbin, 2000) is used to obtain the β (t)|Y }, and Var{α α i (t)|Yi }, where Y denotes all the obβ (t)|Y }, E{α α i (t)|Yi }, Var{β posterior estimates E{β servations and Yi represents data from the ith curve (i = 1, . . . , n). The posterior means are estimates of the functional effects, and Bayesian confidence intervals (Wahba, 1983) for the functional estimates can be constructed using the posterior variances. Because y j is an n × 1 observational vector, the standard vector filtering and smoothing algorithm still requires inversion of the n × n variance–covariance matrices. We adopt the univariate approach of Koopman and Durbin (2000) for efficient estimation. The basic idea of this algorithm is to convert the multivariate state space model into a univariate model of series y11 , . . . , yn1 , y12 , . . . , yn2 , . . . , y1T , . . . , ynT , and apply univariate filtering and smoothing to avoid matrix inversion. Readers are referred to Koopman and Durbin (2000) for more details on the algorithm. The algorithm for our model is implemented in Matlab 6.12. We have assumed ti j = t j and Ti = T in the estimation procedure of this section, for presentation purpose only. In general, measurements may be taken at different indices (e.g. time points) from different curves. Examples are unequally spaced data, irregularly collected data, and none-full period data. The proposed algorithm can be applied with data augmentation, where a collection of all possible indices serves as the observational points t j for all curves. This artificially creates some missing data points yi j , which are regarded as zero with the corresponding F˜ j and e˜ j set to zero in model (3.5).

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We conducted a simulation study to evaluate the performance of the proposed model and estimation procedure. More importantly, we hope to highlight the advantage of using the proposed model over the most commonly used cubic smoothing spline model for the curves, and assess the difference when data are available only on a partial period. Periodic data yi j (i = 1, . . . , 30, j = 1, . . . , 30) were generated on a full period from yi j = wi β(ti j )+ αi (ti j ) + ǫi j , where ǫi j ∼ N (0, 1), wi ∼ U (0, 1), β(ti j ) = A sin 2π ti j , and αi (ti j ) ∼ N (0, b ), where ∞ cos 2πνx j−1 = {σ jk }30 ν=1 (2πν)4 . We assume ti j = t j = 30 , j,k=1 with σ jk = −B4 (t j −tk )/4! and B4 (x) = −48 so both β(t) and αi (t) are periodic curves on [0, 1]. We specified two settings to simulate data (1) A = 5 and b = 25, and (2) A = 2.5 and b = 6.25. We also considered none-full period data by taking off the last 10 data points from each individual’s profile and treating them as unobserved in the models. Cubic spline model and the proposed model with periodic constraint were used to fit the data. A total of 100 simulations were conducted. Figure 2 displays the boxplots of the mean square errors (MSEs) and the coverage probabilities of the 95% Bayesian confidence intervals (Wahba, 1983) for β(t). The MSE is defined as the overall mean of the sum-of-squares of the subject-specific prediction bias. It is shown that in all the settings, the cubic spline estimates (‘Cubic’) have less precision than the periodic constraint approach (‘Periodic’), and on average,

Fig. 2. MSEs and coverage probabilities (CPs). The upper two plots correspond to the simulation model with A = 5 and b = 25, and the lower two plots have A = 2.5 and b = 6.25. All 30 data points were used to obtain the full period results. Only the first 20 data points were included to get partial period estimates.

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the coverage from the periodic approach is closer to the nominal level. When data were only available on partial period, the periodic estimates still outperform the cubic spline estimates because more information was gained by explicitly modeling the periodicity. We can also see that as the amplitudes of the underlying curves increase, the MSEs of both models increase. This is due to the shrinkage toward the mean effect of the random curve predictions. Because the subject-specific deviations are modeled by random effects with zero mean Gaussian prior, the posterior means from these models shrunk toward zero. The absolute shrinkage is proportional to the the amplitudes of the underlying curves. With the current sample size and chosen smoothing parameters, each simulation run takes 33.7 s (on Dual AMD Opteron 250 processors running at 2.4 GHz and 8 Gigabytes SDRAM). Timing results of different sample sizes were obtained to show that the computing times increase linearly with the sample size of the data set. Details are omitted here due to space limit. 5. A PPLICATION In this section, we apply the proposed method to the cortisol data shown in Figure 1. We use m = 2 in this section. The following model is fitted: yi j = I (i
11)β1 (ti j ) + I (i > 11)β2 (ti j ) + αi (ti j ) + ei j ,

ei j ∼ N (0, σe2 ),

(5.6)

Fig. 3. Group-average estimates from the circadian model. The solid lines are the fitted group-average curves with 95% Bayesian confidence intervals for the FM group. The dotted lines are those for the normal group.

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Fig. 4. Subject-specific estimates from the circadian model. The solid lines are observed data. The dotted lines are fitted subject-specific curves with 95% Bayesian confidence intervals. The upper two rows are for FM patients and the lower two rows are for normal subjects.

where yi j ( j = 1, . . . , 24, i = 1, . . . , 22) is the cortisol concentration of the ith subject at the jth hour, I (·) is the indicator function, β1 (t) is the group-average curve of the FM group and β2 (t) is that of the normal group, αi (t) is the subject-specific deviation for the ith subject, and ei j is the measurement error. We fit the circadian pattern model specified in Section (2.2). Figure 3 shows the estimates of the group-average curves together with their 95% Bayesian confidence intervals (Wahba, 1983) for both the

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FM group and the normal group. While there are overlaps between the two group-average curves, it can be seen that FM patients tend to release more cortisol late afternoon (4 pm–9 pm) and early morning (6 am–8 am). This result is similar to the finding by Guo (2002). However, the confidence intervals are in general narrower by incorporating the periodic constraint, which enables us to differentiate the two groups during the early mornings. In the cubic spline approach of Guo (2002), the confidence intervals are wider at both ends because they are estimated with fewer observations. In our periodic approach, the confidence intervals at the boundaries are of similar widths as those in the middle. This is due to the periodic assumption which enables us to borrow the observations from the other end in estimating the curves at the boundary, so there is no boundary points in our case. The data and the fitted subject-specific curves with their 95% Bayesian confidence intervals are shown in Figure 4. The subject-specific curves generally fit the data well. Similar to the fitted group-average curves, the Bayesian confidence intervals of the estimated subject-specific curves are narrower than those in Guo (2002). ACKNOWLEDGMENTS The authors wish to thank the editors, the associate editor, and a referee for helpful comments and suggestions. In addition, Li Qin is grateful to Steve Self, Ross Prentice, and Peter Gilbert for providing helpful comments on an earlier draft of the paper. Li Qin’s research was supported by U.S. National Institute of Health (NIH) Grants NIH 5 U01 AI46703 and NIH 2 R37 AI29168. Wensheng Guo’s research was supported by U.S. NIH Grant R01 CA 84438. R EFERENCES A NSLEY, C. F., KOHN , R. Biometrika 80, 75–88.

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