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Semiparametric Stochastic Mixed Models forLongitudinal DataDaowen Zhang a , Xihong Lin a , Jonathan Raz a & Maryfran Sowers ba Department of Biostatistics , University of Michigan , Ann Arbor , MI , 48109 , USAb Department of Epidemiology , University of Michigan , Ann Arbor , MI , 48109 , USAPublished online: 17 Feb 2012.

To cite this article: Daowen Zhang , Xihong Lin , Jonathan Raz & Maryfran Sowers (1998) Semiparametric Stochastic MixedModels for Longitudinal Data, Journal of the American Statistical Association, 93:442, 710-719

To link to this article: http://dx.doi.org/10.1080/01621459.1998.10473723

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Semiparametric Stochastic Mixed Models forLongitudinal Data

Daowen ZHANG, Xihong LIN, Jonathan RAZ, and MaryFran SOWERS

We consider inference for a semiparametric stochastic mixed model for longitudinal data. This model uses parametric fixed effectsto represent the covariate effects and an arbitrary smooth function to model the time effect and accounts for the within-subjectcorrelation using random effects and a stationary or nonstationary stochastic process. We derive maximum penalized likelihoodestimators of the regression coefficients and the nonparametric function. The resulting estimator of the nonparametric function isa smoothing spline. We propose and compare frequentist inference and Bayesian inference on these model components. We userestricted maximum likelihood to estimate the smoothing parameter and the variance components simultaneously. We show thatestimation of all model components of interest can proceed by fitting a modified linear mixed model. We illustrate the proposedmethod by analyzing a hormone dataset and evaluate its performance through simulations.

KEY WORDS: Correlated data; Nonparametric regression; Penalized likelihood; Restricted maximum likelihood; Smoothingparameter; Smoothing spline; Variance components.

1. INTRODUCTION

Longitudinal studies are used in many fields of research,including epidemiology, clinical trials, and survey sampling.They are often characterized by the dependence of repeatedobservations over time within the same subject. Linearmixed models provide a flexible likelihood framework tomodel longitudinal data parametrically (Diggle, Liang, andZeger 1994; Laird and Ware 1982). The recently developedSAS procedure PROC MIXED greatly increases the popu­larity of linear mixed models.

In the analysis of longitudinal data, the parametric as­sumption in linear mixed models may not always be appro­priate. For example, in a longitudinal hormone study on pro­gesterone (Sowers et al. 1995), urine samples were collectedfrom 34 healthy women in a menstrual cycle and urinaryprogesterone was assayed on alternative days. Progesteroneis a reproductive hormone responsible for normal fertilityand menstrual cycling. The investigators were interested inthe time course of the progesterone level in a menstrualcycle as well as the effects of age and body size on thishormone. Figure 1 plots the log-transformed progesteronelevel as a function of time measured in units in a standard­ized menstrual cycle. (For the definition of the standardizedmenstrual cycle, see Sec. 5.) Figure 1 suggests that the pro­gesterone level varies over time in a complicated manner,and it is difficult to model its time trend using a simple para­metric function. The same is true in other hormone studies(Sowers et al. 1998) and other areas of biomedical research,such as studies of growth (Donnelly, Laird, and Ware 1995)and HIV research on the time course of CD4 counts (Zegerand Diggle 1994). Hence it is of potential interest in these

Daowen Zhang is a postdoctoral fellow, Xihong Lin is Assistant Profes­sor, and Jonathan Raz is Associate Professor, Department of Biostatistics,and MaryFran Sowers is Professor, Department of Epidemiology, Univer­sity of Michigan, Ann Arbor, MI 48109. The work of Lin was supportedin part by a University of Michigan Rackham grant, OVPAMA FacultyCareer Development Award Fund and U.S. National Cancer Institute grantR29 CA76404. The work of Raz was supported in part by National In­stitutes of Health (NIH) grant R29 MH5131O. The work of Sowers wassupported in part by NIH grant ROl NIAMS 40888. The authors grate­fully acknowledge the editor, the associate editor, and the referees for theirhelpful comments and suggestions.

studies to model the time effect nonparametrically whileaccounting for the correlation of observations within thesame subject.

For independent data, there is a rich literature on ker­nel and spline methods for nonparametric and semipara­metric regression (Green and Silverman 1994; Speckman1987). Data-driven methods such as cross-validation (CV)and generalized cross-validation (GCV) are commonly usedto select the smoothing parameter. Wahba (1978) proposeda Bayesian model for spline smoothing. Only limited workhas been done on nonparametric and semiparametric regres­sion for correlated data. Several authors considered kernelsmoothing and spline smoothing for models with a sin­gle nonparametric function with correlated errors (Altman1990; Diggle and Hutchison 1989; Hart 1991; Rice and Sil­verman 1991). Zeger and Diggle (1994) proposed a semi­parametric mixed model for longitudinal data similar to theone we consider in this article. They estimated the nonpara­metric function of time using a kernel smoother with thebandwidth parameter chosen via cross-validation and esti­mated the variance components by extrapolating the autoco­variance function constructed under a parametric model forthe time effect. For inference on the nonparametric functionand the correlation parameters (especially the latter), verylittle has been done by these authors. In terms of modelingwithin-subject serial correlation, Zeger and Diggle (1994)assumed a stationary Gaussian process. This assumptionmay not be appropriate in some applications (Donnelly etal. 1995). For example, in the progesterone data examplementioned earlier, Figure 2 shows the empirical varianceof the log progesterone level as a function of time. One caneasily see that the variance changes over time.

In this article we extend the Zeger and Diggle (1994)model to a more general class of models termed semipara­metric stochastic mixed models. This class of models as­sumes that the mean of the outcome variable is an arbi­trary smooth function of time and parametric functions ofthe covariates. To allow for more flexible within-subject

© 1998 American Statistical AssociationJournal of the American Statistical Association

June 1998, Vol. 93, No. 442, Theory and Methods

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Figure t. Log Progesterone Levels Plotted Against the Number ofDays in a Standardized Menstrual Cycle, With Estimated PopulationMean Curve f(t) and 95% Pointwise Frequentist and Bayesian Confi­dence Intervals (Cis): -, f(t); - - -, FrequentistCI;and---, BayesianCI.

variance--covariance structures, we propose various station­ary and nonstationary stochastic processes to model serialcorrelation. Although our model consists of only a sin­gle nonparametric function, it can be easily generalized toaccommodate interactions between the covariates and thenonparametric time function; see Section 7 for details.

A key feature of this article is that, unlike Zeger andDiggle (1994), we make inference on all model compo­nents within a unified linear mixed model framework. Incontrast to the kernel smoothing approach of Zeger andDiggle (1994), we estimate the nonparametric function oftime using a smoothing spline, which allows us to make in­ference on this nonparametric function as part of the overallinference procedure, The estimators of the regression coef­ficients and the natural cubic spline estimator of the non­parametric function are obtained using maximum penalizedlikelihood. The random effects and the stochastic processare estimated using their conditional means given the data,Frequentist and Bayesian inference on these model compo­nents are proposed and compared. We treat the smoothingparameter as an extra variance component and estimate itjointly with the other variance components using restrictedmaximum likelihood (REML). We show that all model pa­rameters can be estimated from a modified linear mixedmodel.

The article is organized as follows. Section 2 states themodel. Section 3 contains the estimation and inference pro­cedures. Sections 3.1-3.3 present the maximum penalizedlikelihood estimators of the regression coefficients and thenonparametric time function, as well as the empirical Bayesestimators of the random effects and the stochastic process.Section 3.4 considers the biases and the frequentist covari­ances of these estimators under the assumption that thenonparametric time function is a fixed parameter. Section3.5 discusses the estimation procedures in Sections 3.1-3.3from a Bayesian perspective and derives the Bayesian co­variances of the model parameters by assuming the non­parametric time function to be a random function. OurBayesian covariance calculation is motivated by the earlierwork of Wahba (1983), who showed that for independentdata, the Bayesian confidence intervals of the nonparamet-

M

coC

e '"i ~eQ.

0) 0

.3

. I ,I:, ::;),~o'~~~~;~~~~....~. :.1 .... :: ::" 00' .~...... >";:r"-.. 00 ~>...

~o::f~~~o~~~7~~~~~~:::ooo .: .::: ... . ':.. :. I

0'

o 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Days in standardized menstrual cycle

ric function have good coverage probabilities when the truenonparametric function is a fixed smooth function. In thisarticle we are interested in comparing the coverage proba­bilities of the frequentist and Bayesian confidence intervalsof the nonparametric time function in the situation wherethe true time function is a fixed smooth function, whichis the situation of our primary interest. Section 4 proposesa joint estimation procedure for the smoothing parameterand the variance components. Section 5 illustrates applica­tion of the proposed methods to the progesterone data, andSection 6 reports the results from a simulation study de­signed to evaluate the performance of the proposed estima­tion procedure. Section 7 ends the article with a discussion.

2. THE SEMIPARAMETRIC STOCHASTICMIXED MODEL

2.1 The Model Specification

Let the sample consist of m subjects, with the ith sub­ject having ni observations over time. Suppose that Yij (i =1, ... , m, j = 1, ... , ni) is the response for the ith subjectat time point tij and satisfies

Yij = X~f3 + j(tij) + Z~bi + Ui(tij) + Cij, (1)

where f3 is a p x 1 vector of regression coefficients as­sociated with covariates X ij, j (t) is a twice-differentiablesmooth function of time, the b, are independent qx 1 vectorsof random effects associated with covariates Zij, the U,(t)are independent random processes used to model serial cor­relation, and the Cij are independent measurement errors.We assume that b, is distributed as normal (0, D(¢)), whereD is a positive definite matrix depending on a parametervector ¢;Ui(t) is a mean zero Gaussian process with co­variance function cov(Ui(t), Ui(s)) = ,(e, a; t, s) for a spe­cific parametric function ,(-) that depends on a parametervector eand a scalar a used to characterize the varianceand correlation of the process U, (t); and Cij is distributedas normal(O, (72). We further assume b., Ui(t), and Cij to bemutually independent.

Diggle et al. (1994) and Zeger and Diggle (1994) consid­ered a similar model to that given in (1). But their model had

coCe It!co1ijco0)

eQ.

C!~'0co0C Il>

'" ci'C

'">

o 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Days In standardized menstrual cycle

Figure 2. Sample Variances of the Log Progesterone Values Cal­culated by Grouping the Data into 28 toDay Intervals. -, estimatedvariance function curve obtained from fitting model (26).

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only a random intercept and assumed the stochastic processU, (t) to be a stationary Gaussian process. If a parametricfunction is assumed for f(t), then model (1) reduces to thelinear mixed model of Diggle et al. (1994, eq. 5.2.3).

(tm,tm)) and the (j,j')th element (j,j' = 1, ,ni) ofri(ti, t.) being 'Y(e, a; t ij, t ij,); and e = (e[, , e;')T isnorma1(O,(J2I), with I denoting an identity matrix of di-

o ",mmension n = L.."i=l ni·

2.2 The Gaussian Process Specification

Because the within-subject covariance often varies overtime in longitudinal data, we propose to model serial cor­relation using various stationary and nonstationary Gauss­ian stochastic processes U,(t). A specific choice of U,(t)depends on the underlying subject-specific biological pro­cess in a particular application. A useful stochastic processU,(t) to model nonstationary within-subject covariance isthe nonhomogeneous Omstein-Uhlenbeck (NOU) process(Rosenblatt 1974), which assumes the variance function ~(t)

to vary over time and the correlation function to decay ex­ponentially over time corr(Ui(t), Ui(8)) = exp(-alt - 81).For example, we may assume a log-linear model for ~(t)

as log(~(t)) = ~o + 6t. When ~(t) is a constant, the NOUprocess reduces to a stationary Omstein-Uhlenbeck (Ol.I)process (Diggle et al. 1994, sec 5.2).

An extension of the OU/NOU process is to assumevar(Ui(t)) = ~(t) and corr(Ui(t), Ui(8)) = p(a; It - 81) forsome parametric functions ~ (.) and p(.). This includes theGaussian correlation model p(a; It - 81) = exp[-a(t - 8)2](Diggle et al. 1994, sec 5.2). Other useful choices for Ui(t)include (a) a Wiener process (Taylor, Cumberland, and Sy1994); (b) an integrated Wiener process (Wahba 1978),which may lead to each predicted subject-specific curvesimilar to a natural cubic spline; (c) an integrated au (IOU)process (Sandland and McGilchrist 1979), which is useful ingrowth curve analysis and assumes the relative growth rateto be a stationary au process; and (d) an ante-dependenceprocess for equally spaced time points (Digg1e et al. 1994,sec. 5.2).

3.2 Estimation of the Regression Coefficients,Nonparametric Function, Random Effects, andStochastic Process

For given variance components (J = (lj)T,eT,a, (J2)T, thelog-likelihood function of (,8, f), is, apart from a constant,

1l(,8, f; Y) = -"2 log IVI

1 T I-"2 (Y - X,8 - Nf) V- (Y - X,8 - Nf),

where V = diag(VI,o .. ,Vm) and Vi = ZiDZT + I',+ (J2I. Because f(t) is an infinite-dimensional parameter,we consider the maximum penalized likelihood estimator(MPLE) of ,8 and f(t), which leads to a natural cubic splineestimator of f(t) (O'Sullivan, Yandell, and Raynor 1986).Specifically, the MPLE of (,8, f(t)) maximizes

l(,8, f; Y) - ~ (T2

[J"(tW dt = l(,8, f; Y) - ~ fTKf, (4)2Jn 2

where A ~ 0 is a smoothing parameter controlling the bal­ance between the goodness of fit and the roughness of theestimated f(t), TI and T2 specify the range of t, and K is thenonnegative definite smoothing matrix defined in equation(2.3) of Green and Silverman (1994).

For fixed smoothing parameter A and variance compo­nents (J, differentiation of (4) with respect to ,8 and f showsthat the MPLE (/3, f) solves

3. ESTIMATION AND INFERENCE FORTHE MEAN COMPONENTS

3.1 Matrix Notation

where b = (b[, , b;')T is normal(O, V(lj))), withV(¢) diag(D, , D); U (V[, .. 0' U;')T isnorma1(O,r(e, a)), with qe, a) = diag(rl(tl, td, .. 0' I'm

(7)

(6)

where W = V-I = diag(WI, .. 0' W m) and Wi = ViI.Equation (5) has a unique solution if and only if the matrix[X, NT] is of full rank, where T = [1, to] and 1 is an r xlvector of 1'so Note that this condition is also a necessary andsufficient condition for the coefficient matrix of equation(5), which we denote by C, to be positive definite.

To study the theoretical properties of the MPLEs (Sects.3.4 and 3.5), it is useful to derive a closed-form solutionof (5). Denoting two nonnegative definite weight matricesby W x = W - WN(NTWN + AK)-INTW and WI =

W - WX(XTWX)-I XTW, some calculations give theMPLEs of ,8 and f as

Note that both (XTWxX)-1 and (NTWIN + AK)-I arepositive definite given that [X, NT] is of full rank. Thesetwo matrices are in fact the corresponding block-diagonalelements of C- I.

and

(3)Y = X,8 + Nf + Zb + V + e,

Denote Y, = (lil,"" lin.)T, and Xi, Zi, Vi, ei SImI­larly (i = 1, ... ,m). Let to = (t?, ... ,t~)T be a vector of or­dered distinct values of the time points tij (i = 1, ... , m, j =1, ... , ni), and let N, be the incidence matrix for the ithsubject connecting t, = (til,"" tin.) T and to such thatthe (j, l)th element of N, is 1 if tij = t? and 0 otherwise(j = 1, ... , ni, l = 1, ... , r). Then model (1) can be writtenas

where f = (f(t?) , ... , f(t~))T.Further denoting Y = (Y[, ... ,Y;')T, and X, N, e sim­

ilarly, and Z = diag(ZI, 0'" Zm), we have

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Zhang, Lin, Raz, and Sowers: Semiparametric Stochastic Mixed Models 713

Let s, = (Si1,"" Sic,)T be an arbitrary c, x 1 vec­tor of time points. Estimation of the subject-specific ran­dom effects b, and the subject-specific stochastic processVi (Si)(i = 1, ... , m) may proceed by calculating their con­ditional expectations given the data Y i, while estimating {3and f by their MPLEs. This gives

(8)

It is often computationally expensive to solve 13. andh. from the standard normal equation of model (12) (see,e.g., Harville 1977, eqs. 3.1 and 3.3), because of the po­tentially high dimension of the random effects b. and thelack of block-diagonal structure of V. = rB.B; + V, themarginal variance of Y under the mixed-model repr~seAnta­

tion (12). However, one can show that the BLUPs ({3, 0, a)of model (12) are identical to those solving

and

cov(f) = (NTW j N + 'xK)-l

x N TWfVWfN(NTWfN+'xK)-l. (15)

and

and. T) 1E(Vi(Si)) = I'(s., ti)Wd>.Ni(N W j N + >.K - K

- Xi(XTWxX)-lXWxN]f.

The covariances of the MPLEs (13, f) can be easily cal­culated from (6) and (7),

cov(f3) = (XTW.xr'X XTWxVWxX(XTWxX)-l, (14)

3.4 Biases and Covariances of the RegressionCoefficients, Nonparametric Function, RandomEffects, and Stochastic Process

In this section we study the biases and frequentist co­variances of the parameter estimators obtained in Sections3.2 and 3.3, assuming that f(t) is a fixed parameter. Simplecalculations using (6) and (7) show that the biases of theMPLEs 13 and f are

E(f3) -{3 = (XTWxX)-lX

TWxNf

E(f) - f = -'x(NTWfN + 'xK)-lKf.

These biases go to 0 as ,x -i. O. Note that f is a shrinkageestimator of f, because the eigenvalues of 'x(NTW j N +'xK) -1 K are between 0 and 1. Similarly, the biases in theestimators of the random effects hi and the stochastic pro­cess Ui(Si) are

A T T ) 1E(bi) = DiZi Wd,XNi(N WfN +'xK - K

- X i(XTW

xX)-lXWxN]f

(11)

(12)

(10)f = To + Ba,

Y = X{3+NTo +NBa+ Zb+ V +€,

where {3. = ({3T,OT)T are the regression coefficients andb. = (a", bT, VT)T are mutually independent random ef­fects with a distributed as normal(O, 71), and (b, V) havingthe same distributions as those given in Section 3.1. Becausethe design matrix NB does not have a block-diagonal struc­ture, the random effects a and (b, V) hence may be viewedas random effects generated from a crossed design.

It can be easily shown that the best linear unbiased pre­dictors (BLUPs) 13 and f = T8 + Ba from linear mixedmodel (12) are identical to the MPLEs from (5H7). Thesame is true for the estimators of the random effects hiin (8) and the estimator of the stochastic process Ui(Si) in(9). Note that we here represent the MPLE f as a linearcombination of the estimated fixed effects 8 and randomeffects a.

where 0 and a are vectors with dimensions 2 and r - 2, B =L(LTL)-l and L is r x (r - 2) full-rank matrix satisfyingK = LL T and LTT = O. Using the equality fTKf = aTa,we have that the penalized log-likelihood (4) differs by nomore than an additive constant from

lIT-"2 log IVI-"2 (Y - X{3 - NTo - NBa)

1x V-1 (y - X{3 - NTo - NBa) - 27 aT a,

where 7 = 1/'x. Consequently, when j(t) is a natural cubicspline, we can write the semiparametric mixed model (3) asa modified linear mixed model,

[XTWX XTWB ] [{3 ]B!WX: B;WB. + 'xi ;

(9) [XTWY ]= BT• WY , (13)

where the (j,j')th element of ri(Si,ti) is ,(e,O:;Sij,tij')

and fi = N}. where X. = [X, NT] and B. = NB. Note that (13) has astructure parallel to that of (5). An advantage of estimating13 and f from (13), or equivalently from (5), is that it requiresmuch less computation. This is because the dimension ofa is much smaller and is of the same magnitude as thenumber of distinct time points, which is often much lessthan the total number of observations in many longitudinalstudies. Furthermore, calculation of the weight matrix W inthese equations requires only inverting the block-diagonalcovariance matrix V.

and

3.3 The Linear Mixed Model Representation

In view of the quadratic form of the penalty term in thepenalized log-likelihood (4), we show in this section how toderive the MPLEs 13 and f by writing the semiparametricmixed model (3) as a modified linear mixed model. Thislinear mixed model representation provides a foundationfor the estimation procedure of the smoothing parameterand the variance components (Sec. 4).

Following Green (1987), we can write f via a one-to-onelinear transformation as

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For independent data, (4) reduces to (5.4) of Green (987).Under the classical nonparametric model,

(6)

where the e, are independent and follow normal(O, ( 2) , the

covariance of f given in (15) reduces to the familiar forma 2 A 2, where A = (I + 'xK) -1 and ,X = a 2 ,X (Hastie andTibshirani 1991).

The covariances of the estimators of the random effectsb i and the stochastic process U i(s.) are

vals of f have good coverage probabilities when f(t) is afixed smooth function.

We suspect that the same may be true under the semi­parametric stochastic mixed model 0). We hence proposeestimating the covariances of /3 and f by their posterior co­variances under the Bayesian models discussed earlier asan alternative to the covariance estimators given in Section3.4. Some calculations show that the Bayesian covariancesof /3 and f have much simpler forms:

(8)

- Tcovfb, - b.) = D - DZi WiZiD and

+ DZ[WiXiC-1XrWiXiC-

1XrWiZiD(9)

- TCOVB(Ui(Si) - Ui(Si») = T'[s,, s.) - I'(s., ti)Wir(Si, t i)+ r(Si, t i)WiXiC-

1XrWir(Si, ti)T.

Note that these Bayesian covariances of b i and Ui(Si) areidentical to those obtained from the modified linear mixedmodel (2).

and

It is easy to show that these Bayesian covariances are iden­tical to the covariances cov(/3) and cov(f) = cov(T8 + B(a- a)) calculated from the modified linear mixed model (2).Under the classical nonparametric model (6), the Bayesiancovariance of f given in equation (9) reduces to the fa­miliar form a 2 A (Wahba 1983), where A is defined inSection 3.4.

Because the differences of the Bayesian covariance ma­trices in 08H19) and the frequentist covariance matricesin 04HI5), COVB(/3) - cov(/3) and covB(f) - cov(f), arealways nonnegative definite, the Bayesian standard errors of/3 and f hence are often greater than their frequentist coun­terparts. As shown by Wahba 0983, p134), the average ofthe Bayesian variances of f is approximately equal to theaverage of the mean square errors of f, which is the aver­age of the squared biases plus the average of the frequentistvariances. Therefore, an interpretation for larger Bayesianstandard errors of f and /3 could be that they account forthe biases in f and /3.

The frequentist and Bayesian covariances become identi­cal when ,X -l. °or ,X t 00. In Section 5 we compare throughsimulations the coverage probabilities of the resulting fre­quentist and Bayesian confidence interval estimators of fwhen the true f (t) is a fixed smooth function, which is thesituation of our primary interest. Our view of the Bayesianstandard error of f follows that of Berger (980) and Wahba(1983), which is to construct the confidence interval of f un­der a prior and to examine how it performs in the case ofinterest; that is, when f(t) is a fixed parameter.

Compared to the frequentist covariances, the Bayesiancovariances of b i and U i(s.) also have simpler forms andare calculated by assuming that f has a partially improperprior given earlier in this section:

covB(bi - b.)

= D - DZ[WiZiD + DZ[WiXiC-1XrWiZiD

where 60 and 61 have noninformative priors on (-00, +00)and W(t) is the standard Wiener process. Note that theprior specification of fusing (17) is equivalent to assumingf takes the form in (0) with a flat prior for 6, a normalprior normal(O, TI) for a, and B = ~1/2, where ~ is the in­tegrated Wiener covariance matrix evaluated at to. Follow­ing Wahba (978), one can easily show that the posteriormodes and means of ({3, f) under this Bayesian model areidentical to the MPLEs (/3, f), as are the estimators of therandom effects and the stochastic process band U.

An alternative Bayesian formulation to obtain the MPLEs(/3, f) using the posterior modes is based on the finite di­mensional Bayesian model of Green and Silverman (994).Specifically, one can easily see that equation (4) is the pos­terior log-likelihood of ({3, f) if one assumes a flat prior for(3 and a partially improper Gaussian prior for f whose logdensity has kernel -'xfTKf/2.

Under the classical nonparametric model (6), Wahba(1985) suggested estimating the covariance of f using theposterior covariance of f under the Bayesian model (7).She showed that the resulting Bayesian confidence inter-

3.5 Bayesian Formulation and Inference

In this section we first show that the MPLEs (/3, f) andthe random-effects estimators (b, U) obtained in Sections3.2 and 3.3 can be derived within a Bayesian framework,and then derive the Bayesian standard errors of (/3, f) in thespirit of Wahba (983) as an alternative to the frequentiststandard errors given in (4) and OS).

We first derive the MPLEs (/3, f) from a Bayesianprospective. Assume that (3 has a noninformative prior onRP, the p-dimensional Euclidean space, and f(t) has a par­tially improper integrated Wiener prior (Wahba 1978),

f(t) = 60 + (ht +,x-1/21t

W(u) du, (7)

COV(Ui(Si) - Ui(Si))

= I'(s., s.) - I'(s., ti)Wir(Si, tif

and

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(25)

1'I = - Tr (P B BTp B B T)

TT 2 * * * * * * '

_1 ( T {)Y)'ITO. - - Tr P B B P -J 2 * * * * {)()j ,

P * = v:' _ y-1X (XTy-1X )-lXTy-1 (23)* * * ** * **

= y-1 _ y-1 XC-1XTy-1, (24)

Y and C were defined in Section 3.2, and X [X, N].Calculation of P * from (23) is often computationally pro­hibitive, because the n x n covariance matrix Y * does nothave a block-diagonal structure. But it is often computa­tionally feasible to calculate P * from (24), because Y is ablock-diagonal matrix and the dimension of C is about thesame as the number of distinct time points, which is of­ten much lower than the number of total observations. TheFisher scoring algorithm then can be used to solve (21) and(22) for T and 8.

For the classical nonparametric model (16), it can eas­ily be shown that the REML estimator of the smoothingparameter ..\ from (21) reduces to Wahba's (1985) GMLestimator. It is interesting to note that the REML estima­tor of the residual variance 0-

2 from (22) is identical to theconventional estimator &2 = L~l (Yi - j(ti))2/Tr(1 - A)(Green and Silverman 1994, eq. 3.19), where A was givenin Section 3.4. This REML property of &2 provides a moresystematic justification for the use of &2 in the classicalnonparametric model (16), and has not been recognized inthe literature.

The covariance of iJ can be estimated using the corre­sponding block of the inverse of the Fisher informationmatrix obtained from (21) and (22),

1 T 1 A A

-"2 Tr(P*B*B*) +"2 (Y - X{j - Nffy-1

x B*B:y-1(y - x13 - Nf) = ° (21)

where

1 ( {)Y) 1 A A--Tr P - +-(Y-X{j-Nffy-12 * {)()j 2

{)y -1 A A

X {)(). V (Y - X{j - Nf) = 0, (22)J

and

and a Gaussian prior for a as normal(O, TI) in l({j, f; Y) =l({j, d, a; V), and then integrating out ({j, d) and a (Harville1977; Wahba 1985).

Differentiating lR(T,8; Y) with respect to T and 8 andusing the identity y;l(y -X*13J = virv -X13-Nf),the REML estimating equations for T and 8 are

where X* and Y * were defined in Section 3.3. Note that theREML log-likelihood lR(T, 8; Y) in (20) can also be moti-vated from a finite-dimensional Bayesian model discussed andin Section 3.5. Specifically, lR(T, 8; Y) is the marginal log­likelihood of Y obtained by assuming a fiat prior for ({j, d)

4. ESTIMATION OF THE SMOOTHING PARAMETERAND VARIANCE COMPONENTS

We assume in Section 3 that the smoothing parameter ..\and the variance components 8 are known when we makeinference about the mean parameters. But they are oftenunknown in practice and need to be estimated from thedata. Many authors have pointed out that it is crucial tohave a good estimator of the smoothing parameter ..\ toensure a good performance of the estimator j(t) (Greenand Silverman 1994; Wahba 1978). A classical data-drivenapproach to selecting the smoothing parameter is cross­validation, which leaves out one subject's entire data at atime (Rice and Silverman 1991; Zeger and Diggle 1994).~ut this approach is often expensive, and the subsequentinference on the variance components is difficult (Zeger andDiggle 1994).

Under the simple nonparametric model (16) for indepen­dent data, Wahba (1985) proposed estimating the smoothingparameter ..\ using generalized maximum likelihood (GML)by assuming f(t) has a partially improper integrated Wienerprior (17). Speed (1991) pointed out that the GML estimatorof T = 1/..\ is identical to the restricted maximum likelihood(REML) (Harville 1977) estimator of the variance compo­nent T under the linear mixed model Y = Td + Ba + e,where a rv N(O, TI) and e rv N(O, 0-

21), and T and B aredefined in Section 3.3.

REML has also been widely used for selecting thesmoothing parameter in the state-space model literature,where a smoothing spline is formulated using a state­space model and REML or ML is often used to estimatethe smoothing parameter. For example, Ansley, Kohn, andWong (1993) and Wecker and Ansley (1983) estimated thesmoothing parameter using ML and REML in the classicalnonparametric regression model (16) with a fixed unknownsmooth function f (t) using the state-space model formula­tion.

Kohn, Ansley, and Tharm (1991) showed in an extensivesimulation study that the REML estimator of T performswell and often better compared to the CV and aev esti­mators in estimating the nonparametric function under (16).Further simulation studies by Ansley et al. (1993) gave sim­ilar conclusions on the performance of REML and aey.

Motivated by these authors' results and using the con­nection between the smoothing spline and the linear mixedmodels established in Section 3.3, we propose estimating..\ and 8 simultaneously using REML by treating T as anextra variance component in the linear mixed model (12).The REML log-likelihood of (T, 8) is

1 1 hlR(T,8;Y)=--logIY 1--logIXTy-1X I were

2 * 2 * * *

- ~ (Y - X 13 )Ty-1(y - X ?i) (20)2 * * * * f'J. ,

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Table 1. Estimates of the Regression Coefficients. VarianceComponents and Smoothing Parameter in the Progesterone Data

Model Parameter Bayesian Frequentistparameter estimate SE SE

f31 .9247 1.9236 1.9236

f32 -2.9127 2.3762 2.3761¢ .2617 .0719

P .0963 .0617

~o -2.1563 .4469~1 3.6710 1.2214

6 -2.1598 .8071c? .1366 .0163T 7.2976 4.4997

Note that we here are interested mainly in using (25) todraw inference for the variance components () but not forthe smoothing parameter T. Our rationale for using (25) toconstruct the variance of iJ is to account for the loss of in­formation from estimating T. We examine the performanceof this variance estimate of iJ in Section 6 through simula­tions.

5. APPLICATION TO THE PROGESTERONE DATA

We applied the proposed semiparametric stochasticmixed model to analyzing the longitudinal progesteronedata discussed in Section 1. Among the 34 study partici­pants, a total of 492 observations were obtained, with eachwomen contributing from 11 to 28 observations over time.This gave an average of 14.5 observations per woman. Themenstrual cycle lengths of these women ranged from 23 to56 days, with an average of 29.6 days. To overcome theproblem of unequal cycle lengths among the study partici­pants, each woman's menstrual cycle length was standard­ized uniformly to a reference 28-day cycle (Sowers et al.1998).The reason of standardization is that it is biologicallymeaningful to assume that the change of the progesteronelevel for each woman depends on the time during a men­strual cycle relative to her cycle length. The standardizationresulted in 98 distinct time points. A log transformation was

Journal of the American Statistical Association, June 1998

applied to the progesterone level to make the normality as­sumption more plausible.

Figure 1 displays the log-transformed progesterone val­ues during a standardized menstrual cycle. Figure 2 showstheir empirical sample variances, which were calculated bygrouping the data into 28 l-day intervals. Note that be­cause urine samples were analyzed on alternative days, theobservations within every l-day interval came from differ­ent subjects and hence were independent. Examination ofFigures 1 and 2 suggests that both the mean and varianceof the log progesterone level change over time nonlinearly.

Denote by Yij the jth log-transformed progesterone valuemeasured at standardized day tij since menstruation forthe ith woman, and by AGEi and BMIi her age and bodymass index. We considered the following semiparametricstochastic mixed model:

where the bi are independent random intercepts followinga normal(O, 4» distribution, the Ui(t) are mean 0 Gaussianprocesses modeling serial correlation, and the Cij are in­dependent measurement errors following a normal(O, (72)distribution. To allow the variance of Yij to vary over time,we assumed Ui(t) to be a NOD process, which satisfiesvar(Ui(t)) = ~(t) with log(~(t)) = ~o + 6t + 6t2 andcorr(Ui(t),Ui(s)) = plt-sl. Note that here we assume thateach woman's serial correlation is the same on the trans­formed time scale. Other correlation structures are also pos­sible; see Section 7 for more discussion. For the sake ofcomputational stability in fitting model (26), standardizedday-since menstruation was centered at the median 14 daysand divided by 10, whereas age and BMI were centered atthe medians 36 years and 26 kg/m'' and divided by 100.Hence f(t) represents the progesterone profile for the pop­ulation of women with age 36 years and BMI 26 kg/m",

Figure 1 superimposes the MPLE of f (t) and its point­wise 95% frequentist and Bayesian confidence intervals.The levels of progesterone remain relatively low and stablein the first half of a menstrual cycle and increase markedlyafter ovulation. They reach a peak around reference day 23

Number ofsubjects

m = 34

m = 68

Table 2. Simulation Results on Estimates of Model Parameters Based on 500 Replications

Model Relative Empirical Model-based Model-basedparameter bias SE Bayesian SE frequentist SE

f3, -.0739 2.0142 1.9108 1.9107f32 .0077 2.2726 2.3604 2.3604¢ .0091 .0793 .0729

P .1755 .0698 .0637

~o .0883 .5404 .4871~, .1149 1.4560 1.2940

6 .1142 .9261 .8433c? .0292 .0171 .0160

f3, .0583 1.3777 1,3614 1.3613132 .0256 1.6377 1.1817 1.1816¢ .0082 .0450 .0505p .0147 .0442 .0421

~o .0002 .3187 .3079~, -.0382 .8852 .8261~2 -.0612 .5789 .5465c? -.0091 .0123 .0118

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Zhang, Lin, Raz, and Sowers: Semiparametric Stochastic Mixed Models 717

o 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Days in standardized menstrual cycle

-:::::::..~.-::..'::. .

......--- .

m=68

m=34

........--::-:--- .

...g 0"t:l '.

* 0\\~

E \ ~........,., 0~'0w ~<Jl

0

N00

7. DISCUSSION

In this article we have considered the inference for semi­parametric stochastic mixed models for longitudinal data.

Figure 4. Empirical, Frequentist, and Bayesian Pointwise StandardErrors of Estimated Nonparametric Time Functions i(t) Based on 500Replications: -, Empirical SE; - - - , Frequentist SE; and --­Bayesian SE.

gression coefficients and the variance components closelyagree with each other. The Bayesian and frequentist stan­dard errors of the regression coefficient estimates are almostidentical. This may be due to the fact that the bias in /3 isnegligible, and the true fixed nonparametric function f (t)has a large curvature, which results in the values of theestimated smoothing parameter f in all simulated datasetsbeing much larger than the other variance components. Notethat only Bayesian SEs of the variance component estima­tors are given, because their ferquentist counterparts aredifficult to calculate from (22) and the Bayesian SEs per­form very well.

The bias in the estimated nonparametric function f (t)is minimal and becomes smaller as the sample size doubles(Fig. 3). The bias is relatively higher at values of t where thevariance of the outcome variable is larger. Figure 4 showsthat the pointwise model-based Bayesian and frequentistSEs both agree quite well with the empirical SEs. As ex­pected, the frequentist SEs are smaller than the BayesianSEs. All SEs decrease as the sample size increases.

Figure 5 compares the estimated 95% pointwise cover­age probabilities of the frequentist and Bayesian confidenceintervals of f (t). Overall, the coverage probabilities usingthe Bayesian confidence intervals are sightly closer to thenominal value than those obtained from the frequentist con­fidence intervals for both m = 34 and m = 68. The meansover time of the estimated Bayesian and frequentist cov­erage probabilities are 94% and 93% when m = 34 and94.8% and 93.8% when m = 68. These results suggest thatthe Bayesian confidence intervals perform slightly betterthan their frequentist counterparts when the true f(t) is afixed smooth function. This may be due to the fact that theBayesian SEs account for the bias in the MPLE j. Our sim­ulation results are consistent with those of Wahba (1985) forindependent data and suggest that the Bayesian confidenceintervals have good sampling properties for a fixed smoothfunction f(t).

~

E- o

! .'E 0,.,

0III ..'

'".!::giii 9

and then decrease. The frequentist and Bayesian confidenceintervals of the natural cubic spline estimator j (t) are veryclose. This finding reflects our theoretical results in Sec­tion 3.4, because the estimate of smoothing parameter T ismuch larger than the other estimated variance components(Table 1).

Table 1 presents estimates of the regression coefficients,the variance components, and the smoothing parameter. Ageand BMI were found to have no significant effect on proges­terone level. Note that the frequentist and Bayesian standarderrors of the estimates of the regression coefficients are al­most identical. This again reflects our theoretical results inSection 3.4 and may be due to the same reason stated inthe previous paragraph. The estimates of (~o, 6, 6) indi­cate that the variance of the outcome progesterone levelvaries strongly over time. The estimated variance curve su­perimposed in Figure 2 agrees quite well with the empiricalvariances.

Days in standardized menstrual cycle

• Figure 3. Empirical Biases in Estimated Nonparametric Functionsf(t) Based on 500 Replications: -, m = 34; - - - m = 68.

6. A SIMULATION STUDY

We carried out a simulation study to evaluate the perfor­mance of the MPLE estimates of the regression coefficientsand the nonparametric function with the smoothing param­eter estimated using REML and the REML variance com­ponents. The design of the simulation study was identicalto that of the original progesterone data. We generated data(m = 34) from model (26) and set the true regression co­efficients (3, the fixed nonparametric function f(t), and thetrue variance components (J equal to the estimates from theanalysis of the real data. To study the influence of the sam­ple size on the performance of the estimation procedure, werepeated this experiment with the entire design duplicatedand the number of subjects doubled so that m = 68. Wecarried out 500 simulations under each experiment.

Table 2 presents the relative biases, empirical standarderrors, and model-based Bayesian and frequentist standarderrors of the parameter estimates. The relative bias is de­fined as the bias of a parameter estimate divided by itstrue value. The MPLEs of the regression coefficients arenearly unbiased, whereas the REML estimates of the vari­ance components are slightly biased when m = 34 andbecome virtually unbiased when m = 68. The empiricaland model-based standard errors of the estimates of the re-

o 2 4 6 8 10 12 14 16 18 20 22 24 26 28

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Days in standardized menstrual cycle

o 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Figure 5. Estimated Pointwise Frequentist and Bayesian 95% Cov­erage Probabilities of the Values of the True Fixed Function f(t) Basedon 500 Replications: - - - • Frequentist; ---. Bayesian.

Journal of the American Statistical Association, June 1998

depend on which of m and r (or ni) goes to infinity firstand which one goes there faster.

We have discussed using REML to estimate the smooth­ing parameter. Alternative approaches to selecting thesmoothing parameter include cross-validation (CV) (Riceand Silverman 1991) and generalized cross-validation(GCV) (Wahba 1985). Note that GCV so far has not beenwell-defined for correlated data. Unlike REML, GCV posesthe challenge of suffering the same problem as CV-thatis, subsequent inference on the variance components couldbe difficult (Zeger and Diggle 1994). Further research isneeded to compare the performance of the MPLEs (/3, f)using CV and GCV (to be defined) with that using REML.

In view of the fact that in longitudinal studies thevariance-covariance of the outcome variable often variesover time (Donnelly et al. 1995; Taylor et al. 1994), we pro­pose various stationary and nonstationary Gaussian stochas­tic processes to model the within-subject variability. Al­though in some cases the choice of the within-subjectstochastic process does not have fundamental impact onthe major inference of interest, it could be crucial to thevalidity of statistical inference in other situations. Further­more, a reasonable model for the subject-specific processwould help investigators better understand the underlyingbiological process. In the presence of a nonstationary pro­cess, as in the progesterone data, the existing method formodel diagnostics using a variogram (Diggle et al. 1994)is not applicable. More research is needed to develop newdiagnostic tools to examine the appropriateness of the as­sumed covariance structure in such data.

When the variance changes over time in a longitudinalstudy, a question of interest is whether one should specifythe time-varying variance in the random effects part or inthe stochastic process part in model (1). Because the ran­dom effects model the between-subject variation, whereasthe stochastic process models the within-subject variation,our first suggestion is that the decision could be made basedon one's understanding of the underlying biological process,which could provide a guide on the causes of the changeof the variance over time; that is, between or within-subjectvariation. Our second suggestion is to examine the data foreach individual as a single time series to look for nonsta­tionarity. For example, in the progesterone data, it is biolog­ically plausible to assume that the variance changes with themean progesterone level over time. This suggests that thetime-varying variance may be due to the underlying biolog­ical process. Hence we modeled it using a NOU process inthe stochastic process part of the model. This view is sup­ported by examining each woman's data. Specifically, wefound that for most women, the progesterone level is oftenstable during the first half of the cycle when the proges­terone level is low, and becomes variable when the proges­terone level increases during the second half of the cycle.

In the analysis of the progesterone data, we modeled themean progesterone level as well as the serial correlation asa function of time measured in a standardized menstrual cy­cle. This implies that the serial correlation between consec­utive day on the transformed time scale is the same for dif­ferent women. In other words, we assume that, for example,

m=34

m=68

\

...~.\. ,..r...-...._.....'..•.~~../...'..........•.....'....'.. _... rt:'.....I1'~.... / - /.::.:.:\ ,~- . - - ". --::.. -- / ...•..~ "',. ~/-- - .~"""

718

~ 1.00.s...c

0.95-eQ...Cl

0.90~..8

1.00~IIIOl

~ 0.95-E~

0.90w

We used MPLE to estimate the regression coefficients of theparametric fixed effects and to estimate the nonparametricfunction of time as a natural cubic spline. We used REMLto estimate the smoothing parameter and the variance com­ponents simultaneously. A key feature of our approach isthat it allows us to make systematic inference on all modelparameters by representing a semiparametric model as amodified parametric linear mixed model.

Our simulation results show that the proposed methodperforms well in estimating the regression coefficients,the nonparametric function, and the variance components.Bayesian confidence intervals of the nonparametric func­tion f(t) appear to have slightly better coverage probabili­ties compared to their frequentist counterparts in the situa­tion where the true f (t) is a fixed smooth function, whichis the situation of our primary interest. No practical differ­ence is found in our simulation between the Bayesian andfrequentist approaches in calculating the standard errors forthe estimates of the regression coefficients.

Although our simulation study reveals the attractive per­formance of REML in estimating f(t) and (3, the asymp­totic properties of REML still need to be investigated. Thisresearch requires studying consistency and asymptotic nor­mality of (](t), /3, 0), which could be challenging and couldrequire developing innovative statistical theory, as stan­dard asymptotic by simply letting the number of subjectsm --+ 00 might not be sufficient. This is because m --+ 00

does not guarantee that the number of distinct time points rwould also go to infinity. The latter is often required by con­sistency and asymptotic normality of ] (t). To have r --+ 00,

one might need to have the number of time points for eachsubject ni --+ 00 if different subjects are observed at thesame time points. It follows that consistency and asymp­totic normality of ] (t), /3, and () require that both m andr (or ni) go to infinity, and the normalizing constants for](t), /3, and () could well be different. Specifically, it is ofpotential interest to investigate whether the MPLEs of theregression coefficients /3 and the REML estimators of thevariance components 0 are ..;m consistent and whether theconventional convergence rate r-4

/ 5 of the mean squarederror of the MPLE f still applies. The asymptotic could also

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Zhang, Lin, Raz, and Sowers: Semiparametric Stochastic Mixed Models

the correlation of two observations measured I day apart fora woman with a 25-day cycle is the same as the correlationof two observations measured 2 days apart for a womanwith a 50-day cycle. This assumption may be reasonableif one believes the degree of similarity of two consecutiveprogesterone measures for a woman is driven by the timewithin a menstrual cycle relative to cycle length. An alter­native approach suggested by one referee is to model themean progesterone level on the transformed time scale andto model serial correlation on the original time scale. Be­cause 30 out of 34 women had cycle lengths very close tothe reference 28-day cycle (23-32 days) and only 4 womenhad long cycles (39-56 days), we expect that the standard­ization of the lengths of different women's menstrual cycleshas minimal impact on the inference. Because only very fewwomen had long cycles in our data, we found it was dif­ficult to determine which of the two scales was better formodeling serial correlation using the available data.

For simplicity, our model assumes a single nonparamet­ric time function and parametric covariate effects. One caneasily extend our model and inference procedure to accom­modate a more complicated mean structure, such as a modelwith an interaction between a covariate and a nonparamet­ric time function. For example, if we have a single covariatex, then the interaction model can be written as

Yij = Xij(3 + f(tij) + Xijg(tij)

+ Z~bi + Ui(tij) + Cij, (27)

where f(t) and g(t) are smooth nonparametric functionsof times and g(t) is centered for the sake of identifiability.We can use a similar penalized likelihood approach to ob­tain natural cubic smoothing spline estimates of both f(t)and g(t). Specifically, from (10) we can write f and g asf = T~1 + Ba, and g = t082 + Ba2' where to denotes to

* *centered about its mean and al and a2 are independent andfollow normal(O,711) and normal(O,721). Plugging these ex­pressions of f and g into (27), we can write (27) as a linearmixed model similar to (12). Estimation and inference onthe model parameters can then proceed by applying the ap­proach described in this article.

[Received December 1996. Revised October 1997.J

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