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. 2017 Jun 25;19(2):137–152. doi: 10.1093/biostatistics/kxx026

Simple fixed-effects inference for complex functional models

So Young Park 1,✉,, Ana-Maria Staicu 1, Luo Xiao 1, Ciprian M Crainiceanu 2
PMCID: PMC5862370  PMID: 29036541

SUMMARY

We propose simple inferential approaches for the fixed effects in complex functional mixed effects models. We estimate the fixed effects under the independence of functional residuals assumption and then bootstrap independent units (e.g. subjects) to conduct inference on the fixed effects parameters. Simulations show excellent coverage probability of the confidence intervals and size of tests for the fixed effects model parameters. Methods are motivated by and applied to the Baltimore Longitudinal Study of Aging, though they are applicable to other studies that collect correlated functional data.

Keywords: Bootstrap/resampling, Functional data, Measurement error, Smoothing and nonparametric regression

1. Introduction

Rapid advancement in technology and computation has led to an increasing number of studies that collect complex-correlated functional data. In response to these studies research in structured functional data analysis (FDA) has witnessed rapid development. A major characteristic of these data is that they are strongly correlated, as multiple functions are observed on the same observational unit. Many new studies have functional structures including multilevel (Morris and others, 2003; Morris and Carroll, 2006; Di and others, 2009; Crainiceanu and others, 2009), longitudinal (Greven and others, 2010; Chen and Müller, 2012; Scheipl and others, 2015), spatially aligned (Baladandayuthapani and others, 2008; Staicu and others, 2010; Serban and others, 2013), or crossed (Aston and others, 2010; Shou and others, 2015).

While these types of data can have highly complex dependence structures, one is often interested in simple, population-level, questions for which the multi-layered structure of the correlation is just an infinite-dimensional nuisance parameter. For example, the Baltimore Study of Aging (BLSA), which motivated this article, collected physical activity levels from each of many participants at the minute level for multiple consecutive days. Thus, the BLSA activity data exhibit complex within-day and between-day correlations. However, the most important questions in the BLSA tend to be simple; in particular, one may be interested in how age affects the daily patterns of activity or whether the effect is different by gender. In this context, the high complexity and size of the data are just technical inconveniences.

Such simple questions are typically answered by estimating fixed effects in complex functional mixed effects models. Our proposed approach avoids complex modeling and implementation by: (i) estimating the fixed (population-level) effects under the assumption of independence of functional residuals; and (ii) using a nonparametric bootstrap of independent units (e.g. subjects) to construct confidence intervals and conduct tests. A natural question is whether efficiency is lost by ignoring the correlation. While the loss of efficiency is well documented in longitudinal studies with few observations per subject and small dimensional within-subject correlation, little is known about inference when there are many observations per subject with an unknown large dimensional within-subject correlation matrix. An important contribution of this article is to evaluate the performance of bootstrap-based inferential approaches in this particular context. Our view is that estimating large dimensional covariance matrices of functional data may hurt fixed effects estimation by wasting degrees of freedom. Indeed, a covariance matrix for an Inline graphic by Inline graphic matrix of functional data (Inline graphic = number of subjects and Inline graphic = number of subject-specific observations) would require estimation of Inline graphic matrix covariance entries when the covariance matrix is unstructured. When Inline graphic is moderate or large this is a difficult problem. Moreover, the resulting matrix has an unknown low rank and is not invertible.

We will consider cases when multiple functional observations are observed for the same subject. This structure is inspired by many current observational studies, but we will focus on the BLSA, where activity data are recorded at the minute level over multiple consecutive days, resulting in daily activity profiles (each as a function of time of the day) observed for each participant over multiple days. Specifically, we will focus on data from 332 female BLSA participants with age varying between 50 and 90. A total of 1580 daily activity profiles were collected (an average of 4.7 monitoring days per person), where each daily profile consists of 1440 activity counts measured at the minute level. Thus, the activity data considered in this paper is stored in a Inline graphic dimensional matrix. Our primary interest is to conduct inference on the fixed effects of covariates, such as age and body mass index (BMI), on daily activity profiles. Because data from each participant were collected on consecutive days in a short period (about a week on average), age and BMI in the BLSA data are subject-specific but visit-invariant.

While our covariates are time-invariant, we propose methods that can accommodate both time-invariant and time-dependent covariates. Assume that the observed data is of the form Inline graphic, where Inline graphic is the Inline graphicth unit functional response (e.g. Inline graphicth visit) for the Inline graphicth subject, and Inline graphic is the corresponding vector of covariates. This general form applies to all types of functional data discussed above: multilevel, longitudinal, spatially correlated, crossed, etc. The main objective is to make statistical inference for the population-level effects of interest using repeatedly observed functional response data.

A naïve approach to analyze data with such a complex structure is to ignore the dependence over the functional argument Inline graphic, but to account for the dependence across the repeated visits. That is, by assuming that the responses Inline graphic are correlated over Inline graphic and independent over Inline graphic. Longitudinal data analysis literature offers a wide variety of models and methods for estimating the fixed effects and their uncertainty, and for conducting tests (see for example Laird and Ware (1982); Liang and Zeger (1986); Fitzmaurice and others (2012)). These methods allow to account for within-subject correlation, incorporate additional covariates, and make inference about the fixed effects. However, extending these estimation and inferential procedures to functional data is difficult. In the literature this has been addressed by modeling the within- and between-curve dependence structure using functional random effects. These approaches are highly computationally intensive, require inverting high dimensional covariances matrices, and make implicit assumptions about the correlation structures that may not be easy to transport across applications.

Another possible approach is to completely ignore the dependence across the repeated visits Inline graphic, but account for the functional dependence. That is, assume Inline graphic are dependent over Inline graphic, but independent over Inline graphic. Function on scalar/vector regression models can be used to estimate the fixed effects of interest; see for example Faraway (1997); Jiang and others (2011). In this context, testing procedures for hypotheses on fixed effects are available. For example, Shen and Faraway (2004) proposed the functional F statistic for testing hypotheses related to nested functional linear models. Zhang and others (2007) proposed Inline graphic norm based test for testing the effect of a linear combination of time-varying coefficients, and approximate the null sampling distribution using resampling methods. However, failing to account for dependence across visits results in tests with inflated type I error.

In contrast, development of statistical inferential methods for correlated functional data has received less attention. Fully Bayesian inference has been previously considered in the literature for complex designs; see, for example, Morris and Carroll (2006), Morris and others (2006), Morris and others (2011), Zhu and others (2011), and Zhang and others (2016). These approaches take into account both between- and within-function correlations using MCMC simulations of the posterior distribution. In contrast, we focus on a frequentist approach to inference that avoids modeling of the complex correlation structures. In the frequentist framework, Crainiceanu and others (2012) discussed bootstrap-based inferential methods for the difference in the mean profiles of correlated functional data. Staicu and others (2014) proposed a likelihood-ratio type testing procedure, while Staicu and others (2015) considered Inline graphic norm-based testing procedures for testing that multiple group mean functions are equal. Horváth and others (2013) developed inference for the mean function of a functional time series. However, these approaches focus on testing the effect of a categorical variable, and do not handle inference on fixed effects in full generality.

Here we consider a modeling framework that is a direct generalization of the linear mixed model framework from longitudinal data analysis, where scalar responses are replaced with functional ones. We propose to model the fixed effect of a scalar covariate either parametrically or nonparametrically while the error covariance is left unspecified to avoid model complexity. We estimate the fixed effects under the working independence and account for all known sources of data dependence by bootstrapping over subjects. Based on this procedure, we propose confidence bands and Inline graphic norm-based testing for fixed effects parameters. An important contribution of this article is to investigate and confirm the performance of the bootstrap-based inferential approaches when data have a complex functional dependence structure.

2. Modeling framework and estimation

Consider the case when each subject is observed at Inline graphic visits, and data at each visit consist of a functional outcome Inline graphic and a vector of covariates including a scalar covariate of interest, Inline graphic, and additional Inline graphic-dimensional vector of covariates, Inline graphic. We assume that Inline graphic, where Inline graphic is a compact and closed domain; take Inline graphic for simplicity. For convenience, we assume a balanced regular sampling design, i.e. Inline graphic and Inline graphic, though all methods apply to general sampling designs. Furthermore, we assume that Inline graphic is a dense set in the closed domain Inline graphic; this assumption is needed when the fixed effect of Inline graphic is modeled nonparametrically (Ruppert and others, 2003; Fitzmaurice and others, 2012). A common approach for the study of the effect of the covariates on Inline graphic is to posit a model of the type

Yij(t)=μ(t,Xij)+ZijTτ+ϵij(t), (2.1)

where Inline graphic is a time-varying smooth fixed effect of the covariate of interest, Inline graphic, and Inline graphic is a Inline graphic-dimensional parameter quantifying the linear additive fixed effect of the covariate vector, Inline graphic. Inline graphic is a zero-mean random deviation that incorporates both the within- and between-subject variability. Inline graphic can be modeled either parametrically or nonparametrically; see Section 6 (F2.) for some possible mean structures. While technically more difficult to implement, nonparametric smoothing is useful when limited information about the mean structure is available.

Here we present the most complex case where the mean structure for Inline graphic is an unknown bivariate smooth function. We construct a bivariate basis using the tensor product of two univariate B-spline bases, Inline graphic, and Inline graphic, defined on Inline graphic and Inline graphic respectively. The unspecified mean is then expressed as Inline graphicInline graphic, where Inline graphic is the Inline graphic-dimensional vector of Inline graphic’s and Inline graphic is the vector of parameters Inline graphic. Typically, the number of basis functions is chosen sufficiently large to capture the maximum complexity of the mean function and smoothness is induced by a quadratic penalty on the coefficients. There are several penalties for bivariate smoothing; see, for example, Marx and Eilers (2005), Wood (2006), and Xiao and others (2013, 2016). In this article we used the following estimation criterion

argminβ,τ,λi,j,[Yij{B(t,Xij)Tβ+ZijTτ}]2+βTPλβ (2.2)

with a penalty matrix Inline graphic described in Wood (2006) and a vector of smoothing parameters, Inline graphic. Specifically, we used Inline graphic and Inline graphic, where Inline graphic denotes the tensor product, and Inline graphic and Inline graphic are the marginal second order difference matrix and the smoothing parameter for the Inline graphic direction, respectively; Inline graphic and Inline graphic are defined similarly for the Inline graphic direction. Here Inline graphic and Inline graphic are the identity matrices of dimensions Inline graphic and Inline graphic, respectively. For a fixed smoothing parameter, Inline graphic, the minimizer of (2.2) has the form Inline graphic where Inline graphic with Inline graphic the matrix with rows BInline graphic and Inline graphic the matrix obtained by row-stacking of Inline graphic, while the estimated mean is Inline graphic. In this article, the generalized cross validation (GCV) is used to select the optimal smoothing parameters, while other criteria such as the restricted maximum likelihood can be used; relevant literatures on selection of the smoothing parameter include Wahba (1990) and Ruppert and others (2003).

Estimation of the fixed effects in model (2.1) under the working independence assumption is not new; see for example Scheipl and others (2015) and Chen and Müller (2012). However, our approach to inference for the population level fixed effects in the context of structured functional data has not been studied. The novelty of this article consists precisely in filling this gap in the literature. We consider an estimation approach of fixed effects under working independence and a bootstrap of independent units approach to appropriately account for complex correlation.

3. Confidence bands for Inline graphic

We now discuss inference for Inline graphic using confidence bands and formal hypothesis testing. Without loss of generality, assume that the mean structure is Inline graphic, where Inline graphic can be as simple as Inline graphic or as complex as a vector of prespecified basis functions. The mean estimator of interest is Inline graphic. One could study pointwise variability for every pair Inline graphic, that is Inline graphic, or the joint variability for the entire domain Inline graphic, that is Inline graphic. Irrespective of the choice, the variability is fully described by the variability of the parameter estimator Inline graphic.

3.1. Bootstrap algorithms

We consider a flexible dependence structure for Inline graphic that describes both within- and between-subject variability. We make minimal assumption that Inline graphic is independent over Inline graphic but correlated over Inline graphic and Inline graphic, though we do not specify the form of this correlation. Deriving the analytical expression for the sampling variability of the estimator Inline graphic in such contexts is challenging. Instead, we propose to use two bootstrap algorithms: bootstrap of subject-level data and bootstrap of subject-level residuals. These approaches have already been studied and used in nonparametric regression for independent measurements; see, for example, Härdle and Bowman (1988), Efron and Tibshirani (1994), and Hall and others (2013) among many others. Bootstrap of functional data for fixed effects has also been considered, including by Politis and Romano (1994) for weakly dependent processes in Hilbert space, by Cuevas and others (2006) for independent functional data, and by Crainiceanu and others (2012) for paired samples of functional data. However, studying these bootstrap algorithms for functional data with complex correlation is new.

The subject-level bootstrap algorithm for correlated functional data is provided below.

Algorithm 1 Bootstrap of the subject-level data [uncertainty estimation]

  • 1: forInline graphicdo

  • 2: Re-sample the subject indexes from the index set Inline graphic with replacement.

    Let Inline graphic be the resulting sample of Inline graphic subjects.

  • 3: Define the Inline graphicth bootstrap data by: Inline graphic.

  • 4: Using Inline graphic fit the model (2.1) with the mean structure of interest modeled by Inline graphic, by employing criterion (2.2). Let Inline graphic be the corresponding estimate of the parameter of interest; similarly define Inline graphic. end for

  • 5: Calculate the sample covariance of Inline graphic; denote it by Inline graphic.

The bootstrap of subject-level data is more generally applicable, while the bootstrap of subject-level residuals approach relies on two important assumptions: (i) the covariates do not depend on visit, that is Inline graphic and Inline graphic; and (ii) both the correlation and the error variance are independent of covariates. These assumptions ensure that sets of subject-level errors, i.e. Inline graphic for Inline graphic, can be resampled over subjects without affecting the sampling distribution. These assumptions are reasonable when covariates are independent of the visit, as is the case in the BLSA application. Indeed, in BLSA we consider age and BMI, which are time-invariant because repeated measures per subject were collected within a week.

Similarly, we introduce the algorithm for bootstrapping residuals. We start by fitting the model (2.1) with the mean structure of interest modeled by Inline graphic, using the estimation criterion described in (2.2), and calculating the residuals Inline graphic.

Algorithm 2 Bootstrap of the subject-level residuals [uncertainty estimation]

  • 1: forInline graphic

  • 2: Re-sample the subject indexes from the index set Inline graphic with replacement. Let Inline graphic be the resulting sample of subjects. For each Inline graphic denote by Inline graphic the number of repeated time-visits for the Inline graphicth subject selected in Inline graphic.

  • 3: Define the Inline graphicth bootstrap sample of residuals Inline graphic.

  • 4: Define the Inline graphicth bootstrap data by: Inline graphic, where Inline graphic.

  • 5: Using Inline graphic fit the model (2.1) with the mean structure of interest modeled by Inline graphic, by employing criterion (2.2). Let Inline graphic be the corresponding estimate of the parameter of interest; similarly define Inline graphic. end for

  • 6: Calculate the sample covariance of Inline graphic; denote it by Inline graphic.

Based on our numerical investigation (see Section 6) the bootstrap of subject-level residuals has excellent performance and is recommended when the necessary assumptions are satisfied, though the bootstrap of subjects is a good alternative.

3.2. Bootstrap-based inference

For fixed Inline graphic, the variance of the estimator Inline graphic can be estimated as Inline graphic, by using the bootstrap-based estimate of the covariance of Inline graphic. A Inline graphic pointwise confidence interval for Inline graphic can be calculated as Inline graphic, using normal distributional assumption for the estimator Inline graphic, where Inline graphic is the Inline graphic percentile of the standard normal. An alternative is to use the pointwise Inline graphic and Inline graphic quantiles of the bootstrap estimates Inline graphic.

In most cases, it makes more sense to study the variability of Inline graphic, and draw inference about the entire true mean function Inline graphic. Thus, we focus our study on constructing a joint (or simultaneous) confidence band for Inline graphic. Constructing simultaneous confidence bands for univariate smooths has already been discussed in the nonparametric literature. For example, Degras (2009), Ma and others (2012), and Cao and others (2012) proposed asymptotically correct simultaneous confidence bands for different estimators, when data are independently sampled curves; Crainiceanu and others (2012) proposed bootstrap-based joint confidence bands for univariate smooths in the case of functional data with complex error processes. Here, we present an extension of the approach considered by Crainiceanu and others (2012) to bivariate smooth function estimation for general functional correlation structures.

Let Inline graphic and Inline graphic be the evaluation points that are equally spaced in the domains Inline graphic and Inline graphic, respectively. We evaluate the bootstrap estimate Inline graphic of one bootstrap sample at all pairs Inline graphic, and denote by Inline graphic the Inline graphic-dimensional vector with components Inline graphic. Let Inline graphic be the Inline graphic-dimensional matrix obtained by column-stacking Inline graphic for all Inline graphic and Inline graphic. Let Inline graphic as defined above. After adjusting for the bivariate structure of the problem, the main steps of the construction of the joint confidence bands for Inline graphic follow similarly to the ones used in Crainiceanu and others (2012) for univariate smooth parameter functions.

Step 1. Generate a random variable Inline graphic from the multivariate normal with mean Inline graphic and covariance matrix Inline graphic; let Inline graphic for Inline graphic and Inline graphic.

Step 2. Calculate Inline graphic.

Step 3. Repeat Step 1. and Step 2. for Inline graphic, and obtain Inline graphic. Determine the Inline graphic empirical quantile of Inline graphic, say Inline graphic.

Step 4. Construct the Inline graphic joint confidence band by: Inline graphic. Here Inline graphic.

The joint confidence band, in contrast to the pointwise confidence band, can be used as an inferential tool for formal global tests about the mean function, Inline graphic. For example, one can use the joint confidence band for testing the null hypothesis, Inline graphic and for some prespecified function Inline graphic, by checking whether the confidence band Inline graphic contains Inline graphic for all Inline graphic. If the confidence band does not contain Inline graphic for some Inline graphic, then we conclude that there is significant evidence that the true mean function is the prespecified function Inline graphic.

4. Hypothesis testing for Inline graphic

Next, we focus on assessing the effect of the covariate of interest Inline graphic on the mean function. Consider the general case when the model is (2.1) and the average effect is an unspecified bivariate smooth function, Inline graphic. Our goal is to test if the true mean function depends on Inline graphic, that is testing:

H0:μ(t,x)=μ0(t) for all t,x, (4.1)

for some unknown smooth function Inline graphic against Inline graphic varies over Inline graphic for some Inline graphic.

To the best of our knowledge, this type of hypothesis, where the mean function is nonparametric both under the null and alternative hypotheses, has not been studied in FDA. The problem was extensively studied in nonparametric smoothing, where the primary interest centered on significance testing of a subset of covariates in a nonparametric regression model; see, for example, Fan and Li (1996), Lavergne and Vuong (2000), Delgado and Manteiga (2001), Gu and others (2007), and Hall and others (2007). However, all these methods are based on the assumption that observations are independent across sampling units; in our context requiring independence of Inline graphic over Inline graphic and Inline graphic is unrealistic and failing to account for this dependence leads to inflated type I error rates.

To test hypothesis (4.1) we propose a test statistic based on the Inline graphic distance between the mean estimators under the null and alternative hypotheses. Specifically we define it as:

T=XT{μ^A(t,x)μ^0(t)}2dtdx, (4.2)

where Inline graphic and Inline graphic are the estimates of Inline graphic under the null and alternative hypotheses, respectively. In particular, Inline graphic is estimated as in Section 2. The estimator Inline graphic is obtained by modeling Inline graphic for the Inline graphic-dimensional vector Inline graphic and by estimating the mean parameters Inline graphic based on a criterion similar to (2.2).

Deriving the finite sample distribution of the test statistic Inline graphic under the null hypothesis is challenging and we propose to approximate it using the bootstrap. As in Section 3, the smoothing parameter selection is repeated for each bootstrap sample and model, Inline graphic and Inline graphic.

Algorithm 3 Bootstrap approximation of the null distribution of the test statistic, Inline graphic

  • 1: forInline graphic

  • 2: Re-sample the subject indexes from the index set Inline graphic with replacement. Let Inline graphic be the obtained sample of subjects. For each Inline graphic denote by Inline graphic the number of repeated time-visits for the Inline graphicth subject selected in Inline graphic.

  • 3: Define the Inline graphicth bootstrap sample of pseudo-residuals Inline graphic. For each Inline graphic let Inline graphic the corresponding sample of the nuisance covariates for the Inline graphicth subject selected in Inline graphic. Similarly define Inline graphic.

  • 4: Define the Inline graphicth bootstrap data by: Inline graphicInline graphic, where Inline graphic

  • 5: Using Inline graphic fit two models. First, fit model (2.1) with the mean structure modeled by Inline graphic and estimate Inline graphic. Second, fit model (2.1) with the mean model Inline graphic and estimate Inline graphic. Calculate the test statistic Inline graphic using (4.2). end for

  • 6: Approximate the tail probability Inline graphic by the Inline graphic, where Inline graphic is obtained using the original data and Inline graphic is the indicator function.

When the covariates Inline graphic and Inline graphic do not depend on visit, i.e. Inline graphic and Inline graphic, the algorithm can be modified along the lines of the ‘bootstrap of the subject-level residuals’ algorithm.

5. Application to physical activity data

Physical activity measured by wearable devices such as accelerometers provides new insights into the association between activity and health outcomes (Schrack and others, 2014); the complexity of the data also poses serious challenges to current statistical analysis. For example, accelerometers can record activity at the minute level for many days and for hundreds of individuals. Here we consider the physical activity data from the BLSA (Stone and Norris, 1966). Each female participant in the study wore the Actiheart portable physical activity monitor (Brage and others 2006) for 24 h a day for a number of consecutive days; visit duration varied among participants with an average of 4.7 days. Activity counts were measured in 1-min epochs and each daily activity profile has 1440 minute-by-minute activity counts measurements. Activity counts are proxies of activity intensity. Activity counts were log-transformed (more precisely, Inline graphic) because they are highly skewed and then averaged in 30-min intervals. For simplicity, hereafter we refer to the log-transformed counts as log counts. Here we focus on 1580 daily activity profiles from a single visit of 332 female participants who have at least two days of data. Women in the study are aged between 50 and 90 years. Further details on the BLSA activity data can be found in Schrack and others (2014) and Xiao and others (2015).

Our objective is to conduct inference on the marginal effect of age on women’s daily activity after adjusting for BMI. We model the mean log counts as Inline graphic, where Inline graphic and Inline graphic are the age and BMI of the Inline graphicth woman during the visit, Inline graphic is the baseline mean log counts for time Inline graphic within the day for a woman who is Inline graphic-years old, and Inline graphic is the association of BMI with mean log counts for time Inline graphic within the day. We test whether Inline graphic varies solely with Inline graphic. We use the proposed testing statistic, Inline graphic as detailed in Section 4. The estimate Inline graphic is based on the tensor product of Inline graphic cubic basis functions in Inline graphic and Inline graphic cubic basis functions in Inline graphic and the estimate Inline graphic is based on Inline graphic cubic basis functions. Goodness of fit is studied by comparing the observed data with simulated data from the fitted model; see Figure S6 of the supplementary materials available at Biostatistics online. Figure S1 of supplementary material available at Biostatistics online shows the null distribution of the statistic Inline graphic. The observed test statistic is Inline graphic and the corresponding p-value is less than Inline graphic based on Inline graphic MC samples. This indicates that there is strong evidence that daily activity profiles in women vary with age.

Figure 1 displays the estimated baseline activity profile as a function of age, Inline graphic, using the average of all bootstrap estimates. The plot indicates that the average log counts is a decreasing function of age for most times during the day. Furthermore, it depicts two activity peaks, one around 12 pm and the other around 6 pm. The 6-pm peak seems to decrease faster with age, indicating that afternoon activity is more affected by age than morning activity. We use joint confidence band to evaluate the sampling variability of Inline graphic. The joint lower and upper Inline graphic confidence limits based on methods described in Section 3 are displayed in the bottom plots of Figure 1; the plots show that across all ages, the estimated low average activity at night has relatively small variability while the estimated high-average activity during the day has relatively high variability. To visualize the results, we display the estimated activity profile for 60-years-old women, Inline graphic, and the corresponding Inline graphic joint confidence band in Figure 2. Figure S2 of supplementary material available at Biostatistics online displays the estimated association of BMI with mean log counts as a function of time of day; it suggests that women with higher BMI have less activity during the day and evening, albeit more activity at late night and in early morning.

Fig. 1.

Fig. 1.

Heat map of average of bootstrap estimates of log counts as a bivariate function of time of day and age (top left panel); average of bootstrap estimates of log counts for five different age groups (top right panel); and heat maps of joint confidence bands for the estimate in the top left panel (bottom panels). The legend on the right applies to both of the bottom plots.

Fig. 2.

Fig. 2.

Average of bootstrap estimates of log counts as a function of time of day at age 60 and the associated joint confidence bands.

5.1. Validating the testing results via simulation study

We conducted a simulation study designed to closely mimic the BLSA data structure. Specifically, we generated data from model (2.1) with Inline graphic, where Inline graphic is the estimated mean log counts, and Inline graphic is a parameter quantifying the distance from the null and alternative hypotheses. When Inline graphic the true mean profile Inline graphic, whereas when Inline graphic then Inline graphic. The errors Inline graphic are generated with a covariance structure that closely mimics that of the residuals from the BLSA data. Specifically we use the model Inline graphic and the associated model estimates from Xiao and others (2015), where Inline graphic and Inline graphic are subject-specific and subject- and visit-specific random processes with mean zero and Inline graphic is white noise. Inline graphic and Inline graphic are generated uniformly from Inline graphic and Inline graphic, respectively. Sample size is set to be the number of female participants in the BLSA. Estimation is done exactly the same as in our data analysis. Table 1 shows the rejection probabilities in 1000 simulations when Inline graphic and indicates that the empirical size is close to the nominal levels. Figure S3 of supplementary material available at Biostatistics online displays the power in 500 simulations, when Inline graphic. When the true Inline graphic is the estimated bivariate mean log counts of the BLSA data, i.e. Inline graphic, the rejection probability reaches 1.

Table 1.

Empirical type I error of the test statistic Inline graphic based on the Inline graphic MC samples; Mean function is Inline graphic, Inline graphic

Inline graphic, Inline graphic
Inline graphic Inline graphic Inline graphic
0.06 0.11 0.16
(0.01) (0.01) (0.01)

Standard errors are presented in parentheses.

6. Simulation Study

We evaluate the performance of the proposed inferential methods. Data are simulated using the model (2.1) where Inline graphic, Inline graphic. The errors Inline graphic are generated from the model Inline graphic, where for each Inline graphic and Inline graphic the basis coefficients Inline graphic are generated from a multivariate normal distribution with mean zero and covariance Inline graphic, where Inline graphic is a correlation parameter and Inline graphic is the actual time of visit at which Inline graphic is observed; a similar dependence structure has been considered in simulation studies by Park and Staicu (2015) and Islam and others (2016). The residuals Inline graphic are mutually independent with zero mean and variance Inline graphic. The number of repeated measures is fixed at Inline graphic, Inline graphic, and the functions Inline graphic. The subject-specific covariates Inline graphic and Inline graphic are generated from a UniformInline graphic. The grid of points Inline graphic is set as Inline graphic equally spaced points in Inline graphic. The variance of the white noise process Inline graphic is set to Inline graphic, which provides a signal to noise ratio SNRInline graphic equal to Inline graphic.

We consider different combinations of the following factors: F1. number of subjects: (a) Inline graphic, (b) Inline graphic, and (c) Inline graphic; F2. bivariate mean function: (a) Inline graphic for Inline graphic, (b) Inline graphic for Inline graphic, (c) Inline graphic, and (d) Inline graphic, and Inline graphic, with/without the addition of linear effect of nuisance covariate Inline graphic, i.e. Inline graphic (no effect) and Inline graphic; lastly, F3. between-curves correlations: (a) Inline graphic (weak) and (b) Inline graphic (strong).

Confidence bands for model parameters are evaluated in two ways. First, we model the data by assuming the correct model and by evaluating the accuracy of the inferential procedures. Second, we model the data using a bivariate mean, Inline graphic, and evaluate the performance of the confidence bands of Inline graphic for covering the true mean even when the true mean has a simpler structure, i.e. F2 i.(a)–(c). The results for the first case are included in Section B of the supplementary material available at Biostatistics online, whereas those for the second case are presented below, because in the BLSA we used bivariate nonparametric fitting. Estimation is done as detailed in Section 2. We use Inline graphic cubic B-spline basis functions, and select the smoothing parameters via GCV; specifically, for the bivariate smooth, Inline graphic basis functions are used.

The performance of the pointwise and joint confidence bands is evaluated in terms of average coverage probability (ACP), and average length (AL) of the confidence intervals. Specifically, let Inline graphic be the Inline graphic pointwise confidence interval of Inline graphic obtained at the Inline graphic Monte Carlo generation of the data, then

ACPpoint=1NsimGtGxisim=1Nsimgt=1Gtgx=1Gx1{μ(tgt,xgx)(μ^isim,l(tgt,xgx),μ^isim,u(tgt,xgx))}
ALpoint=1NsimGtGxisim=1Nsimgt=1Gtgx=1Gx|μ^isim,l(tgt,xgx)μ^isim,u(tgt,xgx))|,

where Inline graphic and Inline graphic are equi-distanced grid points in the domains Inline graphic, and Inline graphic, respectively. Next, let Inline graphic be Inline graphic joint confidence interval. The AL is calculated as above, while the ACP is calculated as:

ACPμ(t,x)joint=1Nsimisim=1Nsim1{μ(tgt,xgx)(μ^isim,l(tgt,xgx),μ^isim,u(tgt,xgx)): for all gt,gx}.

The performance of the test statistic Inline graphic is evaluated in terms of its size for the nominal levels Inline graphic, Inline graphic, and Inline graphic, and power at Inline graphic. The results for the size are based on Inline graphic MC samples, while the results for ACP and AL of the confidence bands, and power of the test are based on Inline graphic MC samples. For each MC simulation we use Inline graphic bootstrap samples.

Table 2 shows the ACP and AL for the Inline graphic confidence bands based on the bootstrap of subject-level residuals when the sample size Inline graphic and when Inline graphic is modeled nonparametrically regardless of the true mean structure; the results for other nominal coverages (Inline graphic and Inline graphic) are included in Section A of the supplementary material available at Biostatistics online. Overall, the pointwise/joint confidence bands achieve the nominal coverage for all of the mean structures considered. The confidence bands tend to be wider when the between-curves correlation is strong (Inline graphic).

Table 2.

Simulation results for Inline graphic confidence bands based on the bootstrap of subject-level residuals when a nonparametric bivariate function is fitted for Inline graphic; results are based on Inline graphic MC samples

Case True mean function Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
(a) Inline graphic 0.20 0.94 (Inline graphic 0.01) 1.65 (0.01) 0.94 (0.01) 3.22 (0.01)
    0.90 0.94 (Inline graphic 0.01) 2.17 (0.02) 0.93 (0.01) 4.24 (0.01)
  Inline graphic 0.20 0.93 (0.01) 0.14 (Inline graphic 0.01)        
    0.90 0.93 (0.01) 0.14 (Inline graphic 0.01)        
(b) Inline graphic 0.20 0.94 (Inline graphic 0.01) 1.65 (0.01) 0.94 (0.01) 3.22 (0.01)
    0.90 0.94 (Inline graphic 0.01) 2.17 (0.02) 0.93 (0.01) 4.24 (0.01)
  Inline graphic 0.20 0.93 (0.01) 0.14 (Inline graphic 0.01)        
    0.90 0.93 (0.01) 0.14 (Inline graphic 0.01)        
(c) Inline graphic 0.20 0.94 (Inline graphic 0.01) 1.65 (0.01) 0.93 (0.01) 3.23 (0.01)
    0.90 0.94 (Inline graphic 0.01) 2.18 (0.02) 0.93 (0.01) 4.25 (0.01)
  Inline graphic 0.20 0.93 (0.01) 0.14 (Inline graphic 0.01)        
    0.90 0.93 (0.01) 0.14 (Inline graphic 0.01)        
(d) Inline graphic 0.61 0.94 (Inline graphic 0.01) 1.65 (0.01) 0.93 (0.01) 3.23 (0.01)
    0.90 0.94 (Inline graphic 0.01) 2.18 (0.02) 0.93 (0.01) 4.26 (0.01)
  Inline graphic 0.20 0.93 (0.01) 0.14 (Inline graphic 0.01)        
    0.90 0.94 (0.01) 0.14 (Inline graphic 0.01)        

Standard errors are presented in parentheses.

We also investigate the performance of the confidence band when the correct structure of Inline graphic is used; the corresponding results for the bootstrap of subject-level residuals and observations are included in Section B and Section C of the supplementary material available at Biostatistics online. The results show the good coverage of the pointwise/joint confidence bands based on the bootstrap of residuals by subjects for all of the mean structures considered. The bootstrap of observations by subjects leads to equally good coverage when the true effect of the covariate Inline graphic is linear (cases F2 i.(a)–(c)), whereas it leads to slight under–coverage when the true effect of Inline graphic is nonlinear (case F2 i.(d)). However, in the case of a visit-varying covariate Inline graphic the joint confidence band maintains nominal coverage even when the effect of Inline graphic is nonlinear; see Table S9 of the supplementary material available at Biostatistics online. These results indicate that for a time-invariant covariate, Inline graphic, the bootstrap of subject-level residuals is narrower and has better coverage. In terms of computational cost, fitting a nonparametric model is much slower than fitting a parametric model. For example when the true mean F2 i. (c) is used to generate the data, fitting a nonparametric model for Inline graphic bootstrap samples takes 337 s whereas the same procedure for a parametric model takes 50 s; the results are based on Inline graphic MC samples on a computer with a 3.60 Hz processor.

Table 3 shows the empirical size of the proposed testing procedure for testing Inline graphic, where Inline graphic is a smooth effect depending on Inline graphic only. Results indicate that, as sample size increases, the size of the test gets closer to the corresponding nominal levels. In the simulation settings considered, the test attains the correct sizes with sample size Inline graphic, which is the case in our motivating BLSA data application. Including an additional covariate in the model seems to have no effect on the performance of the testing procedure. Figure S4 of supplementary material available at Biostatistics online illustrates the power curves, when the true mean structure deviates from the null hypothesis. It presents the power as a function of the deviation from the null that involves both Inline graphic and Inline graphic, Inline graphic. Here Inline graphic quantifies the departure from the null hypothesis. As expected, for Inline graphic rejection probabilities increase as the departure from the null hypothesis increases, irrespective of the direction in which it deviates. As expected, rejection probabilities increase with the sample size. Our investigation indicates that the strength of the correlation between the functional observations corresponding to the same subject affect the rejection probability: the weaker the correlation, the larger the power. There is no competitive testing method available for this null hypothesis. Lastly we conducted a simulation study to evaluate the robustness of the proposed methods to non-Gaussian error distributions and obtained similar results with those from the Gaussian case; see Section D of the supplementary material available at Biostatistics online.

Table 3.

Empirical Type I error of the test statistic Inline graphic based on the Inline graphic MC samples

    Inline graphic, Inline graphic
    Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic 0.08 (0.01) 0.14 (0.01) 0.21 (0.01)
  Inline graphic 0.09 (0.01) 0.14 (0.01) 0.20 (0.01)
Inline graphic Inline graphic 0.07 (0.01) 0.13 (0.01) 0.17 (0.01)
  Inline graphic 0.08 (0.01) 0.12 (0.01) 0.18 (0.01)
Inline graphic Inline graphic 0.06 (0.01) 0.11 (0.01) 0.16 (0.01)
  Inline graphic 0.06 (0.01) 0.12 (0.01) 0.16 (0.01)
    Inline graphic, Inline graphic
    Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic 0.07 (0.01) 0.15 (0.01) 0.20 (0.01)
  Inline graphic 0.08 (0.01) 0.15 (0.01) 0.21 (0.01)
Inline graphic Inline graphic 0.07 (0.01) 0.13 (0.01) 0.17 (0.01)
  Inline graphic 0.08 (0.01) 0.12 (0.01) 0.18 (0.01)
Inline graphic Inline graphic 0.06 (0.01) 0.11 (0.01) 0.16 (0.01)
  Inline graphic 0.06 (0.01) 0.12 (0.01) 0.16 (0.01)

Supplementary Material

Supplementary Data

Supplementary Material

Supplementary material is available online at http://biostatistics.oxfordjournals.org. Conflict of Interest: None declared.

Acknowledgments

Conflict of Interest: None declared.

Funding

NSF (DMS 1007466 and DMS 1454942 to A.M.S.); NIH (R01 NS085211 and R01 MH086633 to A.M.S); NIH (R01 NS085211, R01 NS060910, R01 HL123407), NIA contracts (HHSN27121400603P and HHSN27120400775P) to C.M.C Data for these analyses were obtained from the Baltimore Longitudinal Study of Aging, performed by the National Institute on Aging.

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