We first discuss the specific results of our analysis, and then assess the fitness of the wavelet-based functional mixed model for analyzing accelerometer data.
We were not surprised by the large daylight savings time (DST) effect we observed in the data. Considering the weather in Boston, naturally children tended to be more active after April 6th than before. In 1997, the sun set in Boston at times ranging from 5:10pm to 6:15pm from February 10th through April 5th, then from 7:16pm through 8:11pm from April 6th through May 28th. This may explain the very large spike in the DST effect between 5pm and 7pm.
In our relative variability analysis, we found that after adjusting for the covariates, the day-to-day dominated the child-to-child variability, especially during the after-school hours. In other words, the difference between the average log profiles of a very active and very inactive child with the same covariate levels is small compared to the variability in activity for a given child from day-to-day, especially after school. This suggests that it is important to get many days per child in order to accurately quantify each child's typical level of activity. While it is already typical in these studies to monitor each child for 4 to 8 days, it may be a better design to obtain even more days per child, especially if this does not result in much increase in cost for the study.
While not directly modeling the bouts, the wavelet-based functional mixed model yields estimates of the probability of bouts at different times of the day for children with different levels of the covariates. It appears that this model does a reasonable job of capturing these bouts, indicating that it is at least somewhat effective in modeling some of the tail behavior in the distributions. However, there is some evidence of attenuation in our model-based estimates of these probabilities, especially during the after-school period in which there is less coordinated activity and more day-to-day variability. These may be related to our underlying assumptions of normality and equal covariances across different groups of children. It is possible in our framework to allow different covariance parameters across different groups of children, and to relax the normality assumptions using scale mixtures of normals. These adaptations would accommodate heavier tails and may do an even better job of capturing these bouts.
The wavelet-based functional mixed model is a powerful tool for analyzing accelerometer data. It allows one to consider the joint functional effects of multiple covariates, and has the ability to model correlations between profiles obtained from the same individual using random effect functions. The functions are allowed to be of arbitrary form, and the within-function covariances are allowed to be nonstationary. The fixed effect functions are adaptively regularized using the principles of wavelet shrinkage. Less adaptive methods using kernels or splines would result in more attenuation of dominant local features in the fixed effects curves, possibly resulting in reduced power for inference.
It is also possible to introduce other random effect functions to account for covariance due to other clustering factors, such as school or neighborhood. In this analysis, we chose to model the schools using fixed effects because there were so few of them, but given more schools, we could use random effects. It would be a good idea to use more schools in these studies, since the school-to-school variability appears to be large, and it would be interesting and important to study a wider range of schools with different socioeconomic makeups.
Our model is linear in that we assume the effects are linear in the covariates. It would be interesting to consider generalizing or testing this assumption. However, this framework is still very flexible since the linear coefficients are allowed to be time-varying, and their functional form over time is left arbitrary. A much less flexible model, for example, would be to allow the overall mean function to be arbitrary, but to force the fixed effects to be constant over time. It would be interesting to develop formal testing procedures for testing whether it is necessary to allow the coefficient to be time-varying for a given covariate; this is a topic for future investigation.
Also, we note that this model is very flexible in its representation of the covariances of the profiles from day-to-day, S, and the covariances of the random effect profiles from child-to-child, Q. Our assumption of independence in the wavelet space reduces the dimensionality of these covariances from T(T +1)/2 to T , yet accommodates a reasonably wide range of nonstationary covariance structures. Different variance components are estimated for each wavelet coeffcient, which allows the day-to-day and child-to-child variability to differ across both scale and location. This accommodates various types of nonstationarity, allowing the day-to-day and child-to-child variances across profiles to vary over time, and also allowing the level of smoothness in the random effect functions and residual error processes to vary over time.
The missing data methods introduced in this paper allow the wavelet-based functional mixed model to be applied to data with incomplete profiles, which are commonly encountered in accelerometer data. In our case study, these methods allowed us to increase the number of profiles in the analysis by a factor of three. The resulting gain in precision is evident if one compares the posterior bounds from the full data and complete case analyses (not shown).
The wavelet-based functional mixed model method possesses some weaknesses and limitations. It is based on linear models, and so assumes additive errors (in our case study, additive on the log scale). Other types of variability may not be adequately picked up by these models. The normality assumption is somewhat restrictive, although as previously mentioned this could be relaxed using scale mixtures. As is true with nonfunctional mixed models, some models can be unstable to fit, especially when there is near-collinearity in the design matrices, or small effective sample sizes for some of the variance components. These problems are exacerbated in the functional context, since we are effectively fitting T simultaneous scalar mixed models. Also, it would be useful to have a global functional test for more formally assessing the significance of the fixed effect functions; this is an area we will investigate in the near future. The method is computationally and memory intensive; the analyses performed for this paper took a total of about 50 computer hours. However, this is not unreasonable considering the time it takes to collect the data. The method is also highly parallelizable, so it could be done in much less time whenever parallel computing resources are available.
While computationally intensive, the method is straightforward to implement. The minimal information one must specify to use our scripts are the Y , X, and Z matrices, the wavelet basis, the number of levels of decomposition, the number of MCMC samples, and the burn-in. By default, maximum likelihood starting values, empirical Bayes regularization parameters, Fisher's information-based proposal variances for the Metropolis-Hastings, and vague priors on the variance components are automatically computed.
Although straightforward to implement, this method is considerably complex, combining wavelets with mixed models and wrapping it all up inside a Markov Chain Monte Carlo. This begs the question of what all this complexity buys, of what this method can do that simpler approaches cannot. This functional data modeling approach can be used to perform the same types of standard analyses found in the existing literature, including analyses of average daily activity levels, 30-minute averages, and probabilities of bouts for different groups of individuals, but these can be done more effciently because we can effectively incorporate incomplete profiles into the analysis. More importantly, the functional approach opens new possibilities in terms of analyses that can be done and types of information that can be extracted from these rich data, for example, allowing us to perform inference on time-varying effects. Given the posterior samples from our model fit, we can perform nearly any type of inference we desire, functional or pointwise, on fixed effect functions, random effect functions, or covariance functions. The Bayesian approach propagates the uncertainty from the different sources of variability in the model and the different parameters estimated throughout any subsequent inference.
The wavelet-based functional mixed model, supplemented with the missing data methods introduced in this paper, comprises a promising tool for extracting information from accelerometer data in activity level studies.