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Testing Statistical Assumptions
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Measuring entrepreneural activity in low-teledensity countries, Int. Jackson, N. Lewis, S. Adams, M. We present LonGP, an additive Gaussian process regression model that is specifically designed for statistical analysis of longitudinal data, which solves these commonly faced challenges. LonGP can model time-varying random effects and non-stationary signals, incorporate multiple kernel learning, and provide interpretable results for the effects of individual covariates and their interactions.
Biomedical research often involves longitudinal studies where individuals are followed over a period of time and measurements are repeatedly collected from the subjects of the study. Longitudinal studies are effective in identifying various risk factors that are associated with an outcome, such as disease initiation, disease onset or any disease-associated molecular biomarker.
There are several classes of longitudinal study designs, including prospective vs.
Also, as the risk factors or covariates can be either static or time-varying, statistical analysis tools need to be versatile enough so that they can be appropriately tailored to every application. Traditionally, analysis of variance ANOVA , general linear mixed effect models LME , and generalised estimating equations are widely used in analysing longitudinal data due to their simplicity and interpretability 1. Although numerous advanced extensions of these statistical techniques have been proposed, longitudinal data analysis is still complicated for several reasons, such as difficulties in choosing covariance structures to model correlated outcomes, handling irregular sampling times and missing values, accounting for time-varying covariates, choosing appropriate nonlinear effects, modelling non-stationary ns signals, and accurate model inference.
Modern statistical methods for timeseries and longitudinal data analysis make less assumptions about the underlying data generating mechanisms. These methods use predominantly non-parametric models, such as splines 2 , and more recently latent stochastic processes, such as Gaussian processes GP 3 , 4. While spline models can implement complex nonlinear functions, they are less efficient in modelling effects of covariate interactions.
GP is a principled, probabilistic approach to learn non-parametric models, where nonlinearity is implemented through kernels 5. A GP modelling framework is adopted in this work due to its flexibility and probabilistic formulation. GPs have become a popular method for non-parametric modelling, especially for time-series data, and a wide variety of kernel functions have been proposed for different modelling tasks. A GP model can be made additive by defining the kernel function to be a sum of kernels. Similarly, a product of two or more kernels is also a valid kernel 5.
Thus, GPs can be made more interpretable and flexible by decomposing the kernel into a sum of individual and product interaction kernels much in the same way, conceptually, as with standard linear models. Here we can view the individual kernels as flexible nonlinear functions, which corresponds to the linear terms in linear regression.
Plate 6 was among the first to formulate additive GPs by proposing a sum of univariate and multivariate kernels in an attempt to balance between model complexity and interpretability. Duvenaud et al. More complex kernel functions and structures were considered later 8.
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Gilboa et al. Bayesian semi-parametric models 4 and additive GP regression together with Bayesian inference methods 11 were proposed in the context of longitudinal study designs. Schulam et al. Computationally efficient model inference for additive GP models AGPM using sparse approximations and variational inference was recently proposed We present LonGP, a flexible and interpretable non-parametric modelling framework together with a versatile software implementation that solves commonly faced challenges in longitudinal data analysis.
LonGP implements an additive GP regression model, with appropriate product kernels, that is specifically designed for longitudinal biomedical data with complex experimental designs.
LonGP inherits the favourable features of GPs and multiple kernel learning. Our method extends previous GP as well as linear mixed effect models in several ways.
Contrary to previous GP methods, LonGP implements a multi-level model that is conceptually similar to the commonly used linear models, and thus enables modelling individual-specific time-varying random effects, for example. LonGP also models ns signals using ns kernel functions and provides interpretable results for the effects of individual covariates and their interactions.
We also develop a fully-Bayesian, predictive inference for LonGP and use that to carry out model selection, i. LonGP with its full functionality is developed as an open-source software tool, which provides great convenience and flexibility of non-parametric longitudinal data analysis for applied research. Linear models and their mixed effect variants have become a standard tool for longitudinal data analysis. However, a number of challenges still persist in longitudinal analysis, e.
GP are a flexible class of models that have become popular in machine learning and statistics.
Regression Analysis: A Constructive Critique
Realizations from a GP correspond to random functions and, consequently, GPs naturally provide a prior for an unknown regression function that is to be estimated from data. Thus, GPs differ from standard regression models in that they define priors for entire nonlinear functions, instead of their parameters. While nonlinear effects can be incorporated into standard linear models by extending the basis functions e. Mean in Eq. Covariance also called the kernel function of the normal distribution defines the smoothness of the function f , i.