Ability to quantify and predict progression of a disease is fundamental for selecting an appropriate treatment. Many clinical metrics cannot be acquired frequently either because of their cost (e.g. MRI, gait analysis) or because they are inconvenient or harmful to a patient (e.g. biopsy, x-ray). In such scenarios, in order to estimate individual trajectories of disease progression, it is advantageous to leverage similarities between patients, i.e. the covariance of trajectories, and find a latent representation of progression. Most of existing methods for estimating trajectories do not account for events in-between observations, what dramatically decreases their adequacy for clinical practice. In this study, we develop a machine learning framework named Coordinatewise-Soft-Impute (CSI) for analyzing disease progression from sparse observations in the presence of confounding events. CSI is guaranteed to converge to the global minimum of the corresponding optimization problem. Experimental results also demonstrates the effectiveness of CSI using both simulated and real dataset.
In mixed-effect models, every trajectory @math is assumed to be composed of two parts: the @math for some @math that remains the same among all patients and a @math that differs for each @math . In its simplest form, we assume where @math is the @math covariance matrix, @math is the standard deviation and @math , @math are functions @math and @math evaluated at the times @math , respectively. Estimations of model parameters @math can be made via expectation maximization (EM) algorithm @cite_1 . Individual coefficients @math can be estimated using the best unbiased linear predictor (BLUP) @cite_4 .
@cite_1 @cite_4
2082246284 2013866489
SUMMARY Models for the analysis of longitudinal data must recogrlize the relationship between serial observations on the same unit. Multivariate models with general covariance structure are often difficult to apply to highly unbalanced data, whereas two-stage random-effects models can be used easily. In two-stage models, the probability distributions for the response vectors of different individuals belong to a single family, but some random-effects parameters vary across individuals, with a distribution specified at the second stage. A general family of models is discussed, which includes both growth models and repeated-measures models as special cases. A unified approach to fitting these models, based on a combination of empirical Bayes and maximum likelihood estimation of model parameters and using the EM algorithm, is discussed. Two examples are taken from a current epidemiological study of the health effects of air pollution. Many longitudinal studies are designed to investigate changes over time in a characteristic which is measured repeatedly for each study participant. In medical studies, the measurement might be blood pressure, cholesterol level, lung volume, or serum glucose. Multiple measurements are obtained from each individual, at different times and possibly under changing experimental conditions. Often, we cannot fully control the circumstances under which the measurements are taken, and there may be considerable variation among individuals in the number and timing of observations. The resulting unbalanced data sets are typically not amenable to analysis using a general multivariate model with unrestricted covariance structure. Statisticians have often analyzed data of this form using some variant of a two-stage model. In this formulation, the probability distribution for the multiple measurements has the same form for each individual, but the parameters of that distribution vary over individuals. The distribution of these parameters, or 'random effects', in the population constitutes the second stage of the model. In a study of changes in lung volume during childhood, for instance, it may be reasonable to assume that the relationship between lung volume and the cube of height is linear for each child, but with linear regression parameters that vary among children. If we assume that the usual linear regression model applies for each child, conditional on the child's individual parameters, and that the regression parameters have a bivariate normal distribution in the population, the marginal distribution of the serial measurements is multivariate normal with a special covariance structure. Mixed linear models are assumed in most animal breeding applications. Convenient methods for computing BLUE of the estimable linear functions of the fixed elements of the model and for computing best linear unbiased predictions of the random elements of the model have been available. Most data available to animal breeders, however, do not meet the usual requirements of random sampling, the problem being that the data arise either from selection experiments or from breeders' herds which are undergoing selection. Consequently, the usual methods are likely to yield biased estimates and predictions. Methods for dealing with such data are presented in this paper.