C
Filomena Pereira-Maxwell in Medical Statistics, 2018
The joint variance of two variables (or random variables). It is estimated as the average cross-product between deviations from the mean, where for each observation, the difference between the observation value for variable x and the mean of x is calculated, and likewise for variable y. The covariances between pairs of variables may be displayed in a covariance or variance-covariance matrix, a symmetrical matrix in which the elements in the main diagonal represent the variances of the variables, and the off-diagonal elements represent the covariance between pairs of variables. With standardized variables (z-scores), the covariance matrix is given by the correlation matrix. Correlation is a standardized covariance (HAMILTON, 1992). The covariance matrix often holds the necessary and sufficient building blocks for statistical analysis.
Methodological Issues in Health Economic Analysis
Demissie Alemayehu, Joseph C. Cappelleri, Birol Emir, Kelly H. Zou in Statistical Topics in Health Economics and Outcomes Research, 2017
An attractive feature of the GPQ methodology is that it can address the interval estimation of an arbitrary function of the mean vectors and covariance matrices in the model (Equation 5.4). For example, suppose that, in addition to the costs, the effectiveness measures are also log-normally distributed. That is, we have the model where is the population mean of the log-transformed effectiveness measures, j = 1, 2. Now, the ICER and INB () have the expressions where is the second diagonal element of the covariance matrix , . The parameter can also be similarly modified. It should be clear that the GPQ methodology can be easily adopted in this scenario in which the costs and effectiveness are both lognormally distributed.
Introduction to Within-Person Analysis and Model Comparisons
Lesa Hoffman in Longitudinal Analysis, 2015
Although the univariate model (or sphericity-based adjustments thereof) is perhaps the most common repeated measures ANOVA model for longitudinal data, a multivariate model could also be used instead. Although both use the same saturated model for the means (i.e., use all possible fixed effects of time to perfectly reproduce the means at each occasion), they differ in what they assume the pattern of variance and covariance over time to be after accounting for all model predictors. The multivariate model cannot be described succinctly by an equation predicting the outcome at any occasion because, unlike the univariate model, it does not include the U0i and eti terms that imply constant variance over time. Instead, the variance at each occasion and the covariances between occasions are all estimated separately. Another way to say this is that the form of the variance–covariance matrix over time is unstructured, meaning that every individual element (variance and covariance) gets to be whatever the data wants it to be. If there are n occasions per person, this will result in (n*[n + 1]) / 2 estimated parameters, or for our current 6-occasion example, 6*7 / 2 = 21 estimated parameters (6 variances and 15 covariances).
Clustering of longitudinal interval-valued data via mixture distribution under covariance separability
Published in Journal of Applied Statistics, 2020
Seongoh Park, Johan Lim, Hyejeong Choi, Minjung Kwak
Covariance separability for matrix variate data is defined as the covariance matrix of the data that is represented as the Kronecker product of two covariance matrices: the covariance matrix of the row vector and that of the column vector. In the above longitudinal interval data (1), the covariance separability implies that (i) the column vectors have a common covariance matrix 13] and Wang and West [39], the mixture model by Viroli [37] and Glanz and Carvalho [19], and the sparse Gaussian graphical model under the matrix normal assumption by Yin and Li [42]. Additionally, we find that some work in the literature has shown that the separable covariance matrix provides a good approximation compared to its nonseparable alternative [31], although this is not always true.
Robust estimation of models for longitudinal data with dropouts and outliers
Published in Journal of Applied Statistics, 2022
Yuexia Zhang, Guoyou Qin, Zhongyi Zhu, Bo Fu
In practice, the covariance matrix 25]. They suggested modeling covariance matrices using additional covariance parameters
A resample-replace lasso procedure for combining high-dimensional markers with limit of detection
Published in Journal of Applied Statistics, 2022
Jinjuan Wang, Yunpeng Zhao, Larry L. Tang, Claudius Mueller, Qizhai Li
The second challenge is encountered when the number of proteins is larger than the sample size. The traditional setting assumes that a fixed number of proteins are profiled on a large sample. To estimate the mean vectors and covariance matrices of multiple biomarkers, a maximum likelihood estimation method (MLE) can be used [9,29]. Additionally, Tomassi et al. [28] recently proposed a likelihood-based sufficient dimension reduction method for analyzing multiple correlated markers with the LOD. However, these methods cannot be applied to high-dimensional proteomic biomarkers since the sample covariance matrices are singular [5,6,15,16,33], and thus cannot be estimated by maximizing the likelihood function which involves the matrix inverse. To overcome this issue, the literature imposed additional assumptions on the covariance matrix. For example, to ensure positive definiteness, the diagonal matrix was used in the Fisher's discriminant problem [6,10,30]. However, the assumption of independence among all the protein profiles is overly simplified and does not represent the true correlation structure. An alternative way is to add penalty terms to the likelihood function, such as the lasso-type penalty term. Besides, the lasso-type penalty imposes a sparsity constraint to the covariance/precision matrix, which is a common assumption for high-dimensional data.
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