High-dimensional mixed data as a combination of both continuous and ordinal variables are widely seen in many research areas such as genomic studies and survey data analysis. Estimating the underlying correlation among mixed data is hence crucial for further inferring dependence structure. We propose a semiparametric latent Gaussian copula model for this problem. We start with estimating the association among ternary-continuous mixed data via a rank-based approach and generalize the methodology to p-level-ordinal and continuous mixed data. Concentration rate of the estimator is also provided and proved. At last, we demonstrate the performance of the proposed estimator by extensive simulations and two case studies of real data examples of algorithmic risk score evaluation and cancer patients survival data.
Purpose: In this paper we consider the estimation of intracluster correlation for ordinal data. We focus on pure-tone audiometry hearing threshold data, where thresholds are measured in 5 decibel increments. We estimate the intracluster correlation for tests from iPhone-based hearing assessment application as a measure of test/retest reliability. Methods: We present a method to estimate the intracluster correlation using mixed effects cumulative logistic and probit models, which assume the outcome data are ordinal. This contrasts with using a mixed effects linear model which assumes that the o…
Precision medicine is an emerging scientific topic for disease treatment and prevention that takes into account individual patient characteristics. It is an important direction for clinical research, and many statistical methods have been recently proposed. One of the primary goals of precision medicine is to obtain an optimal individual treatment rule (ITR), which can help make decisions on treatment selection according to each patient's specific characteristics. Recently, outcome weighted learning (OWL) has been proposed to estimate such an optimal ITR in a binary treatment setting by maximi…
In recent literature, the Gaussian Graphical model (GGM; Lauritzen, 1996),a network of partial correlation coefficients, has been used to capture potential dynamic relationships between observed variables. The GGM can be estimated using regularization in combination with model selection using the extended Bayesian Information Criterion (Foygel and Drton, 2010). I term this methodology GeLasso, and asses its performance using a plausible psychological network structure with both continuous and ordinal datasets.Simulation results indicate that GeLasso works well as an out-of-the-box method to es…
In some contexts, mixture models can fit certain variables well at the expense of others in ways beyond the analyst's control. For example, when the data include some variables with non-trivial amounts of missing values, the mixture model may fit the marginal distributions of the nearly and fully complete variables at the expense of the variables with high fractions of missing data. Motivated by this setting, we present a mixture model for mixed ordinal and nominal data that splits variables into two groups, focus variables and remainder variables. The model allows the analyst to specify a ric…
In the statistical literature, a number of methods have been proposed to ensure valid inference about marginal effects of variables on a longitudinal outcome in settings with irregular monitoring times. However, the potential biases due to covariate-driven monitoring times and confounding have rarely been considered simultaneously, and never in a setting with an ordinal outcome and a continuous exposure. In this work, we propose and demonstrate a methodology for causal inference in such a setting, relying on a proportional odds model to study the effect of the exposure on the outcome. Irregula…
Missing data imputation forms the first critical step of many data analysis pipelines. The challenge is greatest for mixed data sets, including real, Boolean, and ordinal data, where standard techniques for imputation fail basic sanity checks: for example, the imputed values may not follow the same distributions as the data. This paper proposes a new semiparametric algorithm to impute missing values, with no tuning parameters. The algorithm models mixed data as a Gaussian copula. This model can fit arbitrary marginals for continuous variables and can handle ordinal variables with many levels, …
Ordinal cumulative probability models (CPMs) -- also known as cumulative link models -- such as the proportional odds regression model are typically used for discrete ordered outcomes, but can accommodate both continuous and mixed discrete/continuous outcomes since these are also ordered. Recent papers describe ordinal CPMs in this setting using non-parametric maximum likelihood estimation. We formulate a Bayesian CPM for continuous or mixed outcome data. Bayesian CPMs inherit many of the benefits of frequentist CPMs and have advantages with regard to interpretation, flexibility, and exact inf…
A prescription is presented for a new and practical correlation coefficient, $ϕ_K$, based on several refinements to Pearson's hypothesis test of independence of two variables. The combined features of $ϕ_K$ form an advantage over existing coefficients. First, it works consistently between categorical, ordinal and interval variables. Second, it captures non-linear dependency. Third, it reverts to the Pearson correlation coefficient in case of a bi-variate normal input distribution. These are useful features when studying the correlation between variables with mixed types. Particular emphasis is…
Continuous response variables often need to be transformed to meet regression modeling assumptions; however, finding the optimal transformation is challenging and results may vary with the choice of transformation. When a continuous response variable is measured repeatedly for a subject or the continuous responses arise from clusters, it is more challenging to model the continuous response data due to correlation within clusters. We extend a widely used ordinal regression model, the cumulative probability model (CPM), to fit clustered continuous response variables based on generalized estimati…
This paper introduces a general framework for estimating variance components in the linear mixed models via general unbiased estimating equations, which include some well-used estimators such as the restricted maximum likelihood estimator. We derive the asymptotic covariance matrices and second-order biases under general estimating equations without assuming the normality of the underlying distributions and identify a class of second-order unbiased estimators of variance components. It is also shown that the asymptotic covariance matrices and second-order biases do not depend on whether the re…
Penalized EM algorithm and copula skeptic graphical models for inferring networks for mixed variables
本文研究了高斯copula图模型中稀疏精度矩阵的估计方法，对连续变量、二元变量、计数变量和离散序变量的网络进行重构。我们提出两种方法：$\ell_1$用Monte Carlo期望最大化算法（copula EM glasso）惩罚扩展秩似然，用copula Gaussian图模型的成对copula估计惩罚copula怀疑。所提出的方法有助于推断网络所产生的非正态和混合变量。我们通过对乳腺癌基因组和临床数据以及玉米遗传学数据的模拟研究和分析来证明我们的方法的性能。
The functional linear model is a popular tool to investigate the relationship between a scalar/functional response variable and a scalar/functional covariate. We generalize this model to a functional linear mixed-effects model when repeated measurements are available on multiple subjects. Each subject has an individual intercept and slope function, while shares common population intercept and slope function. This model is flexible in the sense of allowing the slope random effects to change with the time. We propose a penalized spline smoothing method to estimate the population and random slope…
Many real-world datasets contain missing entries and mixed data types including categorical and ordered (e.g. continuous and ordinal) variables. Imputing the missing entries is necessary, since many data analysis pipelines require complete data, but this is challenging especially for mixed data. This paper proposes a probabilistic imputation method using an extended Gaussian copula model that supports both single and multiple imputation. The method models mixed categorical and ordered data using a latent Gaussian distribution. The unordered characteristics of categorical variables is explicitl…
Covariate-adjusted win statistics in randomized clinical trials with ordinal outcomes
Shenzhen Technology University（深圳理工大学）；Feinberg School of Medicine, Northwestern University（西北大学费恩伯格医学院）；University of Wisconsin - Madison（威斯康星大学麦迪逊分校）；Yale University School of Public Health（耶鲁大学公共卫生学院）；Yale Center for Analytical Sciences, Yale University School of Public Health（耶鲁大学公共卫生学院分析科学中心）；University of Washington（华盛顿大学）；Clinical Learning, Evidence, and Research (CLEAR) Center, University of Washington（华盛顿大学临床学习、证据与研究（CLEAR）中心）；Section of Cardiovascular Medicine, Department of Internal Medicine, Yale School of Medicine（耶鲁医学院内科学系心血管医学部）；Cardiovascular Medicine Analytics Center, Yale School of Medicine（耶鲁医学院心血管医学分析中心）；Center for Methods in Implementation and Prevention Science, Yale School of Public Health（耶鲁大学公共卫生学院实施与预防科学方法中心）
An iteratively reweighted least squares (IRLS) method is proposed for estimating polyserial and polychoric correlation coefficients in this paper. It iteratively calculates the slopes in a series of weighted linear regression models fitting on conditional expected values. For polyserial correlation coefficient, conditional expectations of the latent predictor is derived from the observed ordinal categorical variable, and the regression coefficient is obtained using weighted least squares method. In estimating polychoric correlation coefficient, conditional expectations of the response variable…
Extending generalized estimating equations (GEE) to ordinal response data requires a conversion of the ordinal response to a vector of binary category indicators. That leads to a rather complicated association structure, and the introduction of large matrices when the number of categories and dimension of the cluster are large. To allow a richer specification of working correlation assumptions, we adopt the weighted scores method which is essentially an extension of the GEE approach, since it can also be applied to families that are not in the GLM class. The weighted scores method stems from t…
Clustering of mixed-type datasets can be a particularly challenging task as it requires taking into account the associations between variables with different level of measurement, i.e., nominal, ordinal and/or interval. In some cases, hierarchical clustering is considered a suitable approach, as it makes few assumptions about the data and its solution can be easily visualized. Since most hierarchical clustering approaches assume variables are measured on the same scale, a simple strategy for clustering mixed-type data is to homogenize the variables before clustering. This would mean either rec…
Personalized medicine has gained much popularity recently as a way of providing better healthcare by tailoring treatments to suit individuals. Our research, motivated by the UK INTERVAL blood donation trial, focuses on estimating the optimal individualized treatment rule (ITR) in the ordinal treatment-arms setting. Restrictions on minimum lengths between whole blood donations exist to safeguard donor health and quality of blood received. However, the evidence-base for these limits is lacking. Moreover, in England, the blood service is interested in making blood donation both safe and sustainab…
Poverty is a multidimensional concept often comprising a monetary outcome and other welfare dimensions such as education, subjective well-being or health, that are measured on an ordinal scale. In applied research, multidimensional poverty is ubiquitously assessed by studying each poverty dimension independently in univariate regression models or by combining several poverty dimensions into a scalar index. This inhibits a thorough analysis of the potentially varying interdependence between the poverty dimensions. We propose a multivariate copula generalized additive model for location, scale a…
We present `latentcor`, an R package for correlation estimation from data with mixed variable types. Mixed variables types, including continuous, binary, ordinal, zero-inflated, or truncated data are routinely collected in many areas of science. Accurate estimation of correlations among such variables is often the first critical step in statistical analysis workflows. Pearson correlation as the default choice is not well suited for mixed data types as the underlying normality assumption is violated. The concept of semi-parametric latent Gaussian copula models, on the other hand, provides a uni…
We consider three problems in high-dimensional Gaussian linear mixed models. Without any assumptions on the design for the fixed effects, we construct an asymptotic $F$-statistic for testing whether a collection of random effects is zero, derive an asymptotic confidence interval for a single random effect at the parametric rate $\sqrt{n}$, and propose an empirical Bayes estimator for a part of the mean vector in ANOVA type models that performs asymptotically as well as the oracle Bayes estimator. We support our results with numerical simulations and provide comparisons with oracle estimators. …
For estimating area-specific parameters (quantities) in a finite population, a mixed model prediction approach is attractive. However, this approach strongly depends on the normality assumption of the response values although we often encounter a non-normal case in practice. In such a case, transforming observations to make them suitable for normality assumption is a useful tool, but the problem of selecting suitable transformation still remains open. To overcome the difficulty, we here propose a new empirical best predicting method by using a parametric family of transformations to estimate a…
Multivariate Functional Principal Component Analysis for Mixed-Type mHealth Data: An Application to Mood Disorders
Texas A&M University（德克萨斯农工大学）；University of South Carolina（南卡罗来纳大学）；National Institute of Mental Health（美国国家精神卫生研究所）；Johns Hopkins Bloomberg School of Public Health（约翰霍普金斯大学布隆伯格公共卫生学院）
Linear Mixed Effects (LME) models have been widely applied in clustered data analysis in many areas including marketing research, clinical trials, and biomedical studies. Inference can be conducted using maximum likelihood approach if assuming Normal distributions on the random effects. However, in many applications of economy, business and medicine, it is often essential to impose constraints on the regression parameters after taking their real-world interpretations into account. Therefore, in this paper we extend the classical (unconstrained) LME models to allow for sign constraints on its o…
Inferring dependence structure through undirected graphs is crucial for uncovering the major modes of multivariate interaction among high-dimensional genomic markers that are potentially associated with cancer. Traditionally, conditional independence has been studied using sparse Gaussian graphical models for continuous data and sparse Ising models for discrete data. However, there are two clear situations when these approaches are inadequate. The first occurs when the data are continuous but display non-normal marginal behavior such as heavy tails or skewness, rendering an assumption of norma…
London School of Economics and Political Science（伦敦政治经济学院）；University of Minnesota（明尼苏达大学）
Matrix completion is a class of machine learning methods that concerns the prediction of missing entries in a partially observed matrix. This paper studies matrix completion for mixed data, i.e., data involving mixed types of variables (e.g., continuous, binary, ordinal). We formulate it as a low-rank matrix estimation problem under a general family of non-linear factor models and then propose entrywise consistent estimators for estimating the low-rank matrix. Tight probabilistic error bounds are derived for the proposed estimators. The proposed methods are evaluated by simulation studies and …
