DOCSLIB.ORG

Linear Mixed Model Selection by Partial Correlation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Mixed Model Selection by Partial Correlation LINEAR MIXED MODEL SELECTION BY PARTIAL CORRELATION Audry Alabiso A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2020 Committee: Junfeng Shang, Advisor Andy Garcia, Graduate Faculty Representative Hanfeng Chen Craig Zirbel ii ABSTRACT Junfeng Shang, Advisor Linear mixed models (LMM) are commonly used when observations are no longer independent of each other, and instead, clustered into two or more groups. In the LMM, the mean response for each subject is modeled by a combination of fixed effects and random effects. The fixed effects are characteristics shared by all individuals in the study; they are analogous to the coefficients of the linear model. The random effects are specific to each group or cluster and help describe the correlation structure of the observations. Because of this, linear mixed models are popular when multiple measurements are made on the same subject or when there is a natural clustering or grouping of observations. Our goal in this dissertation is to perform fixed effect selection in the high-dimensional linear mixed model. We generally define high-dimensional data to be when the number of potential predictors is large relative to the sample size. High-dimensional data is common in genomic and other biological datasets. In the high-dimensional setting, selecting the fixed effect coefficients can be difficult due to the number of potential models to choose from. However, it is important to be able to do so in order to build models that are easy to interpret. Many current techniques for fixed effect selection in the high-dimensional LMM are based on the penalized log likelihood. However, adding a penalized term to the log likelihood results in a non-convex optimization problem which requires numerical methods to solve and a data dependent tuning parameter to select the amount of regularization. In contrast to the penalized likelihood, the partial correlation is based on the marginal mea- sures of association between each predictor and the conditioned response. Techniques based on the partial correlation have two main advantages to those based on penalized likelihoods: no data dependent tuning parameter is required to select the fixed effects and the partial correlation is not influenced by strong correlation between covariates. In this dissertation we propose using the par- tial correlation between the response variable conditioned on the random effects to select fixed effects in the LMM. This is an extension of variable selection using partial correlation developed iii by Buhlmann,¨ Kalisch, and Maathuis (2010) to the linear mixed model by conditioning the re- sponse variable on the random effects. At the time of this writing, selection methods using partial correlation have not been attempted to select fixed effects in the linear mixed model. This dissertation proposes a two stage procedure for selecting the fixed effects in the high- dimensional linear mixed model. In the first stage, we use the partial correlation to perform an initial fixed effect variable screening procedure in order to estimate an initial linear mixed model. In the second stage, we use the initial linear mixed model to predict the values of the random effects using the Best Linear Unbiased Predictor (BLUP). These predicted values are used to condition the response variable by subtracting the group-specific random effects from the response. After conditioning on the random effects, the observations are effectively independent, and we select variables using the partial correlation between the covariates and the conditioned response. In this dissertation, we show that this procedure consistently selects fixed effects in the linear mixed model. To use the partial correlation to select variables in the LMM, we require the assumption of par- tial faithfulness on the design matrix X. The partial faithfulness assumption in the LMM describes the relationship between the response conditioned on the random effects and the coefficients of the fixed effects of the LMM. Partial faithfulness in the LMM says that the fixed effect coefficient is equal to zero if and only if the partial correlation between the conditioned response and a predictor under consideration is equal to zero for some set of controlling variables. We present theoretical results that demonstrate that when partial faithfulness holds for the LMM, the relationship between the partial correlation and the coefficients of the fixed effects holds. We investigate the performance of this method in a variety of simulated high-dimensional sce- narios, including non-normal distributions of the random effects. We find that the method is effec- tive at selecting the active set of variables even in the presence of many covariates. Through these simulations, we observe that the proposed technique selects variables quickly and with few false positives, especially in the case where the covariates are highly correlated with each other. We also apply the method to a real high-dimensional dataset regarding the production of riboflavin. iv ACKNOWLEDGMENTS I would like to gratefully acknowledge my advisor, Dr. Junfeng Shang for all the help she has given me throughout this process. I cannot begin to express my appreciation to her for encouraging my independence as a researcher, for helping me stay focused on the goal and for the countless amount of time and feedback she has given me at all stages of this dissertation. This manuscript is immeasurably better because of her invaluable advice. Beyond this dissertation, I am also thankful for her help with my research statement for my job applications and for working with me remotely over the last few semesters. A special thanks to my committee, Dr. Hanfeng Chen, Dr. Craig Zirbel, and Dr. Andy Garcia. Your feedback and insight has greatly helped me improve my dissertation. I would like to give additional thanks to Dr. Zirbel for his help and feedback on my job applications. To Sima Sharghi, Kelsey Meyer, Kim Brooks, Yi-Ching Lee, who have made my time in Bowling Green so much better. Thanks for coming into my life and enriching it so much. Extra thanks to Sherri Stefanovic, whose constant encouragement and optimism kept me going. To my dad, my siblings, Michele and Aaron, and my in-laws, Be and Cam, I hope I made you all proud. Thank you for all the support and encouragement, I don’t have the words to express my appreciation for everything you all have done for me. Finally, I would like to give my eternal gratitude to my husband, La. I could not have done any of this without you. Whenever I doubted, you always had faith. I could write an entire dissertation on how much you mean to me. v TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION . 1 1.1 Motivation . 1 1.2 Objectives . 3 1.3 Outline . 6 CHAPTER 2 LINEAR MIXED MODELS (LMMs) . 7 2.1 Introduction . 7 2.2 Linear Model . 7 2.3 Linear Mixed Model . 9 2.3.1 Notation and Estimation . 9 2.3.2 Prediction of Random Effects . 12 2.3.3 Random Intercepts and Random Slopes . 13 2.3.4 Covariance Structures for Longitudinal Data . 14 2.3.5 Random Effects with Skew t Distributions . 16 CHAPTER 3 MODEL SELECTION IN THE LM AND LMM . 19 3.1 Introduction . 19 3.1.1 High-Dimensional and Ultrahigh-Dimensional Data . 19 3.2 Penalized Likelihood Methods . 22 3.2.1 LASSO . 22 3.2.2 ALASSO . 23 3.3 Marginal Methods . 24 3.3.1 Sure Independence Screening . 25 3.3.2 Multiple Hypothesis Testing . 26 3.3.3 Variable Selection via Partial Correlation . 26 vi CHAPTER 4 VARIABLE SELECTION USING THE THRESHOLDED PARTIAL COR- RELATION CONDITIONAL ALGORITHM . 28 4.1 Partial Faithfulness in the Linear Model . 29 4.1.1 Definitions . 29 4.2 Partial Faithfulness in the Linear Mixed Model . 30 4.2.1 Proof of Theorem 4.2.4 . 32 4.3 The TPCc Algorithm for the LMM . 35 4.4 Asymptotic Theory of the TPCc . 41 4.4.1 Lemma 1 . 43 4.4.2 Lemma 2 . 43 4.4.3 Lemma 3 . 43 CHAPTER 5 SIMULATION STUDY . 58 5.1 Setup . 58 5.2 Normally Distributed Random Effects . 60 5.2.1 Compound Symmetric . 60 5.2.2 AR1 ...................................... 73 5.2.3 Random Slope . 83 5.3 Random Effects From a skew t Distribution . 88 CHAPTER 6 APPLICATION . 92 6.1 Description of Data . 92 6.2 Analysis . 92 6.2.1 A Permutation Test for Variable Selection . 96 CHAPTER 7 CONCLUDING REMARKS AND FUTURE WORKS . 99 7.1 Concluding Remarks . 99 7.2 Future Works . 100 vii BIBLIOGRAPHY . 102 CHAPTER A APPENDIX SELECTED R PROGRAMS . 105 A.1 Selected Code for Chapter 5: Simulation Study . 105 A.1.1 Code to Create Random Intercept or Random Slope Data . 105 A.1.2 Code to Analyze Random Intercept Data . 107 A.1.3 Code to Create AR1 Data . 108 A.1.4 Code to Analyze AR1 Data . 109 A.1.5 Code to Create skew t Data . 110 A.1.6 Code to Analyze skew t Data . 111 A.2 Selected Code for Chapter 6: Application . 112 A.2.1 Code for Riboflavin Analysis . 112 A.2.2 Code for Permutation Test . 114 viii LIST OF FIGURES Figure Page 2.1 Representation of Random Intercepts and Slopes. 14 5.1 CS False Positive Number by Dimension, ρx = 0:0. 69 5.2 CS Overfit by Dimension, ρx = 0:0. ......................... 69 5.3 CS Median Model Error by Dimension, ρx = 0:0................... 70 5.4 CS Correct Fit by Dimension, ρx = 0:0. ....................... 70 5.5 CS False Positive Number by Dimension, ρx = 0:3. 70 5.6 CS Overfit by Dimension, ρx = 0:3.
Load more

Recommended publications
  • The Locally Gaussian Partial Correlation Håkon Otneim∗ Dag Tjøstheim†
    The Locally Gaussian Partial Correlation Håkon Otneim∗ Dag Tjøstheim† Abstract It is well known that the dependence structure for jointly Gaussian variables can be fully captured using correlations, and that the conditional dependence structure in the same way can be described using partial correlations. The partial correlation does not, however, characterize conditional dependence in many non-Gaussian populations. This paper introduces the local Gaussian partial correlation (LGPC), a new measure of conditional dependence. It is a local version of the partial correlation coefficient that characterizes conditional dependence in a large class of populations. It has some useful and novel properties besides: The LGPC reduces to the ordinary partial correlation for jointly normal variables, and it distinguishes between positive and negative conditional dependence. Furthermore, the LGPC can be used to study departures from conditional independence in specific parts of the distribution. We provide several examples of this, both simulated and real, and derive estimation theory under a local likelihood framework. Finally, we indicate how the LGPC can be used to construct a powerful test for conditional independence, which, again, can be used to detect Granger causality in time series. 1 Introduction Estimation of conditional dependence and testing for conditional independence are extremely important topics in classical as well as modern statistics. In the last two decades, for instance, there has been a very intense development using conditional dependence in probabilistic network theory. This comes in addition to conditional multivariate time series analysis and copula analysis. For jointly Gaussian variables, conditional dependence is measured by the partial correlation coefficient. Given three jointly Gaussian stochastic variables X1, X2 and X3, X1 and X2 are conditionally independent given X3 if and only if the partial correlation between X1 and X2 given X3 is equal to zero.
    [Show full text]
  • Estimation of Correlation Coefficient in Data with Repeated Measures
    Paper 2424-2018 Estimation of correlation coefficient in data with repeated measures Katherine Irimata, Arizona State University; Paul Wakim, National Institutes of Health; Xiaobai Li, National Institutes of Health ABSTRACT Repeated measurements are commonly collected in research settings. While the correlation coefficient is often used to characterize the relationship between two continuous variables, it can produce unreliable estimates in the repeated measure setting. Alternative correlation measures have been proposed, but a comprehensive evaluation of the estimators and confidence intervals is not available. We provide a comparison of correlation estimators for two continuous variables in repeated measures data. We consider five methods using SAS/STAT® software procedures, including a naïve Pearson correlation coefficient (PROC CORR), correlation of subject means (PROC CORR), partial correlation adjusting for patient ID (PROC GLM), partial correlation coefficient (PROC MIXED), and a mixed model (PROC MIXED) approach. Confidence intervals were calculated using the normal approximation, cluster bootstrap, and multistage bootstrap. The performance of the five correlation methods and confidence intervals were compared through the analysis of pharmacokinetics data collected on 18 subjects, measured over a total of 76 visits. Although the naïve estimate does not account for subject-level variability, the method produced a point estimate similar to the mixed model approach under the conditions of this example (complete data). The mixed model approach and corresponding confidence interval was the most appropriate measure of correlation as the method fully specifies the correlation structure. INTRODUCTION The correlation coefficient ρ is often used to characterize the linear relationship between two continuous variables. Estimates of the correlation (r) that are close to 0 indicate little to no association between the two variables, whereas values close to 1 or -1 indicate a strong association.
    [Show full text]
  • Partial Correlation Lecture
    Analytical Paleobiology Workshop 2018 Partial Correlation Instructor Michał Kowalewski Florida Museum of Natural History University of Florida [email protected] Partial Correlation Partial correlation attempts to estimate relationship between two variables after accounting for other variables. Synthetic Example Z <- rnorm(100, 20, 3) # let Z (‘human impact’) be independent X <- rnorm(100, 20, 1) + Z # ley X (‘vegetation density’) be strongly dependent on Z Y <- rnorm(100, 20, 1) +Z # let Y (‘animal diversity’) be also strongly dependent on Z Note that in this example X (vegetation density) and Y (animal diversity) are not interrelated directly However, they appear strongly correlated because they are both strongly influenced by Z One way to re-evaluate X and Y while minimizing effects of Z is partial correlation Computing Partial Correlation For a simplest case of three linearly related variables (x, y, z) partial correlation of x and y given z can be computed from residuals of OLS linear regression models z ~ x and z ~ y 1. Compute residuals for x (dependent) ~ z (independent) model 2. Compute residuals for y (dependent) ~ z (independent) model 3. Compute correlation between the residuals rxy_z = cor(lm(X ~ Z)$residuals), lm(Y ~ Z)$residuals)) What are residuals? OLS residuals for y = vegetation density Partial correlation XY_Z is -0.081 (recall that r for XY was 0.901) partial correlations human.impact versus vegetation.density 0.72651264 human.impact versus animal.diversity 0.71126584 vegetation.density versus animal.diversity -0.08147584 Partial and Semi-Partial Correlation “The partial correlation can be explained as the association between two random variables after eliminating the effect of all other random variables, while the semi-partial correlation eliminates the effect of a fraction of other random variables, for instance, removing the effect of all other random variables from just one of two interesting random variables.
    [Show full text]
  • Regression Analysis and Statistical Control
    CHAPTER 4 REGRESSION ANALYSIS AND STATISTICAL CONTROL 4.1 INTRODUCTION Bivariate regression involves one predictor and one quantitative outcome variable. Adding a second predictor shows how statistical control works in regression analysis. The previous chap- ter described two ways to understand statistical control. In the previous chapter, the outcome variable was denoted , the predictor of interest was denoted , and the control variable was Y X1 distribute called X2. 1. We can control for an X variable by dividing data into groups on the basis of X 2 or 2 scores and then analyzing the X1, Y relationship separately within these groups. Results are rarely reported this way in journal articles; however, examining data this way makes it clear that the nature of an X1, Y relationship can change in many ways when you control for an X2 variable. 2. Another way to control for an X2 variable is obtaining a partial correlation between X1 and Y, controlling for X2. This partial correlation is denoted r1Y.2. Partial correlations are not often reported in journalpost, articles either. However, thinking about them as correlations between residuals helps you understand the mechanics of statistical control. A partial correlation between X1 and Y, controlling for X2, can be understood as a correlation between the parts of the X1 scores that are not related to X2, and the parts of the Y scores that are not related to X2. This chapter introduces the method of statistical control that is most widely used and reported. This method involves using both X1 and X2 as predictors of Y in a multiple linear regression.
    [Show full text]
  • Financial Development and Economic Growth: a Meta-Analysis
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Valickova, Petra; Havranek, Tomas; Horváth, Roman Working Paper Financial development and economic growth: A meta-analysis IOS Working Papers, No. 331 Provided in Cooperation with: Leibniz Institute for East and Southeast European Studies (IOS), Regensburg Suggested Citation: Valickova, Petra; Havranek, Tomas; Horváth, Roman (2013) : Financial development and economic growth: A meta-analysis, IOS Working Papers, No. 331, Institut für Ost- und Südosteuropaforschung (IOS), Regensburg, http://nbn-resolving.de/urn:nbn:de:101:1-201307223036 This Version is available at: http://hdl.handle.net/10419/79241 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under
    [Show full text]
  • A Tutorial on Regularized Partial Correlation Networks
    Psychological Methods © 2018 American Psychological Association 2018, Vol. 1, No. 2, 000 1082-989X/18/$12.00 http://dx.doi.org/10.1037/met0000167 A Tutorial on Regularized Partial Correlation Networks Sacha Epskamp and Eiko I. Fried University of Amsterdam Abstract Recent years have seen an emergence of network modeling applied to moods, attitudes, and problems in the realm of psychology. In this framework, psychological variables are understood to directly affect each other rather than being caused by an unobserved latent entity. In this tutorial, we introduce the reader to estimating the most popular network model for psychological data: the partial correlation network. We describe how regularization techniques can be used to efficiently estimate a parsimonious and interpretable network structure in psychological data. We show how to perform these analyses in R and demonstrate the method in an empirical example on posttraumatic stress disorder data. In addition, we discuss the effect of the hyperparameter that needs to be manually set by the researcher, how to handle non-normal data, how to determine the required sample size for a network analysis, and provide a checklist with potential solutions for problems that can arise when estimating regularized partial correlation networks. Translational Abstract Recent years have seen an emergence in the use of networks models in psychological research to explore relationships of variables such as emotions, symptoms, or personality items. Networks have become partic- ularly popular in analyzing mental illnesses, as they facilitate the investigation of how individual symptoms affect one-another. This article introduces a particular type of network model: the partial correlation network, and describes how this model can be estimated using regularization techniques from statistical learning.
    [Show full text]
  • Correlation: Measure of Relationship
    Correlation: Measure of Relationship • Bivariate Correlations are correlations between two variables. Some bivariate correlations are nondirectional and these are called symmetric correlations. Other bivariate correlations are directional and are called asymmetric correlations. • Bivariate correlations control for neither antecedent variables (previous) nor intervening (mediating) variables. Example 1: An antecedent variable may cause both of the other variables to change. Without the antecedent variable being operational, the two observed variables, which appear to correlate, may not do so at all. Therefore, it is important to control for the effects of antecedent variables before inferring causation. Example 2: An intervening variable can also produce an apparent relationship between two observed variables, such that if the intervening variable were absent, the observed relationship would not be apparent. • The linear model assumes that the relations between two variables can be summarized by a straight line. • Correlation means the co-relation, or the degree to which two variables go together, or technically, how those two variables covary. • Measure of the strength of an association between 2 scores. • A correlation can tell us the direction and strength of a relationship between 2 scores. • The range of a correlation is from –1 to +1. • -1 = an exact negative relationship between score A and score B (high scores on one measure and low scores on another measure). • +1 = an exact positive relationship between score A and score B (high scores on one measure and high scores on another measure). • 0 = no linear association between score A and score B. • When the correlation is positive, the variables tend to go together in the same manner.
    [Show full text]
  • Kernel Partial Correlation Coefficient
    Kernel Partial Correlation Coefficient | a Measure of Conditional Dependence Zhen Huang, Nabarun Deb, and Bodhisattva Sen∗ 1255 Amsterdam Avenue New York, NY 10027 e-mail: [email protected] 1255 Amsterdam Avenue New York, NY 10027 e-mail: [email protected] 1255 Amsterdam Avenue New York, NY 10027 e-mail: [email protected] Abstract: In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of conditional dependence between two random variables Y and Z given a third variable X, all taking values in general topological spaces. The popu- lation version of any of these nonparametric measures | defined using the theory of reproducing kernel Hilbert spaces (RKHSs) | captures the strength of conditional de- pendence and it is 0 if and only if Y and Z are conditionally independent given X, and 1 if and only if Y is a measurable function of Z and X. Thus, our measure | which we call kernel partial correlation (KPC) coefficient | can be thought of as a nonparamet- ric generalization of the classical partial correlation coefficient that possesses the above properties when (X; Y; Z) is jointly normal. We describe two consistent methods of es- timating KPC. Our first method of estimation is graph-based and utilizes the general framework of geometric graphs, including K-nearest neighbor graphs and minimum spanning trees. A sub-class of these estimators can be computed in near linear time and converges at a rate that automatically adapts to the intrinsic dimensionality of the underlying distribution(s). Our second strategy involves direct estimation of con- ditional mean embeddings using cross-covariance operators in the RKHS framework.
    [Show full text]
  • And Partial Regression Coefficients
    Part (Semi Partial) and Partial Regression Coefficients Hervé Abdi1 1 overview The semi-partial regression coefficient—also called part correla- tion—is used to express the specific portion of variance explained by a given independent variable in a multiple linear regression an- alysis (MLR). It can be obtained as the correlation between the de- pendent variable and the residual of the prediction of one inde- pendent variable by the other ones. The semi partial coefficient of correlation is used mainly in non-orthogonal multiple linear re- gression to assess the specific effect of each independent variable on the dependent variable. The partial coefficient of correlation is designed to eliminate the effect of one variable on two other variables when assessing the correlation between these two variables. It can be computed as the correlation between the residuals of the prediction of these two variables by the first variable. 1In: Neil Salkind (Ed.) (2007). Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. Address correspondence to: Hervé Abdi Program in Cognition and Neurosciences, MS: Gr.4.1, The University of Texas at Dallas, Richardson, TX 75083–0688, USA E-mail: [email protected] http://www.utd.edu/∼herve 1 Hervé Abdi: Partial and Semi-Partial Coefficients 2 Multiple Regression framework In MLR, the goal is to predict, knowing the measurements col- lected on N subjects, a dependent variable Y from a set of K in- dependent variables denoted {X1,..., Xk ,..., XK } . (1) We denote by X the N × (K + 1) augmented matrix collecting the data for the independent variables (this matrix is called augmented because the first column is composed only of ones), and by y the N × 1 vector of observations for the dependent variable.
    [Show full text]
  • Second-Order Accurate Inference on Simple, Partial, and Multiple Correlations Robert J
    Journal of Modern Applied Statistical Methods Volume 5 | Issue 2 Article 2 11-1-2005 Second-Order Accurate Inference on Simple, Partial, and Multiple Correlations Robert J. Boik Montana State University{Bozeman, [email protected] Ben Haaland University of Wisconsin{Madison, [email protected] Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Boik, Robert J. and Haaland, Ben (2005) "Second-Order Accurate Inference on Simple, Partial, and Multiple Correlations," Journal of Modern Applied Statistical Methods: Vol. 5 : Iss. 2 , Article 2. DOI: 10.22237/jmasm/1162353660 Available at: http://digitalcommons.wayne.edu/jmasm/vol5/iss2/2 This Invited Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright c 2006 JMASM, Inc. November 2006, Vol. 5, No. 2, 283{308 1538{9472/06/$9 5.00 Invited Articles Second-Order Accurate Inference on Simple, Partial, and Multiple Correlations Robert J. Boik Ben Haaland Mathematical Sciences Statistics Montana State University{Bozeman University of Wisconsin{Madison This article develops confidence interval procedures for functions of simple, partial, and squared multiple correlation coefficients. It is assumed that the observed multivariate data represent a random sample from a distribution that possesses finite moments, but there is no requirement that the distribution be normal. The coverage error of conventional one-sided large sample intervals decreases at rate 1=pn as n increases, where n is an index of sample size.
    [Show full text]
  • Semipartial (Part) and Partial Correlation
    Semipartial (Part) and Partial Correlation This discussion borrows heavily from Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, by Jacob and Patricia Cohen (1975 edition; there is also an updated 2003 edition now). Overview. Partial and semipartial correlations provide another means of assessing the relative “importance” of independent variables in determining Y. Basically, they show how much each variable uniquely contributes to R2 over and above that which can be accounted for by the other IVs. We will use two approaches for explaining partial and semipartial correlations. The first relies primarily on formulas, while the second uses diagrams and graphics. To save paper shuffling, we will repeat the SPSS printout for our income example: Regression Descriptive Statistics Mean Std. Deviation N INCOME 24.4150 9.78835 20 EDUC 12.0500 4.47772 20 JOBEXP 12.6500 5.46062 20 Correlations INCOME EDUC JOBEXP Pearson Correlation INCOME 1.000 .846 .268 EDUC .846 1.000 -.107 JOBEXP .268 -.107 1.000 Model Summary Adjusted Std. Error of Model R R Square R Square the Estimate 1 .919a .845 .827 4.07431 a. Predictors: (Constant), JOBEXP, EDUC ANOVAb Sum of Model Squares df Mean Square F Sig. 1 Regression 1538.225 2 769.113 46.332 .000a Residual 282.200 17 16.600 Total 1820.425 19 a. Predictors: (Constant), JOBEXP, EDUC b. Dependent Variable: INCOME Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts 95% Confidence Interval for B Correlations Collinearity Statistics Model B Std. Error Beta t Sig. Lower Bound Upper Bound Zero-order Partial Part Tolerance VIF 1 (Constant) -7.097 3.626 -1.957 .067 -14.748 .554 EDUC 1.933 .210 .884 9.209 .000 1.490 2.376 .846 .913 .879 .989 1.012 JOBEXP .649 .172 .362 3.772 .002 .286 1.013 .268 .675 .360 .989 1.012 a.
    [Show full text]
  • The Pennsylvania State University the Graduate School THRESHOLDED PARTIAL CORRELATION APPROACH for VARIABLE SELECTION in LINEAR
    The Pennsylvania State University The Graduate School THRESHOLDED PARTIAL CORRELATION APPROACH FOR VARIABLE SELECTION IN LINEAR MODELS AND PARTIALLY LINEAR MODELS A Dissertation in Statistics by Lejia Lou c 2013 Lejia Lou Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2013 The dissertation of Lejia Lou was reviewed and approved∗ by the following: Runze Li Distinguished Professor of Statistics Dissertation Advisor and Chair of Committee David Hunter Professor of Statistics Head of Department of Statistics Bing Li Professor of Statistics Rongling Wu Professor of Public Health Science Aleksandra Slavkovic Chair of Graduate Program in Statistics ∗Signatures are on file in the Graduate School. Abstract This thesis is concerned with variable selection in linear models and partially linear models for high-dimensional data analysis. With the development of technology, it is crucial to identify a small subset of covariates that exhibits the strongest relationship with the response. Researchers have made much effort on developing variable selection methodologies, such as regularized techniques, including least absolute shrinkage and selection operator (LASSO, Tibshirani, 1996) and penalized least squares estimate with smoothly clipped absolute deviation penalty (SCAD, Fan and Li, 2001). Different from those regularization methods for variable selection in linear mod- els, Buhlmann et al. (2010) proposed the PC-simple algorithm to select significant variables. As they showed, under some conditions and proper choice of the sig- nificance level, the PC-simple algorithm can consistently identify the true active set with probability approaching 1 when the response and covariates are jointly normally distributed. In Chapter 3, we study the performance of the PC-simple algorithm under a non-normal distribution.
    [Show full text]