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The Effects of Multicollinearity in Multilevel Models Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2013 The Effects of Multicollinearity in Multilevel Models Patrick Carl Clark Jr. Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Industrial and Organizational Psychology Commons Repository Citation Clark, Patrick Carl Jr., "The Effects of Multicollinearity in Multilevel Models" (2013). Browse all Theses and Dissertations. 740. https://corescholar.libraries.wright.edu/etd_all/740 This Dissertation is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. THE EFFECTS OF MULTICOLLINEARITY IN MULTILEVEL MODELS A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy By PATRICK C. CLARK B.A., Indiana University of Pennsylvania, 2007 M.S., Wright State University, 2010 _________________________________________ 2013 Wright State University WRIGHT STATE UNIVERSITY GRADUATE SCHOOL I HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER MY SUPERVISION BY Patrick C. Clark ENTITLED The Effects of Multicollinearity in Multilevel Models BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy. ___________________________________ David LaHuis, Ph.D. Dissertation Director ___________________________________ Scott Watamaniuk, Ph.D. Graduate Program Director, ___________________________________ R. William Ayres, Ph.D. Interim Dean, Graduate School Committee on Final Examination ___________________________________ David LaHuis, Ph.D. ___________________________________ Debra Steele-Johnson, Ph.D. ___________________________________ Gary Burns, Ph.D. ___________________________________ Valerie Shalin, Ph.D. ABSTRACT Clark, Patrick C. Ph.D., Industrial/Organizational Psychology Ph.D. program, Wright State University, 2013. The Effects of Multicollinearity in Multilevel Models. This study examined a method for calculating the impact of multicollinearity on multilevel modeling. The major research questions concerned a) how the simulation design factors affect (multilevel variance inflation factor) MVIF, b) how MVIF affects standard errors of regression coefficients, and c) how MVIF affects significance of regression coefficients. Monte Carlo simulations were conducted to address these questions. Predictor relationships were manipulated in order to simulate multicollinearity. Findings indicate that a) increases in relationships among Level 1 predictors and also relationships among Level 2 predictors led to increased MVIF for those specific variables, b) as MVIF increases for a predictor, the standard errors for the regression coefficients also increase., and c) when MVIF values for the regression coefficients were 5 or higher, margins of error were around .20, and therefore any coefficients around .20 or lower will become non-significant. iii TABLE OF CONTENTS Page I. INTRODUCTION AND PURPOSE……………………………………………………1 Multilevel Modeling………………………………………………………………2 Multicollinearity in OLS Regression……………………………………………...5 Measures of Multicollinearity in OLS Regression………………………………..6 Multicollinearity in Multilevel Models…………………………………………..11 Multilevel Variance Inflation Factor…………………………………………….14 II. METHOD……………………………………………………………………………..17 Multilevel Model Structure………………………………………………………17 Simulation Design Factors……………………………………………………….18 Analyses………………………………………………………………………….20 III. RESULTS……………………………………………………………………………20 Effect of Design Factors on MVIF………………………………………………21 MVIF and Predictor Standard Errors…………………………………………….22 MVIF and Significance of Regression Coefficients……………………………..25 IV. DISCUSSION………………………………………………………………………..30 Implications………………………………………………………………………34 Limitations and Future Research………………………………………………...36 Conclusion……………………………………………………………………….37 iv LIST OF FIGURES Figure Page 1a. Margins of error in each MVIF range based on a multilevel model with only Level 1 variables………………………...46 1b. Standard error means in each MVIF range based on a multilevel model with only Level 1 variables…………………...47 2a. Margins of error in each MVIF range based on a multilevel model with only Level 2 variables………………………...48 2b. Standard error means in each MVIF range based on a multilevel model with only Level 2 variables…………………...49 3a. Margins of error in each MVIF range based on a multilevel model with a Level 2 variable predicting the slopes of all three Level 1 variables………………………...50 3b. Standard error means in each MVIF range based on a multilevel model with a Level 2 variable predicting the slopes of all three Level 1 variables………………………...51 4a. Margins of error in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of one Level 1 variable………………………52 4b. Standard error means in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of one Level 1 variable………………………53 5a. Margins of error in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of all three Level 1 variables………………………...54 5b. Standard error means in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of all three Level 1 variables……………………….55 6a. Margins of error of cross-level interaction coefficients in each v TABLE OF CONTENTS (Continued) MVIF range based on a multilevel model with a Level 2 variable predicting the slopes of all three Level 1 variables……………………………..56 6b. Standard error means of cross-level interaction coefficients in each MVIF range based on a multilevel model with a Level 2 variable predicting the slopes of all three Level 1 variables………………………..57 7a. Margins of error of cross-level interaction coefficients in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of one Level 1 variable………………………..58 7b. Standard error means of cross-level interaction coefficients in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of one Level 1 variable………………………...59 8a. Margins of error of cross-level interaction coefficients in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of all three Level 1 variables………………………...60 8b. Standard error means of cross-level interaction coefficients in each MVIF range based on a multilevel model with all three Level 2 variable predicting the slopes of all three Level 1 variables………………………………61 vi LIST OF TABLES Table Page 1. Simulated correlation matrix of three Level 1 variables………………………………41 2. Simulated inverse correlation matrix of three Level 1 Variables. VIFs are on the diagonals and highlighted…………………………………...42 3. Simulated correlation matrix of three Level 1 variables, one Level 2 variable, and three cross-level interactions………………………………43 4. Simulated inverse correlation matrix of three Level 1 variables, one Level 2 variable, and three cross-level interactions. MVIFs are on the diagonals and highlighted………………………………44 5. MVIFs, parameter estimates, standard errors, and t-values for the simulated multilevel model…………………………………….45 Appendix A1. Plan for analyses…………………………………………………………………….62 A2. Standard error ranges, means, margins of error, and number of conditions falling in each MVIF range based on a model with Level 1 variables only………………………………..63 A3. Standard error ranges, means, margins of error, and number of conditions falling in each MVIF range based on a model with Level 2 variables only………………………………..64 A4. Standard error ranges, means, margins of error, and number of conditions falling in each MVIF range based on a model with a Level 2 variable predicting the slopes of all three Level 1 variables………………………………..65 A5. Standard error ranges, means, margins of error, and number of conditions falling in each MVIF range based on a model with all three Level 2 variables predicting the slopes of one Level 1 variable…………………………………..66 vii TABLE OF CONTENTS (Continued) A6. Standard error ranges, means, margins of error, and number of conditions falling in each MVIF range based on a model with all three Level 2 variables predicting the slopes of all three Level 1 variables………………………………….67 A7. Standard error ranges, means, margins of error, and number of conditions for cross-level interactions falling in each MVIF range based on a model with a Level 2 variable predicting the slopes of all three Level 1 variables……………………………..68 A8. Standard error ranges, means, margins of error, and number of conditions for cross-level interactions falling in each MVIF range based on a model with all three Level 2 variables predicting the slopes of one Level 1 variable………………………………..69 A9. Standard error ranges, means, margins of error, and number of conditions for cross-level interactions falling in each MVIF range based on a model with all three Level 2 variables predicting the slopes of all three Level 1 variables……………………………71 BIBLIOGRAPHY………………………………………………………………………..39 viii The Effects of Multicollinearity in Multilevel Models Multicollinearity occurs when one or more of the predictor variables highly correlates with the other predictor variables in a regression equation (Cohen, Cohen, West, & Aiken, 2003). Multicollinearity is an issue that has been widely
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