# Anatomy of Correlational Magnitude Transformations in Latency and Discretization Contexts in Monte-Carlo Studies

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## Abstract

This chapter is concerned with the assessment of correlational magnitude changes when a subset of the continuous variables that may marginally or jointly follow nearly any distribution in a multivariate setting is dichotomized or ordinalized. Statisticians generally regard discretization as a bad idea on the grounds of power , information , and effect size loss. Despite this undeniable disadvantage and legitimate criticism, its widespread use in social, behavioral, and medical sciences stems from the fact that discretization could yield simpler, more interpretable, and understandable conclusions, especially when large audiences are targeted for the dissemination of the research outcomes. We do not intend to attach any negative or positive connotations to discretization, nor do we take a position of advocacy for or against it. The purpose of the current chapter is providing a conceptual framework and computational algorithms for modeling the correlation transitions under specified distributional assumptions within the realm of discretization in the context of the latency and threshold concepts. Both directions (identification of the pre-discretization correlation value in order to attain a specified post-discretization magnitude, and the other way around) are discussed. The ideas are developed for bivariate settings; a natural extension to the multivariate case is straightforward by assembling the individual correlation entries. The paradigm under consideration has important implications and broad applicability in the stochastic simulation and random number generation worlds. The proposed algorithms are illustrated by several examples; feasibility and performance of the methods are demonstrated by a simulation study.

## Keywords

Absolute Average Deviation Normal Mixture Polychoric Correlation Gaussian Copula Unbiased Measure## References

- Allozi, R., & Demirtas, H. (2016). Modeling Correlational Magnitude Transformations in Discretization Contexts, R package
**CorrToolBox**. https://cran.r-project.org/web/packages/CorrToolBox. - Amatya, A., & Demirtas, H. (2015). Simultaneous generation of multivariate mixed data with Poisson and normal marginals.
*Journal of Statistical Computation and Simulation*,*85*, 3129–3139.Google Scholar - Barbiero, A., & Ferrari, P.A. (2015). Simulation of Ordinal and Discrete Variables with Given Correlation Matrix and Marginal Distributions, R package
**GenOrd**. https://cran.r-project.org/web/packages/GenOrd. - Cario, M. C., & Nelson, B. R. (1997).
*Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix (Technical Report)*. Department of Industrial Engineering and Management Services: Northwestern University, Evanston, IL, USA.Google Scholar - Demirtas, H. (2004a). Simulation-driven inferences for multiply imputed longitudinal datasets.
*Statistica Neerlandica*,*58*, 466–482.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H. (2004b). Assessment of relative improvement due to weights within generalized estimating equations framework for incomplete clinical trials data.
*Journal of Biopharmaceutical Statistics*,*14*, 1085–1098.MathSciNetCrossRefGoogle Scholar - Demirtas, H. (2005). Multiple imputation under Bayesianly smoothed pattern-mixture models for non-ignorable drop-out.
*Statistics in Medicine*,*24*, 2345–2363.MathSciNetCrossRefGoogle Scholar - Demirtas, H. (2006). A method for multivariate ordinal data generation given marginal distributions and correlations.
*Journal of Statistical Computation and Simulation*,*76*, 1017–1025.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H. (2007a). Practical advice on how to impute continuous data when the ultimate interest centers on dichotomized outcomes through pre-specified thresholds.
*Communications in Statistics-Simulation and Computation*,*36*, 871–889.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H. (2007b). The design of simulation studies in medical statistics.
*Statistics in Medicine*,*26*, 3818–3821.MathSciNetCrossRefGoogle Scholar - Demirtas, H. (2008). On imputing continuous data when the eventual interest pertains to ordinalized outcomes via threshold concept.
*Computational Statistics and Data Analysis*,*52*, 2261–2271.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H. (2009). Rounding strategies for multiply imputed binary data.
*Biometrical Journal*,*51*, 677–688.MathSciNetCrossRefGoogle Scholar - Demirtas, H. (2010). A distance-based rounding strategy for post-imputation ordinal data.
*Journal of Applied Statistics*,*37*, 489–500.MathSciNetCrossRefGoogle Scholar - Demirtas, H. (2016). A note on the relationship between the phi coefficient and the tetrachoric correlation under nonnormal underlying distributions.
*American Statistician*,*70*, 143–148.MathSciNetCrossRefGoogle Scholar - Demirtas, H. (2017). Concurrent generation of binary and nonnormal continuous data through fifth order power polynomials,
*Communications in Statistics- Simulation and Computation*.*46*, 344–357.Google Scholar - Demirtas, H., Ahmadian, R., Atis, S., Can, F. E., & Ercan, I. (2016a). A nonnormal look at polychoric correlations: Modeling the change in correlations before and after discretization.
*Computational Statistics*,*31*, 1385–1401.Google Scholar - Demirtas, H., Arguelles, L. M., Chung, H., & Hedeker, D. (2007). On the performance of bias-reduction techniques for variance estimation in approximate Bayesian bootstrap imputation.
*Computational Statistics and Data Analysis*,*51*, 4064–4068.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H., & Doganay, B. (2012). Simultaneous generation of binary and normal data with specified marginal and association structures.
*Journal of Biopharmaceutical Statistics*,*22*, 223–236.Google Scholar - Demirtas, H., Freels, S. A., & Yucel, R. M. (2008). Plausibility of multivariate normality assumption when multiply imputing non-Gaussian continuous outcomes: A simulation assessment.
*Journal of Statistical Computation and Simulation*,*78*, 69–84.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H., & Hedeker, D. (2007). Gaussianization-based quasi-imputation and expansion strategies for incomplete correlated binary responses.
*Statistics in Medicine*,*26*, 782–799.MathSciNetCrossRefGoogle Scholar - Demirtas, H., & Hedeker, D. (2008a). Multiple imputation under power polynomials.
*Communications in Statistics- Simulation and Computation*,*37*, 1682–1695.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H., & Hedeker, D. (2008b). Imputing continuous data under some non-Gaussian distributions.
*Statistica Neerlandica*,*62*, 193–205.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H., & Hedeker, D. (2008c). An imputation strategy for incomplete longitudinal ordinal data.
*Statistics in Medicine*,*27*, 4086–4093.MathSciNetCrossRefGoogle Scholar - Demirtas, H., & Hedeker, D. (2011). A practical way for computing approximate lower and upper correlation bounds.
*The American Statistician*,*65*, 104–109.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H., & Hedeker, D. (2016). Computing the point-biserial correlation under any underlying continuous distribution.
*Communications in Statistics- Simulation and Computation*,*45*, 2744–2751.MathSciNetCrossRefzbMATHGoogle Scholar - Demirtas, H., Hedeker, D., & Mermelstein, J. M. (2012). Simulation of massive public health data by power polynomials.
*Statistics in Medicine*,*31*, 3337–3346.MathSciNetCrossRefGoogle Scholar - Demirtas, H., & Schafer, J. L. (2003). On the performance of random-coefficient pattern-mixture models for non-ignorable drop-out.
*Statistics in Medicine*,*22*, 2553–2575.CrossRefGoogle Scholar - Demirtas, H., Shi, Y., & Allozi, R. (2016b). Simultaneous generation of count and continuous data, R package PoisNonNor. https://cran.r-project.org/web/packages/PoisNonNor.
- Demirtas, H., Wang, Y., & Allozi, R. (2016c) Concurrent generation of binary, ordinal and continuous data, R package BinOrdNonNor. https://cran.r-project.org/web/packages/BinOrdNonNor.
- Demirtas, H., & Yavuz, Y. (2015). Concurrent generation of ordinal and normal data.
*Journal of Biopharmaceutical Statistics*,*25*, 635–650.CrossRefGoogle Scholar - Emrich, J. L., & Piedmonte, M. R. (1991). A method for generating high-dimensional multivariate binary variates.
*The American Statistician*,*45*, 302–304.Google Scholar - Farrington, D. P., & Loeber, R. (2000). Some benefits of dichotomization in psychiatric and criminological research.
*Criminal Behaviour and Mental Health*,*10*, 100–122.CrossRefGoogle Scholar - Ferrari, P. A., & Barbiero, A. (2012). Simulating ordinal data.
*Multivariate Behavioral Research*,*47*, 566–589.CrossRefGoogle Scholar - Fleishman, A. I. (1978). A method for simulating non-normal distributions.
*Psychometrika*,*43*, 521–532.CrossRefzbMATHGoogle Scholar - Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données.
*Annales de l’Université de Lyon Section A*,*14*, 53–77.zbMATHGoogle Scholar - Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., Bornkamp, B., Maechler, M., & Hothorn, T. (2016). Multivariate normal and t distributions, R package mvtnorm. https://cran.r-project.org/web/packages/mvtnorm.
- Headrick, T. C. (2002). Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions.
*Computational Statistics and Data Analysis*,*40*, 685–711.MathSciNetCrossRefzbMATHGoogle Scholar - Headrick, T. C. (2010).
*Statistical Simulation: Power Method Polynomials and Other Transformations Boca Raton*. FL: Chapman and Hall/CRC.zbMATHGoogle Scholar - Hoeffding, W. (1994). Scale-invariant correlation theory. In N. I. Fisher & P. K. Sen (Eds.),
*The Collected Works of Wassily Hoeffding*(the original publication year is 1940) (pp. 57–107). New York: Springer.Google Scholar - Inan, G., & Demirtas, H. (2016). Data generation with binary and continuous non-normal components, R package BinNonNor. https://cran.r-project.org/web/packages/BinNonNor.
- MacCallum, R. C., Zhang, S., Preacher, K. J., & Rucker, D. D. (2002). On the practice of dichotomization of quantitative variables.
*Psychological Methods*,*7*, 19–40.CrossRefGoogle Scholar - R Development Core Team. (2016). R: A Language and Environment for Statistical Computing. http://www.cran.r-project.org.
- Revelle, W. (2016). Procedures for psychological, psychometric, and personality researchmultivariate normal and t distributions, R package psych. https://cran.r-project.org/web/packages/psych.
- Vale, C. D., & Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions.
*Psychometrika*,*48*, 465–471.CrossRefzbMATHGoogle Scholar - Yucel, R. M., & Demirtas, H. (2010). Impact of non-normal random effects on inference by multiple imputation: A simulation assessment.
*Computational Statistics and Data Analysis*,*54*, 790–801.Google Scholar