Abstract
In this article, we present a new approach to determining direction of effects in binary variables. A variable is considered explanatory if it explains the probability distribution of another variable. In cross-classifications of binary variables, the univariate probability distribution of variables can be considered explained if omitting the univariate effects of this variable does not lead to an ill-fitting model. Directional (non-hierarchical) log-linear models are introduced that allow statements concerning the direction of association in binary data. Cases in which variables of latent linear regression processes are partially observed as binary variables reveal a close conceptual link between the proposed log-linear approach and existing direction of effect methodology for metric variables. A Monte Carlo study is presented that shows that the proposed approach has good power and enables researchers to distinguish the correct model from the incorrect, reverse model. The approach can be extended to multiple explanatory and multiple outcome variables. Empirical data examples from research on aggression development in adolescence illustrate the proposed direction dependence approach.
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Notes
We are well aware that, when using a nominal significance level of 5%, both, the main effect of T and the LR-test for Model 2, fail to reject the null hypothesis in the strict sense. However, in the present context, we are in favor of interpreting (1) T as being sufficiently non-uniform (with \(Pr\left( {T = 1} \right) \) = .584) and (2) the difference in LR-test statistics (3.68 vs. 0.46) as being substantively meaningful.
References
Agresti A (2013) Categorical data analysis, 3rd edn. Wiley, New York
Bradley JV (1978) Robustness? Br J Math Stat Psychol 31:144–152. doi:10.1111/j.2044-8317.1978.tb00581.x
Christensen R (1990) Log-linear models. Springer, New York
Cohen J (1983) The cost of dichotomization. Appl Psychol Meas 7:249–253. doi:10.1177/014662168300700301
Cohen J (1988) Statistical power analysis for the behavioral sciences, 2nd edn. Lawrence Erlbaum Associates, New Jersey
Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman & Hall, London
DeCoster J, Iselin AMR, Gallucci M (2009) A conceptual and empirical evaluation of justifications of dichotomization. Psycho Methods 14:349–366. doi:10.1037/a0016956
Dodge Y, Rousson V (2000) Direction dependence in a regression line. Commun Stat Theory Methods 32:2053–2057. doi:10.1080/03610920008832589
Dodge Y, Rousson V (2001) On asymmetric properties of the correlation coefficient in the regression setting. Am Stat 55:51–54. doi:10.1198/000313001300339932
Dodge Y, Yadegari I (2010) On direction of dependence. Metrika 72:139–150. doi:10.1007/s00184-009-0273-0
Dodge Y, Rousson V (2016) Statistical inference for direction of dependence in linear models. In: Wiedermann W, von Eye A (eds) Statistics and causality. Wiley, Hoboken
Fienberg SE (1981) The analysis of cross-classified categorical data, 2nd edn. MIT Press, Cambridge
Finkelstein JW, von Eye A, Preece MA (1994) The relationship between aggressive behavior and puberty in normal adolescents: a longitudinal study. J Adolesc Health 15:319–326. doi:10.1016/1054-139X(94)90605-X
Fitzgerald P, Knuiman MW, Divitini ML, Bartholomew HC (1999) Effect of dichotomising a continuous variable on the assessment of familial aggregation: an empirical study using body mass index data from the Busselton Health Study. J Epidemiol Biostat 4:321–327
Fitzsimons GJ (2008) Death to dichotomizing. J Consum Res 35:5–8. doi:10.1086/589561
Goodman LA (1973) The analysis of multidimensional contingency tables when some variables are posteriori to others: a modified path analysis approach. Biometrika 60:179–192. doi:10.1093/biomet/60.1.179
Gupta K, Nguyen T, Pardo L (2007) On Christensen’s conjecture. Stat Pap 48:313–319. doi:10.1007/s00362-007-0379-2
Hagenaars JA (1998) Categorical causal modeling: latent class analysis and directed log-linear models with latent variables. Sociol Methods Res 26:436–486. doi:10.1177/0049124198026004002
Holland PW (1988) Causal inference, path analysis, and recursive structural equation models (with discussion). Sociol Methodol 18:449–493. doi:10.2307/271055
Homburg C, Dobratz A (1992) Covariance structure analysis via specification searches. Stat Pap 33:119–142. doi:10.1007/bf02925318
Hosmer DW, Lemeshow S (2002) Applied logistic regression, 2nd edn. Wiley, New York
Imai K, Keele L, Tingley D (2010) A general approach to causal mediation analysis. Psychol Methods 15:309–334. doi:10.1037/a0020761
Inazumi T, Washio T, Shimizu S, Suzuki J, Yamamoto A, & Kawahara Y (2014) Causal discovery in a binary exclusive-or skew acyclic model: BExSAM. arXiv:1401.5636
Kawasaki S, Zimmermann KF (1964) Measuring relationships in the log-linear probability model by some compact measures of association. Stat Pap 22:82–109. doi:10.1007/bf02933546
Kendall MG, Stuart A (1961) The advanced theory of statistics (vol. 2): inference and relationship. Hafner, New York
Kim D, Kim JM (2013) Analysis of directional dependence using asymmetric copula-based regression models. Journal of Statistical Computation and Simulation. doi:10.1080/00949655.2013.779696
Knoke D, Burke PJ (1980) Log-linear models. Sage, Thousand Oaks
MacCallum RC, Zhang S, Preacher KJ, Rucker DD (2002) On the practice of dichotomization of quantitative variables. Psychol Methods 7:19–40. doi:10.1037//1082-989X.7.1.19
McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman & Hall, London
Mair P, von Eye A (2007) Application scenarios for nonstandard log-linear models. Psychol Methods 12:139–156. doi:10.1037/1082-989X.12.2.139
Marszalek JM, Barber C, Kohlhart J, Holmes CB (2011) Sample size in psychological research over the past 30 years. Percept Motor Skills 112(2):331–348. doi:10.2466/03.11.PMS.112.2.331-348
Maxwell SE, Delaney HD (1993) Bivariate median splits and spurious statistical significance. Psychol Bull 113:181–190. doi:10.1037/0033-2909.113.1.181
Nelder JA (1974) Log linear models for contingency tables: a generalization of classical least squares. J R Stat Soc C 23:323–329. doi:10.2307/2347125
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York
Nigg JT, Knottnerus GM, Martel MM, Nikolas M, Cavanagh K, Karmaus W, Rappley MD (2008) Low blood lead levels associated with clinically diagnosed attention-deficit/ hyperactivity disorder and mediated by weak cognitive control. Biol Psychiatry 63:325–331. doi:10.1016/j.biopsych.2007.07.013
Paul LA (2012) Counterfactual theories. In: Beebee H, Hitchcock C, Menzies P (eds) The Oxford handbook of causation. Oxford University Press, Oxford, pp 158–184
Paul LA, Hall N (2013) Causation: a user’s guide. Oxford University Press, Oxford
Pearson K (1900) On the correlation of characters not quantitatively measurable. R Soc Philos Trans Ser A 195:1–47. doi:10.1098/rsta.1900.0022
Pornprasertmanit S, Little TD (2012) Determining directional dependency in causal associations. Int J Behav Dev 36:313–322. doi:10.1177/0165025412448944
R Core Team (2017). R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
Rindskopf D (1990) Nonstandard log-linear models. Psychol Bull 108:150–162. doi:10.1037/0033-2909.108.1.150
Robins JM (1986) A new approach to causal inference in mortality studies with sustained exposure periods–application to control of the healthy worker survivor effect. Math Model 7:1393–1512. doi:10.1016/0270-0255(86)90088-6
Shimizu S, Hoyer PO, Hyvärinen A, Kerminen AJ (2006) A linear non-gaussian acyclic model for causal discovery. J Mach Learn Res 7:2003–2030
Vermunt J (1996) Causal log-linear modeling with latent variables and missing data. In: Engel U, Reinecke J (eds) Analysis of change: advanced techniques in panel data analysis. de Gruyter, Berlin, pp 35–60
von Eye A, & DeShon RP (2008) Characteristics of measures of directional dependence—a Monte Carlo study. Interstat. http://interstat.statjournals.net/YEAR/2008/articles/0802002.pdf. Accessed March 13, 2013
von Eye A, DeShon RP (2012) Directional dependence in developmental research. Int J Behav Dev 36:303–312. doi:10.1177/0165025412439968
von Eye A, & Mair P (2012) On the effects of dichotomizing information. In EE Barragan, AFM Martínez, LEN Barajas, & CC Covarrubias (eds), Memorias del XXIV Foro National de Estadistica
von Eye A, Mun E-Y (2013) Log-linear modeling—concepts, interpretation and applications. Wiley, New York
von Eye A, Wiedermann W (2013) On direction of dependence in latent variable contexts. Educ Psychol Meas 74:5–30. doi:10.1177/0013164413505863
von Wright GH (1971) Explanation and understanding. Cornell University Press, Ithaca
Walwyn R (2005) Moments. In: Everitt BS, Howell DC (eds) Handbook of statistics in social science. John Wiley, Chichester, pp 1258–1260
Westfall PH (2011) Improving power by dichotomising (even under normality). Stat Biopharm Res 3:353–362. doi:10.1198/sbr.2010.09055
Wiedermann W, Hagmann M, Kossmeier M, & von Eye A (2013) Resampling techniques to determine direction of effects in linear regression models. InterStat. http://interstat.statjournals.net/YEAR/2013/articles/1305002.pdfs. Accessed July 24, 2013
Wiedermann W, Hagmann M, von Eye A (2015) Significance tests to determine the direction of effects in linear regression models. Br J Math Stat Psychol 68:116–141. doi:10.1111/bmsp.12037
Wiedermann W, von Eye A (2015a) Direction dependence analysis: a confirmatory approach for testing directional theories. Int J Behav Dev 39:570–580. doi:10.1177/0165025415582056
Wiedermann W, von Eye A (2015b) Direction of effects in multiple linear regression model. Multivar Behav Res 50:23–40. doi:10.1080/00273171.2014.958429
Wiedermann W, Artner R, von Eye A (2017) Heteroscedasticity as a basis for direction dependence in reversible linear regression models. Multiv Behav Res 52:222–241. doi:10.1080/00273171.2016.1275498
Yamaguchi K (2012) Log-linear causal analysis of cross-classified categorical data. Sociol Methodol 42:257–285. doi:10.1177/0081175012460661
Yamaguchi K (2016) Log-linear causal analysis of cross-classified categorical data. In: Wiedermann W, von Eye A (eds) Statistics and causality. Wiley, Hoboken, pp 311–331
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Appendix: R code for estimating bivariate directional log-linear models
Appendix: R code for estimating bivariate directional log-linear models
The function model.fit can be used to compute the likelihood ratio goodness of fit test:
The following code can be used to specify the design matrices and estimate the competing directional log-linear models:
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Wiedermann, W., von Eye, A. Log-linear models to evaluate direction of effect in binary variables. Stat Papers 61, 317–346 (2020). https://doi.org/10.1007/s00362-017-0936-2
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DOI: https://doi.org/10.1007/s00362-017-0936-2