Sage Journals: Discover world-class research

Abstract

In this article, we discuss and propose methods that may be of use to determine direction of dependence in non-normally distributed variables. First, it is shown that standard regression analysis is unable to distinguish between explanatory and response variables. Then, skewness and kurtosis are discussed as tools to assess deviation from normality. Deviation from normality can be used to assess direction of dependence. This proposition is based on the fact that the response variable will always have less skew than the independent variable (Dodge & Rousson, 2000). It has been shown that the cube of the Pearson correlation coefficient can be calculated as the ratio of the skewness measures of the correlated variables. Because correlations cannot exceed the interval (−1.0; +1.0), directional dependence of the two correlated variables can be determined by the ratio that results in a correlation that stays within this interval. It is also proposed that other measures of deviation from normality can be used to determine directional dependence; for example, kurtosis. Recommendations are given for making decisions concerning directional dependence. Empirical data examples from developmental research on violence against women and on attention deficit hyperactivity disorder (ADHD) illustrate the use of the methodology. Cross-sectional and temporal directional dependence are discussed, and the effects of onset of a causal agent and termination of a causal agent are illustrated.

Keywords

causality development directional dependency research

Get full access to this article

View all access options for this article.

References

American Psychiatric Association. (1994). Diagnostic and statistical manual of mental disorders (4th ed.). Washington, DC: American Psychiatric Association.

Baltes

P. B.

Reese

H. W.

Nesselroade

J. R.

(1977). Life-span developmental psychology: Introduction to research methods. Monterey, CA: Brooks/Cole.

Bogat

G. A.

Levendosky

A. A.

DeJonghe

Davidson

W. S.

von Eye

(2004). Pathways of suffering: The temporal effects of domestic violence on women’s mental health. Maltrattamento e abuso all’infanzia, 6, 97–112.

Bollen

K. A.

(1989). Structural equations with latent variables. New York, NY: John Wiley & Sons.

Cox

D. R.

Wermuth

(2001). Some statistical aspects of causality. European Sociological Review, 17, 65–74.

D’Agostino

R. B.

(1971). Transformation to normality of the null distribution of g1. Biometrika, 57, 341–348.

D’Agostino

R. B.

Belanger

D’Agostino

R. B.

Jr. (1990). A suggestion for using powerful and informative tests of normality. The American Statistician, 44, 316–321.

D’Agostino

R. B.

Pearson

E. S.

(1973). Testing departures from normality: 1: Fuller empirical results for the distribution of b₂ and √b₁ . Biometrika, 60, 613–622.

Dodge

Rousson

(2000). Direction dependence in a regression line. Communications in Statistics—Theory and Methods, 32, 2053–2057.

10.

Dodge

Rousson

(2001). On asymmetric properties of the correlation coefficient in the regression setting. The American Statistician, 55, 51–54.

11.

Dodge

Yadegari

(2010). On direction of dependence. Metrika, 72, 130–150.

12.

Falk

Well

A. D.

(1996). Correlation as probability of common descent. Multivariate Behavioral Research, 31, 219–238.

13.

Hogg

R. V.

Tanis

E. A.

(1993). Probability and statistical inference (4th ed.). New York, NY: Macmillan.

14.

Hume

(1777). An enquiry concerning human understanding. Oxford, UK: Clarendon Press.

15.

Lucas

R. E.

Diener

(2001). Understanding extraverts’ enjoyment of social situations: The importance of pleasantness. Journal of Personality and Social Psychology, 81, 343–356.

16.

Lucas

R. E.

Fujita

(2000). Factors influencing the relation between extraversion and pleasant affect. Journal of Personality and Social Psychology, 79, 1039–1056.

17.

Lynd-Stevenson

R. M.

(2007). Concerns regarding the traditional paradigm for causal research: The unified paradigm and causal research in scientific psychology. Review of General Psychology, 11, 286–304.

18.

Mardia

K. V.

(1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530.

19.

Mardia

K. V.

(1980). Tests of univariate and mutivariate normality. In Krishnaiah

P. R.

(Ed.), Handbook of statistics (Vol. 1, pp. 279–320). Amsterdam, the Netherlands: North Holland.

20.

Micceri

(1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166.

21.

Muddapur

M. V.

(2003). On directional dependence in a regression line: Communications in statistics. Theory and Methods, 32, 2053–2057.

22.

Myers

R. H.

Montgomery

D. C.

Vining

G. G.

(2002). Generalized linear models, with applications in engineering and the sciences. New York, NY: Wiley.

23.

Nigg

J. T.

Knottnerus

G. M.

Martel

M. M.

Nikolas

Cavanagh

Karmaus

Rappley

M. D.

(2008). Low blood lead levels associated with clinically diagnosed attention-deficit/hyperactivity disorder and mediated by weak cognitive control. Biological Psychiatry, 63, 325–331.

24.

Pearson

(1895). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London, 91, 343–358.

25.

Rodgers

J. L.

Nicewander

W. A.

(1988). Thirteen ways to look at the correlation coefficient. The American Statistician, 42, 59–66.

26.

Rovine

M. J.

von Eye

(1997). A 14th way to look at a correlation coefficient: Correlation as the proportion of matches. The American Statistician, 51, 42–46.

27.

Shimizu

Hoyer

P. O.

Hyvärinen

Kerminen

(2006). A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7, 2003–2030.

28.

Shimizu

Kano

(2006). Use of non-normality in structural equation modeling: Application to direction of causation. Journal of Statistical Planning and Inference, 136, 3481–3491.

29.

Sungur

E. A.

(2005). A note on directional dependence in regression setting. Communications in Statistics—Theory and Methods, 34, 1957–1965.

30.

Sungur

E. A.

Çelebioglu

(2011). Copulas with directional dependence property. Gazi University Journal of Science, 24, 415–424.

31.

von Eye

(2006). Comparing tests of multinormality under sparse data conditions—A Monte Carlo study. InterStat. Retrieved from http://interstat.statjournals.net/

32.

von Eye

DeShon

R. P.

(2008). Characteristics of measures of directional dependence—A Monte Carlo study. InterStat. Retrieved from http://interstat.statjournals.net/

33.

von Eye

Gardiner

J. C.

(2004). Locating deviations from multivariate normality. Understanding Statistics, 3, 313–331.

34.

von Eye

Bogat

G. A.

(2006). Multinormality and symmetry: A comparison of two statistical tests. Psychology Science, 48, 419–435.

35.

Walwyn

(2005). Moments. In Everitt

B. S.

Howell

D. C.

(Eds.), Handbook of statistics in social science (pp. 1258–1260). Chichester, UK: John Wiley.

Directional dependence in developmental research

Abstract

Keywords

Get full access to this article

References