Quantile Regression Analysis and Other Alternatives to Ordinary Least Squares Regression
A Methodological Comparison on Corporal Punishment
Abstract
Data analyses by classical ordinary least squares (OLS) regression techniques often employ unrealistic assumptions, fail to recognize the source and nature of heterogeneity, and are vulnerable to extreme observations. Therefore, this article compares robust and non-robust M-estimator regressions in a statistical demonstration study. Data from the Erlangen-Nuremberg Development and Prevention Project are used to model risk factors for physical punishment by fathers of 485 elementary school children. The Corporal Punishment Scale of the Alabama Parenting Questionnaire was the dependent variable. Fathers’ aggressiveness, dysfunctional parent-child relations, various other parenting characteristics, and socio-demographic variables served as predictors. Robustness diagnostics suggested the use of trimming procedures and outlier diagnostics suggested the use of robust estimators as an alternative to OLS. However, a quantile regression analysis provided more detailed insights beyond the measures of central tendency and detected sources of considerable heterogeneity in the risk structure of father’s corporal punishment. Advantages of this method are discussed with regard to methodological and content issues.
References
(1995). Parenting stress index (PSI). Odessa, FL: Psychological Assessment Resources (PAR).
(2005). An alternative to Cohen’s standardized mean difference effect size: A robust parameter and confidence interval in the two independent groups case. Psychological Methods, 10, 317–328.
(2002). Ordinary physical punishment. Is it harmful? Comment on Gershoff (2002). Psychological Bulletin, 128, 580–589.
(1984). The determinants of parenting: A process model. Child Development, 55, 83–96.
(1993). Etiology of child maltreatment: A developmental-ecological analysis. Psychological Bulletin, 114, 413–434.
(2006). Misshandlung von Kindern: Risikofaktoren und Schutzfaktoren. In
[Child abuse: Risk and protective factors] Kindesmisshandlung und Vernachlässigung.[Child abuse and neglect] (pp. 317–346). Göttingen, Germany: Hogrefe.(2004). Evaluating the subtle impact of a ban on corporal punishment of children in Germany. Child Abuse Review, 13, 292–311.
(2003). A gentle introduction to quantile regression for ecologists. Ecological Environment, 8, 412–420.
(1988). Statistical power analysis for the behavioral sciences (2nd ed.). Mahwah, NJ: Erlbaum.
(2008). Physical punishment, culture and rights: Current issues for professionals. Journal of Developmental and Behavioral Pediatrics, 29, 55–66.
(2008). Modern robust statistical methods: An easy way to maximize the accuracy and power of your research. American Psychologist, 63, 591–601.
(1989). Das Freiburger Persönlichkeitsinventar – revidierte Fassung (FPI-R).
[The Freiburg Personality Inventory – revised version (FPI-R)] Göttingen, Germany: Hogrefe.(2005). Childhood origins of antisocial behavior. Clinical Psychology and Psychotherapy, 12, 177–190.
. (1994). Soziale Schichtung und Lebenschance in Deutschland.
[Social stratification and life chances in Germany] Stuttgart, Germany Enke.(2002). Corporal punishment by parents and associated child behaviors and experiences: A meta-analytic and theoretical review. Psychological Bulletin, 128, 539–579.
(2010). More harm than good: A summary of scientific research on the intended and unintended effects of corporal punishment. Law and Contemporary Problems, 73, 31–56.
(2008). Applied econometrics with R. Heidelberg, Germany: Springer-Verlag.
(2005). Quantile regression. New York, NY: Cambridge University Press.
(2009). quantreg: Quantile Regression. R package version 4.44.
(1978). Regression quantiles. Econometrica, 46, 33–50.
(1982). Robust tests for heteroscedasticity based on regression quantiles. Econometrica, 50, 43–61.
(2009). Trajectories of physical discipline: Antecedents and developmental outcomes. Child Development, 80, 1385–1402.
(2010). Are spanking injunctions scientifically supported? Law and Contemporary Problems, 73, 57–87.
(2005). Comparing child outcomes of physical punishment and alternative disciplinary tactics: A meta-analysis. Clinical Child and Family Psychology Review, 80, 1–37.
(2008). Violence and serious theft: Risk and promotive factors from childhood to early adulthood. Mahwah, NJ: Erlbaum.
(2003). Resilience and protective factors. In , Early prevention of adult antisocial behaviour (pp. 130–204). New York, NY: Cambridge University Press.
(2012). Direct protective and buffering protective factors in the development of youth violence. American Journal of Preventive Medicine, 43(Suppl. 2), 8–23.
(2009). Universal prevention of antisocial development: Short- and long-term effects of a child- and parent-oriented program. Monatsschrift für Kriminologie und Strafrechtsreform.
[Monthly Journal for Criminology and Penal Law Reform] 2-3, 289–307.(1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample performance. Journal of Econometrics, 29, 305–325.
(2006). Robust statistics. Theory and methods. New York, NY: Wiley.
(1977). Data analysis and regression. New York, NY: Addison-Wesley.
(1991). Estimation of monotonic regression models under quantile restrictions. In , Nonparametric and semiparametric methods in econometrics and statistics (pp. 357–384). Cambridge, UK: Cambridge University Press.
(1984). Least median of squares regression. Journal of the American Statistical Assocication, 79, 871–880.
(1997). Family effects on individual differences in development. In , Perspectives on adjustment, risk, and disorder (pp. 115–137). New York, NY: Cambridge University Press.
(1995). Familien gestern und heute.
[Families yesterday and today] München, Germany: Quintessenz.(2002). Experimental and quasi-experimental designs for generalized causal inference. Boston, MA: Houghton-Mifflin.
(1996). Assessment of parenting practices in families of elementary school-age children. Journal of Clinical Child Psychology, 25, 317–329.
(2001). Beating the devil out of them: Corporal punishment in american families and its effects on children (2nd ed). New Brunswick, NJ: Transaction.
(2010a). Criminogenic effects of corporal punishment by parents. In , Transnational Criminology Manual (pp. 373–390). Amsterdam, Netherlands: Wolf Legal Publishing.
(2010b). Prevalence, societal causes, and trends in corporal punishment by parents in world perspective. Law and Contemporary Problems, 73, 1–30.
(2003). Violence by children against mothers in relation to violence between parents and corporal punishment by parents. Journal of Comparative Family Studies, 34, 41–60.
(2002). Modern applied statistics with S (4th ed.). New York, NY: Springer-Verlag.
(2008). Hands-on intermediate econometrics using R. Hackensack, NJ: World Scientific Publishers.
(2000). The centercept: An estimable and meaningful parameter. Psychological Science, 11, 434–436.
(1992). The cycle of violence. Washington, DC: National Institute of Justice.
(1998). How many discoveries have been lost by ignoring modern statistical methods? American Psychologist, 53, 300–314.
(2011). Modern statistics for the social and behavioral sciences: A practical introduction. Boca Raton, FL: CRC Press.
(2001). Econometric applications of high-breakdown robust regression techniques. Economics Letters, 71, 1–8.
(2004). Econometric computing with HC and HAC covariance matrix estimators. Journal of Statistical Software, 11, 1–17. Retrieved from www.jstatsoft.org/v11/i10/
