Skip to main content
SpringerLink
Log in

Group-mean-centering independent variables in multi-level models is dangerous

  • Published:
Quality & Quantity Aims and scope Submit manuscript

Abstract

Group-mean centering of independent variables in multi-level models is widely practiced and widely recommended. For example, in cross-national studies of educational performance, family background is scored as a deviation from the country mean for student’s family background. We argue that this is usually a serious mis-specification, introducing bias and random measurement error with all their attendant vices. We examine five diverse examples of “real world” analyses using large, high quality datasets on topics of broad interest in the social sciences. In all of them, consistent with much (but not all) of the technical literature, group-mean centering substantially distorts results. Moreover the distortions are large, substantively important differences pointing towards seriously incorrect interpretations of important social processes. We therefore recommend that group-mean centering be abandoned.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Theory implies that the natural log of the number of books is the proper measure (Evans et al. 2014: Hypothesis 1) so we treat that as the natural units. Group-mean centered scoring is then the natural log minus the country mean of the natural log.

  2. Hox (2010) recommends a minimum sample size of 30 level-2 cases per uncorrelated level-2 predictor to avoid potential bias in the level-2 variance estimates, so this sample size is sufficient.

  3. Multi-level models can be written in two mathematically equivalent ways, either as two equations (one for the individual level and another for the second level) or as a single equation combining both (obtained by substituting the second-level equation into the individual-level equation). For present purposes the single combined equation is simpler and clearer.

  4. In our example that would be an individual effect for mother's education and a national level effect for GDP per capita but no further national level effects linked to mother's education. In a different and more complex model envisioning a national level contextual effect of mother's education in addition to its individual effect—perhaps on the argument that nations where most mothers are well educated typically has better quality teachers and spends more on its schools—things would be different. Models of this sort are taken up in Appendix A.

  5. This is the same logic that applies to mobility scores (Blau and Duncan 1967, Ch. 5) which is why they are only occasionally optimal (for example Kelley and Kelley 2009, Table 6.6).

  6. When there are other individual-level variables in addition to X, the effect of random error will usually be to reduce X's effect but often also to increase the effect of some other correlated variables. For example, random error in measuring parents' occupational status will typically increase the effect of parents' education.

  7. Models which have a set of individual level variables and also include all of the corresponding aggregate variables are of this sort. An example would be a model with age, sex, education, occupation, supervision, income, political party, and vote as individual-level variables which also includes all the corresponding aggregate level variables—viz country mean age, country mean gender, country mean education, country mean occupation, country mean percent supervisors, country mean income, country mean party ID, and country mean vote. Then group-mean centering the individual level variables gives a model that is just a mathematically equivalent reparameterization of the model with the individual level and aggregate level variables scored conventionally. Nothing gained but, in principle, nothing lost. However the model may not be reliably estimable because it has so many aggregate level variables (Hox's rule of thumb asks for 30 separate countries for each (uncorrelated) country-level variable, so for 240 countries in our example—rather more countries than actually exist). Moreover many of those aggregate level variables are not conceptually sensible.

References

  • Albright, J.J., and Marinova, D.M.: Estimating multilevel models using SPSS, Stata, SAS, and R. Indiana University, pp. 1–35 (2010)

  • Bickel, R.: Multilevel Analysis for Applied Research. The Guilford Press, New York (2007)

    Google Scholar 

  • Blalock, H.M.: Some implications of random measurement error for causal inferences. Am. J. Sociol. 71, 37–47 (1965)

    Article  Google Scholar 

  • Blalock, H.M.: The identification problem and theory building: The case of status inconsistency. Am. Soc. Rev. 31(1), 52–61 (1966)

    Article  Google Scholar 

  • Blalock, H.M.: Contextual effects models: Theoretical and methodological issues. Annu. Rev Sociol. 10, 353–372 (1984)

    Article  Google Scholar 

  • Blau, P.M., Duncan, O.D.: The American occupational structure. Free Press, New York (1967)

    Google Scholar 

  • Bollen, K.A.: Structural Equations With Latent Variables. Wiley, New York (1989a)

    Book  Google Scholar 

  • Bollen, K.A.: The consequences of measurement error. In: Bollen, K.A. (ed.) Structural Equations With Latent Variables, pp. 151–169. Wiley, New York (1989b)

    Google Scholar 

  • Breznau, N., Lykes, V., Kelley, J., Evans, M.D.R.: A clash of civilizations? Preferences for religious political leaders in 81 nations. J. Sci. Study Relig. 50, 671–691 (2011)

    Article  Google Scholar 

  • Chiu, M.M., Chow, B.W.Y.: Classroom discipline across forty-one countries: school, economic, and cultural differences. J. Cross Cult. Psychol. 42, 516–533 (2011)

    Article  Google Scholar 

  • Cohen, J., Cohen, P.: Applied multiple regression/correlation analysis for the behavioral sciences, 2nd edn. Lawrence Erlbaum, Hillsdale (1983)

    Google Scholar 

  • Davis, J.A.: The campus as a frog pond: an application of the theory of relative deprivation to career decisions of college men. Am. J. Sociol. 72, 17–31 (1966)

    Article  Google Scholar 

  • Diener, E., Tay, L.: The religion paradox: if religion makes people happy, why are so many dropping out? J. Pers. Soc. Psychol. 101, 1278–1290 (2011)

    Article  Google Scholar 

  • DiPrete, T.A., Forristal, J.D.: Multilevel models: methods and substance. Annu Rev. Sociol. 20, 331–357 (1994)

    Article  Google Scholar 

  • Duncan, O.D.: Methodological issues in the analysis of social mobility. In: Smelser, N.J., Lipset, S.M. (eds.) Social structure and mobility in economic development, pp. 51–97. Aldine, Chicago (1966)

    Google Scholar 

  • Duncan, O.D., Hodge, R.W.: Education and occupational mobility a regression analysis. Am. J. Sociol. 68, 629–644 (1963)

    Article  Google Scholar 

  • Enders, C., Tofighi, D.: Centering predictor variables in cross-sectional multilevel models: a new look at an old issue. Psychol. Methods 12, 121–138 (2007)

    Article  Google Scholar 

  • Evans, M.D.R., Kelley, J.: Effect of Family Structure on Life Satisfaction: Australian Evidence. Soc. Indic. Res. 69, 303–353 (2004)

    Article  Google Scholar 

  • Evans, M.D.R., Kelley, J., Sikora, J.: Scholarly culture and academic performance in 42 nations. Soc. Forces 92(4), 1573–1605 (2014)

    Article  Google Scholar 

  • Evans, M.D.R., Kelley, J., Sikora, J., Treiman, D.J.: Family scholarly culture and educational success: evidence from 27 nations. Res. Soc. Stratif. Mobil. 28, 171–197 (2010)

    Article  Google Scholar 

  • Hawkes, R.K.: Some methodological problems in explaining social mobility. Am. Sociol. Rev. 37, 294–300 (1972)

    Article  Google Scholar 

  • Hodge, R.W., Kraus, V., et al.: Intergeneration occupational mobility and income. Soc. Sci. Res. 15(4), 297–322 (1986)

    Article  Google Scholar 

  • Hofmann, D.A., Griffin, M.A., Gavin, M.B.: The application of hierarchical linear modeling to organizational research. In: Klein, K.J., Kozlowski, S.W.J. (eds.) Multilevel Theory, Research, and Methods in Organizations: Foundations, Extensions, and New Directions, pp. 467–511. Jossey-Bass, San Francisco (2000)

    Google Scholar 

  • Hox, J.J.: Multilevel Analysis: Techniques and Applications, 2nd edn. Routledge, New York (2010)

    Google Scholar 

  • Jackson, E.F., Curtis, R.F.: Effects of vertical obility and status inconsistency: A body of negative evidence. Am. Soc. Rev. 37(6), 701–713 (1972)

    Article  Google Scholar 

  • Joreskog, K.G.: A general method for analysis of covariance structures. Biometrika 57, 239–252 (1970)

    Article  Google Scholar 

  • Kelley, J.: Causal chain models for the socioeconomic career. Am. Sociol. Rev. 38, 481–493 (1973)

    Article  Google Scholar 

  • Kelley, J.: Methods and pitfalls in the analysis of social mobility: class of origin, class of destination, and mobility per se. In: Turner, F.C. (ed.) Social Mobility and Political Attitudes: Comparative Perspectives, pp. 233–251. Transaction Publishers, New Brunswick (1992)

    Google Scholar 

  • Kelley, J., de Graaf, N.D.: National context, parental socialization, and religious belief: results from 15 nations. Am. Sociol. Rev. 62, 639–659 (1997)

    Article  Google Scholar 

  • Kelley, S.M.C., Kelley, C.G.E.: Subjective social mobility: Data from 30 Nations. In: Haller, M., Jowell, R., Smith, T. (eds.) Charting the globe: The international social survey programme 1984-2009, chap. 6, pp. 106–124. Routledge, New York (2009)

    Google Scholar 

  • Kenny, D.A., Kashy, D.A., Bolger, N.: Data analysis in social psychology. In: Gilbert, D.T., Fiske, S.T., Lindzey, G. (eds.) The Handbook of Social Psychology, 4th edn, pp. 233–268. McGraw-Hill Companies, Inc, New York (1998)

    Google Scholar 

  • Kreft, I.G.G., de Leew, J., Aiken, L.S.: The effect of different forms of centering in hierarchical linear models. Multivar. Behav. Res. 30, 1–21 (1995)

    Article  Google Scholar 

  • Kromrey, J.D., Foster-Johnson, L.: Mean centering in moderated multiple regression: much ado about nothing. Educ. Psychol. Measur. 58, 42–67 (1998)

    Article  Google Scholar 

  • Krymkowski, D.H.: Measurement in the comparative study of the process of stratification. Soc. Sci. Res. 17, 191–205 (1988)

    Article  Google Scholar 

  • Lopez-Turley, R.N.: Is relative deprivation beneficial? The effects of richer and poorer neighbors on children’s outcomes. J. Community Psychol. 30, 671–686 (2002)

    Article  Google Scholar 

  • Maas, C., Hox, J.: Sufficient sample sizes for multilevel modeling. Methodology 1(3), 86–92 (2005)

    Article  Google Scholar 

  • Marks GN.: Are school-SES effects statistical artefacts? Evidence from longitudinal population data. Oxford Rev. Educ. 41, in press (2015)

  • Nezlek, J.B.: Multilevel random coefficient analysis of event-and-interval-contingent data in social and personality psychology research. Pers. Soc. Psychol. Bull. 27, 771–785 (2001)

    Article  Google Scholar 

  • Nezlek, J.B., Zyzniewski, L.E.: Using hierarchical linear modeling to analyze grouped data. Group Dyn. Theory, Res. Pract. 2(4), 313–320 (1998)

    Article  Google Scholar 

  • O’Connor, S., Fischer, R.: Predicting societal corruption across time: values, wealth, or institutions? J. Cross Cult. Psychol. 43, 644–659 (2011)

    Article  Google Scholar 

  • Olsen, M.E., Tully, J.C.: Socioeconomic-ethnic status inconsistency and preference for political change. Am. Soc. Rev. 37(5), 560–574 (1972)

    Article  Google Scholar 

  • Paccagnella, O.: Centering or not centering in multilevel models? The role of the group mean and the assessment of group effects. Eval. Rev. 30, 66–85 (2006)

    Article  Google Scholar 

  • Perna, L.W., Titus, M.A.: The relationship between parental involvement and social capital and college enrollment: an examination of racial/ethnic group differences. J. Higher Educ. 76, 485–518 (2005)

    Article  Google Scholar 

  • Preacher, K.: A primer on interaction effects in multiple regression analysis. University of North Carolina, Chapel Hill (2003)

    Google Scholar 

  • Raudenbush, S.W., Bryk, A.S.: Hierarchical linear models: applications and data analysis methods. Sage Publications, Thousand Oaks (2002)

    Google Scholar 

  • Roscigno, V.J., Crowley, M.L.: Rurality, institutional disadvantage, and achievement/attainment. Rural Sociol. 66, 268–293 (2001)

    Article  Google Scholar 

  • Ross, C.E., Mirowsky, J.: A comparison of life-event-weighting schemes: change, undesirability, and effect-proportional indices. J. Health Soc. Behav. 20, 166–177 (1979)

    Article  Google Scholar 

  • Runciman, W.G.: Relative deprivation and social justice. Penguin, Harmondsworth (1966)

    Google Scholar 

  • Ryabov, I., van Hook, J.: School segregation and academic achievement on Hispanic children. Soc. Sci. Res. 36, 767–788 (2007)

    Article  Google Scholar 

  • Sampson, R.J., Raudenbush, S.W.: Seeing disorder: neighborhood stigma and the social construction of “broken windows”. Soc. Psychol. Q. 67, 319–342 (2004)

    Article  Google Scholar 

  • Snijders, Tom A.B., Bosker, R.J.: Multilevel analysis, 2nd edn. Sage Publications, Thousand Oaks (2012)

    Google Scholar 

  • Treiman, D.J.: Status discrepancies and prejudice. Am. J. Soc. 71(6), 651–654 (1966)

    Article  Google Scholar 

  • Treiman, D.J., Terrell, K.: Sex and the process of status attainment: a comparison of working women and men. Am. Sociol. Rev. 40, 174–200 (1975)

    Article  Google Scholar 

  • Tucker, J.S., Sinclair, R.R., Thomas, J.L.: The multilevel effects of occupational stressors on soldiers’ well-being, organizational attachment, and readiness. J. Occup. Health Psychol. 10, 276–299 (2005)

    Article  Google Scholar 

  • Zagorski, K., Kelley, J., Evans, M.D.R.: Economic development and happiness: evidence from 32 nations. Polish Sociol. Rev. 2010, 3–20 (2010)

    Google Scholar 

  • Zagorski, K., Evans, M.D.R., Kelley, J., Piotrowska, K.: Does national income inequality affect individuals' quality of life in Europe? inequality, happiness, finances, and health. Soc. Indic. Res. 117, 1089–1110 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Nevada, Reno, USA

    Jonathan Kelley, M. D. R. Evans, Jennifer Lowman & Valerie Lykes

Authors
  1. Jonathan Kelley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. D. R. Evans
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jennifer Lowman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Valerie Lykes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan Kelley.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 38 kb)

Appendix: grand mean centering

Appendix: grand mean centering

Advantages have sometimes been claimed for scoring variables as deviations from their sample means, sometimes called “grand-mean centering”.

  1. (1)

    In times past, it sometimes afforded real gains in the precision of estimation when variables were measured on very different scales. Some writers still claim that grand-mean centering reduces multicollinearity, particularly when the regression includes many interactions, and most especially when these are cross-level interactions, (Bickel 2007; Preacher 2003). However, that computational advantage evaporated long ago with improvements in computer hardware and software. Nonetheless, the practice persists in some subfields, e.g. social psychology. Formal statistical analysis and practical experience in many applications make it clear that the hoped-for benefits in reduction of collinearity and enhanced precision of estimation of interaction effects do not hold (Cohen and Cohen 1983, p. 865; Kromrey and Foster-Johnson 1998).

  2. (2)

    Another putative advantage of grand mean centering sometimes mentioned in the literature is that it allows one to interpret the intercept as the predicted mean on the dependent variable when all the predictors are set to zero (Paccagnella 2006), but this “advantage” became nugatory with the incorporation of calculation of predicted values/regression simulations for any combinations of values on predictors that one chooses (for example using Stata’s “predict” and “margins” commands).

  3. (3)

    It is also sometimes said that grand-mean centering facilitates regression coefficient interpretation, particularly for cross-level interactions when a variable is continuous (Bickel 2007; Kenny et al. 1998; Hox 2010), although the interpretation of regression effects, especially with interactions, is much better clarified with the use of predicted values/regression simulations and confidence bands around regression effects. With modern software, this is straightforward.

  4. (4)

    Hox (2010) reports that convergence tends to be achieved more frequently and analyses run faster using grand-mean centering, which could be real advantages, although slow convergence in the original metrics is often an important sign that a model is not stable and will not replicate well across datasets.

  5. (5)

    The once-touted advantage of interpretability for variables without an original metric has vanished in favor of the use of predicted-value graphics for interpretation and the use of scoring procedures with a stronger logic, such as effect-proportional scaling (Krymkowski 1988; Ross and Mirowsky 1979; Treiman and Terrell 1975) or related ordinal-probit-based methods (Evans and Kelley 2004).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kelley, J., Evans, M.D.R., Lowman, J. et al. Group-mean-centering independent variables in multi-level models is dangerous. Qual Quant 51, 261–283 (2017). https://doi.org/10.1007/s11135-015-0304-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11135-015-0304-z

Keywords

Search

Navigation