Journal of Statistical Theory and Practice

, Volume 12, Issue 1, pp 66–81 | Cite as

The simultaneous assessment of normality and homoscedasticity in linear fixed effects models

  • Ye Yang
  • Thomas MathewEmail author


This article investigates the problem of simultaneously testing the normality and homoscedasticity assumptions in a linear fixed effects model when we have grouped data. This has been facilitated by the assumption of a smooth alternative to the normal distribution. The smooth alternative is specified using Legendre polynomials, and the score statistic is derived under two scenarios: a common smooth alternative across the different groups, or different smooth alternatives across the different groups. A data-driven approach available in the literature is used for determining the order of the polynomials. For the null distribution of the score statistic, the accuracy of the asymptotic chi-squared distribution is numerically investigated under a one-way fixed effects model with balanced and unbalanced data. The results are illustrated with an example.


Balanced data Legendre polynomials one-way fixed effects model score test smooth alternative unbalanced data 

AMS Subject Classification

62F03 62J20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andrews, D. Ε., and Α. Μ. Herzberg. 1985. Data: A collection of problems from many fields for students and research workers. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
  2. Bogdan, M. 1996. Data driven smooth tests for bivariate normality. Preprint IM PWr 42/96, Technical University of Wroclaw, Wroclaw, Poland.Google Scholar
  3. Bogdan, M. 1999. Data driven smooth tests for bivariate normality. Journal of Multivariate Analysis 68:26–53.MathSciNetCrossRefGoogle Scholar
  4. Chang, C.-H., N. Pal, and J.-J. Lin. 2016. A revisit to test the equality of variances of several populations. Communications in Statistics—Simulation and Computation, to appear. MathSciNetCrossRefGoogle Scholar
  5. Inglot, T., and T. Ledwina. 1996. Asymptotic optimality of data-driven Neymans tests for uniformity. Annals of Statistics 24:1982–2019.MathSciNetCrossRefGoogle Scholar
  6. Inglot, T., W. Kallenberg, and T. Ledwina. 1997. Data driven smooth tests for composite hypotheses. Annals of Statistics 25:1222–1250.MathSciNetCrossRefGoogle Scholar
  7. Janic, Α., and T. Ledwina. 2009. Data-driven smooth tests for a location-scale family revisited. Journal of Statistical Theory and Practice 3:645–664.MathSciNetCrossRefGoogle Scholar
  8. Kallenberg, W., and T. Ledwina. 1995. Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests. Annals of Statistics 23:1594–1608.MathSciNetCrossRefGoogle Scholar
  9. Kallenberg, W., and T. Ledwina. 1997a. Data-driven smooth tests for composite hypotheses: Comparison of powers. Journal of Statistical Computation and Simulation 59:101–121.MathSciNetCrossRefGoogle Scholar
  10. Kallenberg, W., and T. Ledwina. 1997b. Data-driven smooth tests when the hypothesis is composite. Journal of the American Statistical Association 92:1094–1104.MathSciNetCrossRefGoogle Scholar
  11. Kallenberg, W., T. Ledwina, and E. Rafajlowicz. 1997. Testing bivariate independence and normality. Sankhyã, Series A 59:42–59.MathSciNetzbMATHGoogle Scholar
  12. Ledwina, T. 1994. Data-driven version of Neymans smooth test of fit. Journal of the American Statistical Association 89:1000–1005.MathSciNetCrossRefGoogle Scholar
  13. Mardia, K. V., and J. T. Kent. 1991. Rao score tests for goodness of fit and independence. Biometrika 78:355–363.MathSciNetCrossRefGoogle Scholar
  14. Montgomery, D. C. 2012. Design and analysis of experiments, 8th Ed. New York, NY: John Wiley.Google Scholar
  15. Neyman, J. 1937. Smooth test for goodness of fit. Skandinaviska Aktuarietidskrift 20:149–199.zbMATHGoogle Scholar
  16. Peña, Ε. Α., and Ε. Η. Slate. 2006. Global validation of linear model assumptions. Journal of the American Statistical Association 101:341–354.MathSciNetCrossRefGoogle Scholar
  17. Rayner, J. C. W., O. Thas, and D. J. Best. 2009. Smooth tests of goodness of fit: Using R. New York, NY: John Wiley.CrossRefGoogle Scholar
  18. Reaven, G. M., and R G. Miller. 1979. An attempt to define the nature of chemical diabetes using a multidimensional analysis. Diabetologia 16:17–24.CrossRefGoogle Scholar
  19. Yang, Y. 2016. The simultaneous assessment of normality and homoscedasticity in some linear models. Doctoral dissertation submitted to the University of Maryland Baltimore County, Baltimore, MD.Google Scholar

Copyright information

© Grace Scientific Publishing 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland Baltimore CountyBaltimoreUSA

Personalised recommendations