Computational Statistics

, Volume 32, Issue 4, pp 1453–1480 | Cite as

Testing the equality of several linear regression models

  • Filipe J. Marques
  • Carlos A. Coelho
  • Paulo C. Rodrigues
Original Paper
  • 204 Downloads

Abstract

The linear regression models are widely used in different research fields, and often there is the need to analyze if there are similarities between two or more different linear models or to verify if a given relation between two variables remains the same in different intervals of time, in particular in cases where small differences might make a big difference. Motivated by these problems the authors consider a test of equality of k linear regression models which is a simultaneous test of equality of slopes, intercepts and variances. In order to overcome the extreme difficulties that exist in the use of the exact distribution of the likelihood ratio test (LRT) statistic and to make this test reliable and easy to use, we propose the use of near-exact distributions to approximate the distribution of the LRT statistic, under \(H_0\), in the balanced case, and of new asymptotic approximations for the unbalanced case. The near-exact approximations are built by approximating one factor of an adequate factorization of the characteristic function of the logarithm of the LRT statistic and may be easily implemented. The asymptotic approximations are developed using an expansion for the ratio of gamma functions. The quality of these approximations is analyzed and confirmed. Power studies are conducted in order to better assess the performance of the test. Finally to illustrate the applicability of the test we consider a real data set of gross domestic product at market prices and final consumption expenditure in European countries and one tests the existence of similarities between countries.

Keywords

Characteristic function Hypotheses testing Mixtures Linear models Near-exact distributions 

Notes

Acknowledgements

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).

References

  1. Azaisa J-M, Delmas C, Rabier C-E (2014) Likelihood ratio test process for quantitative trait locus detection. Statistics 48:787–801CrossRefMATHMathSciNetGoogle Scholar
  2. Box GEP (1949) A general distribution theory for a class of likelihood criteria. Biometrika 36:317–346CrossRefMATHMathSciNetGoogle Scholar
  3. Bretz F, Hothorn T, Westfall P (2008) Multiple comparison procedures in linear models. In: Brito P (ed) COMPSTAT 2008. Proceedings in computational statistics. Physica-Verlag, HeidelbergGoogle Scholar
  4. Chen HJ, Yuan SP (1982) On concurrence of several linear regressions and applications to problems of enzyme kinetics. Statistics 11:395–409MathSciNetGoogle Scholar
  5. Coelho CA (1998) The generalized integer Gamma distribution—a basis for distributions in multivariate statistics. J Multivar Anal 64:86–102CrossRefMATHMathSciNetGoogle Scholar
  6. Coelho CA (2004) The generalized near-integer gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent beta random variables. J Multivar Anal 89:191–218CrossRefMATHMathSciNetGoogle Scholar
  7. Coelho CA, Marques FJ (2009) The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics. Linear Algebra Appl 430:2592–2606CrossRefMATHMathSciNetGoogle Scholar
  8. Coelho CA, Marques FJ (2010) Near-exact distributions for the independence and sphericity likelihood ratio test statistics. J Multivar Anal 101:583–593CrossRefMATHMathSciNetGoogle Scholar
  9. Coelho CA, Arnold BC, Marques FJ (2010) Near-exact distributions for certain likelihood ratio test statistics. J Stat Theory Pract 4:711–725CrossRefMathSciNetGoogle Scholar
  10. Coelho CA, Alberto RP (2012) On the distribution of the product of independent beta random variables applications. Technical report, CMA 12Google Scholar
  11. Deaton A (1992) Understanding consumption. Oxford University Press, OxfordCrossRefGoogle Scholar
  12. Johansen S (2000) A Bartlett correction factor for tests on the cointegrating relations. Econom Theory 16:740–778CrossRefMATHMathSciNetGoogle Scholar
  13. Krugman PR, Obstfeld M, Melitz MJ (2012) International economics: theory and policy. Pearson, HarlowGoogle Scholar
  14. Luke YL (1969) The special functions and their approximations. Academic Press Inc, LondonMATHGoogle Scholar
  15. Marques FJ, Coelho CA (2013) Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions. Comput Stat 28:2091–2115CrossRefMATHMathSciNetGoogle Scholar
  16. Marques FJ, Coelho CA, Arnold BC (2010) A general near-exact distribution theory for the most common likelihood ratio test statistics used in multivariate analysis. Test 20:180–203CrossRefMATHMathSciNetGoogle Scholar
  17. Moschopoulos PG (1986) New representations for the distribution function of a class of likelihood ratio criteria. J Stat Res 20(1&2):13–20MathSciNetGoogle Scholar
  18. Moschopoulos PG (1989) Tests of hypotheses on concomitant variables in linear models. Commun Stat Theory Methods 18(5):1735–1746CrossRefMATHMathSciNetGoogle Scholar
  19. Nielsen HB, Rahbek A (2007) The likelihood ratio test for cointegration ranks in the I(2) model. Econom Theory 23:615–637CrossRefMATHMathSciNetGoogle Scholar
  20. Nkurunziza S (2008) Likelihood ratio test for a special predator–prey system. Statistics 42:149–166CrossRefMATHMathSciNetGoogle Scholar
  21. Park J, Sinha B, Shah A, Xu D, Lin J (2015) Likelihood ratio tests for interval hypotheses with applications. Commun Stat 44(11):2351–2370CrossRefMATHMathSciNetGoogle Scholar
  22. Paternoster R, Brame R, Mazerolle P, Piquero A (1998) Using the correct statistical test for the equality of regression coefficients. Criminology 36:859–866CrossRefGoogle Scholar
  23. Solomon H, Stephens MA (1978) Approximations to density functions using Pearson curves. J Am Stat Assoc 73:153–160CrossRefGoogle Scholar
  24. Stöckl D, Dewitte K, Thienpont LM (1998) Validity of linear regression in method comparison studies: is it limited by the statistical model or the quality of the analytical input data? Clin Chem 44:2340–2346Google Scholar
  25. Tricomi FG, Erdélyi A (1951) The asymptotic expansion of a ratio of gamma functions. Pac J Math 1:133–142CrossRefMATHMathSciNetGoogle Scholar
  26. Wilks SS (1938) The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann Math Stat 9:60–62CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Filipe J. Marques
    • 1
    • 4
  • Carlos A. Coelho
    • 1
    • 4
  • Paulo C. Rodrigues
    • 2
    • 3
  1. 1.Center for Mathematics and Applications (CMA)NOVA University of LisbonLisbonPortugal
  2. 2.Federal University of BahiaSalvadorBrazil
  3. 3.CASTUniversity of TampereTampereFinland
  4. 4.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal

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