Testing the equality of several linear regression models

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.

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References

  1. Azaisa J-M, Delmas C, Rabier C-E (2014) Likelihood ratio test process for quantitative trait locus detection. Statistics 48:787–801

    Article  MATH  MathSciNet  Google Scholar 

  2. Box GEP (1949) A general distribution theory for a class of likelihood criteria. Biometrika 36:317–346

    Article  MATH  MathSciNet  Google 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, Heidelberg

  4. Chen HJ, Yuan SP (1982) On concurrence of several linear regressions and applications to problems of enzyme kinetics. Statistics 11:395–409

    MathSciNet  Google Scholar 

  5. Coelho CA (1998) The generalized integer Gamma distribution—a basis for distributions in multivariate statistics. J Multivar Anal 64:86–102

    Article  MATH  MathSciNet  Google 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 89:191–218

    Article  MATH  MathSciNet  Google 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 430:2592–2606

    Article  MATH  MathSciNet  Google Scholar 

  8. Coelho CA, Marques FJ (2010) Near-exact distributions for the independence and sphericity likelihood ratio test statistics. J Multivar Anal 101:583–593

    Article  MATH  MathSciNet  Google Scholar 

  9. Coelho CA, Arnold BC, Marques FJ (2010) Near-exact distributions for certain likelihood ratio test statistics. J Stat Theory Pract 4:711–725

    Article  MathSciNet  Google Scholar 

  10. Coelho CA, Alberto RP (2012) On the distribution of the product of independent beta random variables applications. Technical report, CMA 12

  11. Deaton A (1992) Understanding consumption. Oxford University Press, Oxford

    Google Scholar 

  12. Johansen S (2000) A Bartlett correction factor for tests on the cointegrating relations. Econom Theory 16:740–778

    Article  MATH  MathSciNet  Google Scholar 

  13. Krugman PR, Obstfeld M, Melitz MJ (2012) International economics: theory and policy. Pearson, Harlow

    Google Scholar 

  14. Luke YL (1969) The special functions and their approximations. Academic Press Inc, London

    Google 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 28:2091–2115

    Article  MATH  MathSciNet  Google 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 20:180–203

    Article  MATH  MathSciNet  Google Scholar 

  17. Moschopoulos PG (1986) New representations for the distribution function of a class of likelihood ratio criteria. J Stat Res 20(1&2):13–20

    MathSciNet  Google Scholar 

  18. Moschopoulos PG (1989) Tests of hypotheses on concomitant variables in linear models. Commun Stat Theory Methods 18(5):1735–1746

    Article  MATH  MathSciNet  Google Scholar 

  19. Nielsen HB, Rahbek A (2007) The likelihood ratio test for cointegration ranks in the I(2) model. Econom Theory 23:615–637

    Article  MATH  MathSciNet  Google Scholar 

  20. Nkurunziza S (2008) Likelihood ratio test for a special predator–prey system. Statistics 42:149–166

    Article  MATH  MathSciNet  Google 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):2351–2370

    Article  MATH  MathSciNet  Google Scholar 

  22. Paternoster R, Brame R, Mazerolle P, Piquero A (1998) Using the correct statistical test for the equality of regression coefficients. Criminology 36:859–866

    Article  Google Scholar 

  23. Solomon H, Stephens MA (1978) Approximations to density functions using Pearson curves. J Am Stat Assoc 73:153–160

    Article  Google 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 44:2340–2346

    Google Scholar 

  25. Tricomi FG, Erdélyi A (1951) The asymptotic expansion of a ratio of gamma functions. Pac J Math 1:133–142

    Article  MATH  MathSciNet  Google Scholar 

  26. Wilks SS (1938) The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann Math Stat 9:60–62

    Article  MATH  Google Scholar 

Download references

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).

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Correspondence to Filipe J. Marques.

Appendix: Empirical power values

Appendix: Empirical power values

Balanced case

See Tables 14, 15, 16, 17, 18 and 19.

Table 14 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\alpha _1=\alpha _2=0\) and significance level 0.1
Table 15 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\beta _1=\beta _2=2\) and significance level 0.1
Table 16 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\sigma _1^2=\sigma _2^2=1\) and significance level 0.1
Table 17 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\alpha _1=\alpha _2=0\) and significance level 0.01
Table 18 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\beta _1=\beta _2=2\) and significance level 0.01
Table 19 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\sigma _1^2=\sigma _2^2=1\) and significance level 0.01

Unbalanced case

See Tables 20, 21, 22, 23, 24 and 25.

Table 20 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\alpha _1=\alpha _2=0\) and significance level 0.1
Table 21 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\beta _1=\beta _2=2\) and significance level 0.1
Table 22 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\sigma _1^2=\sigma _2^2=1\) and significance level 0.1
Table 23 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\alpha _1=\alpha _2=0\) and significance level 0.01
Table 24 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\beta _1=\beta _2=2\) and significance level 0.01
Table 25 Empirical power values of the test, assuming \(\alpha _1=0\), \(\beta _1=2\) and \(\sigma _1^2=1\) and for \(\sigma _1^2=\sigma _2^2=1\) and significance level 0.01

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Marques, F.J., Coelho, C.A. & Rodrigues, P.C. Testing the equality of several linear regression models. Comput Stat 32, 1453–1480 (2017). https://doi.org/10.1007/s00180-016-0703-1

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Keywords

  • Characteristic function
  • Hypotheses testing
  • Mixtures
  • Linear models
  • Near-exact distributions