Program for multiple linear regression with permutation test. The program includes regression through the origin.
Pierre LegendreDecember 1999
Département de Sciences Biologiques
Université de Montréal
This program computes a multiple linear regression and performs tests of significance of the equation parameters using permutations. In this new version, the regression line can be forced through the origin. Permutation testing is recommended when the residuals of the regression equation are not normally distributed; it does not solve the problems caused by heteroscedasticity. Two permutation methods are available in the program:
 Permutation of the raw data.
 Permutation of residuals of the full regression model (ter Braak 1990, 1992).
 For data with nonnormal error structure, permutation tests had type I error closer to the nominal significance level alpha and greater power than the normaltheory ttest.
 Permutation of raw data and permutation of residuals gave asymptotically equivalent results in most situations and provided good approximative tests for partial regression coefficients.
 Permutation of raw data resulted in unstable (and often inflated) type I error when the covariable contained an extreme outlier, whether or not there was collinearity between predictor variables, or the data were normal or nonnormal. This problem was not amended with increasing sample sizes.
 The presence of outliers in the covariable did not adversely affect the tests based on permutation of residuals.
Program availability

Macintosh version
 Fortran source code for Macintosh (file Regression_test.f), which can be compiled using a Fortran compiler. The user may modify the Parameter statement at the beginning of the program, which fixes the size (nmax, mmax) of the largest data matrix which may be analysed.
 Compiled versions of the program for any Macintosh computer, including MacOS X
 Program documentation, in Adobe Acrobat format
 Sample files
 32bit DOS version (The executable file is a Win32 "console" executable, not a DOS executable. Therefore it cannot run under plain DOS, nor in a DOS window under Windows 3.x, only in Windows 95/98 or Windows NT consoles)
References:
 Anderson, M. J. & P. Legendre. 1999. An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model. Journal of Statistical Computation and Simulation 62: 271303.
 Legendre, P. & Legendre, L. 1998. Numerical Ecology, 2nd English edition. Elsevier Science BV, Amsterdam. xv + 853 pages.
 "Legendre_Desdevises_JTB_in_press.pdf"
 Sokal, R. R. & F. J. Rohlf. 1995. Biometry  The principles and practice of statistics in biological research. 3rd edition. W. H. Freeman, New York.
 ter Braak, C. J. F. 1990. Update notes: CANOCO version 3.10. Agricultural Mathematics Group, Wageningen.
 ter Braak, C. J. F. 1992. Permutation versus bootstrap significance tests in multiple regression and ANOVA. 7986 in: K.H. Jöckel, G. Rothe & W. Sendler [eds.] Bootstrapping and related techniques. SpringerVerlag, Berlin.