Multiple regression through the origin

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Description

Function lmorigin computes a multiple linear regression and performs tests of significance of the equation parameters (F-test of R-square and t-tests of regression coefficients) using permutations.

The regression line can be forced through the origin. Testing the significance in that case requires a special permutation procedure. This option was developed for the analysis of independent contrasts, which requires regression through the origin. A permutation test, described by Legendre & Desdevises (2009), is needed to analyze contrasts that are not normally distributed.

Usage

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lmorigin(formula, data, origin=TRUE, nperm=999, method=NULL, silent=FALSE)

Arguments

formula

A formula specifying the bivariate model, as in lm and aov.

data

A data frame containing the two variables specified in the formula.

origin

origin = TRUE (default) to compute regression through the origin; origin = FALSE to compute multiple regression with estimation of the intercept.

nperm

Number of permutations for the tests. If nperm = 0, permutation tests will not be computed. The default value is nperm = 999. For large data files, the permutation test is rather slow since the permutation procedure is not compiled.

method

method = "raw" computes t-tests of the regression coefficients by permutation of the raw data. method = "residuals" computes t-tests of the regression coefficients by permutation of the residuals of the full model. If method = NULL, permutation of the raw data is used to test the regression coefficients in regression through the origin; permutation of the residuals of the full model is used to test the regression coefficients in ordinary multiple regression.

silent

Informative messages and the time to compute the tests will not be written to the R console if silent=TRUE. Useful when the function is called by a numerical simulation function.

Details

The permutation F-test of R-square is always done by permutation of the raw data. When there is a single explanatory variable, permutation of the raw data is used for the t-test of the single regression coefficient, whatever the method chosen by the user. The rationale is found in Anderson & Legendre (1999).

The print.lmorigin function prints out the results of the parametric tests (in all cases) and the results of the permutational tests (when nperm > 0).

Value

reg

The regression output object produced by function lm.

p.param.t.2tail

Parametric probabilities for 2-tailed tests of the regression coefficients.

p.param.t.1tail

Parametric probabilities for 1-tailed tests of the regression coefficients. Each test is carried out in the direction of the sign of the coefficient.

p.perm.t.2tail

Permutational probabilities for 2-tailed tests of the regression coefficients.

p.perm.t.1tail

Permutational probabilities for 1-tailed tests of the regression coefficients. Each test is carried out in the direction of the sign of the coefficient.

p.perm.F

Permutational probability for the F-test of R-square.

origin

TRUE is regression through the origin has been computed, FALSE if multiple regression with estimation of the intercept has been used.

nperm

Number of permutations used in the permutation tests.

method

Permutation method for the t-tests of the regression coefficients: method = "raw" or method = "residuals".

var.names

Vector containing the names of the variables used in the regression.

call

The function call.

Author(s)

Pierre Legendre, Universite de Montreal

References

Anderson, M. J. and Legendre, P. (1999) An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model. Journal of Statistical Computation and Simulation, 62, 271–303.

Legendre, P. and Desdevises, Y. (2009) Independent contrasts and regression through the origin. Journal of Theoretical Biology, 259, 727–743.

Sokal, R. R. and Rohlf, F. J. (1995) Biometry - The principles and practice of statistics in biological research. Third edition. New York: W. H. Freeman.

Examples

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## Example 1 from Sokal & Rohlf (1995) Table 16.1
## SO2 air pollution in 41 cities of the USA
data(lmorigin.ex1)
out <- lmorigin(SO2 ~ ., data=lmorigin.ex1, origin=FALSE, nperm=99)
out

## Example 2: Contrasts computed on the phylogenetic tree of Lamellodiscus
## parasites. Response variable: non-specificity index (NSI); explanatory
## variable: maximum host size. Data from Table 1 of Legendre & Desdevises
## (2009).
data(lmorigin.ex2)
out <- lmorigin(NSI ~ MaxHostSize, data=lmorigin.ex2, origin=TRUE, nperm=99)
out

## Example 3: random numbers
y <- rnorm(50)
X <- as.data.frame(matrix(rnorm(250),50,5))
out <- lmorigin(y ~ ., data=X, origin=FALSE, nperm=99)
out

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