Model II regression
Description
This function computes model II simple linear regression using the following methods: ordinary least squares (OLS), major axis (MA), standard major axis (SMA), and ranged major axis (RMA). The model only accepts one response and one explanatory variable.
Usage
1 
Arguments
formula 
A formula specifying the bivariate model, as in

data 
A data frame containing the two variables specified in the formula. 
range.y, range.x 
Parametres for ranged major axis regression
(RMA). If 
nperm 
Number of permutations for the tests. If 
Details
Model II regression should be used when the two variables in the
regression equation are random, i.e. not controlled by the
researcher. Model I regression using least squares underestimates
the slope of the linear relationship between the variables when they
both contain error. Ordinary least squares (OLS) is, however, appropriate
in some cases as a model II regression model; see the “Model
II User's guide, R edition” which you can read using command
vignette("mod2user")
.
The model II regression methods of ordinary least squares (OLS),
major axis (MA), standard major axis (SMA), and ranged major axis
(RMA) are described in Legendre and Legendre (1998, Section
10.3.2). OLS, MA, and SMA are also described in Sokal and Rohlf
(1995). The PDF document “Model II User's guide, R edition”
provided with this function contains a tutorial for model II
regression, and can be read with command
vignette("mod2user")
.
The plot
function plots the data points together with one of the
regression lines, specified by method="OLS"
, method="MA"
(default),
method="SMA"
, or method="RMA"
, and its 95 percent confidence interval.
Value
The default output provides the regression output. It draws
information from a list, produced by function lmodel2
, which
contains the following elements:
y 
The response variable. 
x 
The explanatory variable. 
regression.results 
A table with rows corresponding to the four regression methods. Column 1 gives the method name, followed by the intercept and slope estimates, the angle between the regression line and the abscissa, and the permutational probability (onetailed, for the tail corresponding to the sign of the slope estimate). 
confidence.intervals 
A table with rows corresponding to the four regression methods. The method name is followed by the parametric 95 the intercept and slope estimates. 
eigenvalues 
Eigenvalues of the bivariate dispersion, computed during major axis regression. 
H 
The H statistic used for computing the confidence interval of the major axis slope. Notation following Sokal and Rohlf (1995). 
n 
Number of objects. 
r 
Correlation coefficient. 
rsquare 
Coefficient of determination (Rsquare) of the OLS regression. 
P.param 
2tailed parametric Pvalue for the test of r and the OLS slope. 
theta 
Angle between the two OLS regression lines,

nperm 
Number of permutations for the permutation tests. 
epsilon 
Any value smaller than epsilon is considered to be zero. 
info.slope 
Information about the slope notation when r = 0. 
info.CI 
Information about the confidence limits notation when the slope is infinite. 
call 
Call of the function. 
Note
The package exports only the main functions lmodel2
,
plot.lmodel2
and lines.lmodel2
. Much of the work is
done by internal functions which are not directly visible, but you
can use triple colon to see or directly use these functions (e.g.,
lmodel2:::print.lmodel2
). Internal functions that perform
essential parts of the analysis are MA.reg
, SMA.reg
,
CLma
, CLsma
and permutest.lmodel2
.
Author(s)
Pierre Legendre, Departement de Sciences Biologiques, Universite de Montreal
References
Legendre, P. and L. Legendre. 1998. Numerical ecology, 2nd English edition. Elsevier Science BV, Amsterdam.
Sokal, R. R. and F. J. Rohlf. 1995. Biometry – The principles and practice of statistics in biological research. 3rd edition. W. H. Freeman, New York.
See Also
A tutorial (file “Model II User's guide, R edition”) is provided
with this function, and can be read within R session using command
vignette("mod2user", package="lmodel2")
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  ## The example data files are described in more detail in the
## \dQuote{Model II User's guide, R edition} tutorial.
## Example 1 (surgical unit data)
data(mod2ex1)
Ex1.res < lmodel2(Predicted_by_model ~ Survival, data=mod2ex1, nperm=99)
Ex1.res
plot(Ex1.res)
## Example 2 (eagle rays and Macomona)
data(mod2ex2)
Ex2.res < lmodel2(Prey ~ Predators, data=mod2ex2, "relative", "relative", 99)
Ex2.res
op < par(mfrow = c(1,2))
plot(Ex2.res, "SMA")
plot(Ex2.res, "RMA")
par(op)
## Example 3 (cabezon spawning)
op < par(mfrow = c(1,2))
data(mod2ex3)
Ex3.res < lmodel2(No_eggs ~ Mass, data=mod2ex3, "relative", "relative", 99)
Ex3.res
plot(Ex3.res, "SMA")
plot(Ex3.res, "RMA")
par(op)
## Example 4 (highly correlated random variables)
op < par(mfrow=c(1,2))
data(mod2ex4)
Ex4.res < lmodel2(y ~ x, data=mod2ex4, "interval", "interval", 99)
Ex4.res
plot(Ex4.res, "OLS")
plot(Ex4.res, "MA")
par(op)
# Example 5 (uncorrelated random variables)
data(mod2ex5)
Ex5.res < lmodel2(random_y ~ random_x, data=mod2ex5, "interval", "interval", 99)
Ex5.res
op < par(mfrow = c(2,2))
plot(Ex5.res, "OLS")
plot(Ex5.res, "MA")
plot(Ex5.res, "SMA")
plot(Ex5.res, "RMA")
par(op)
## Example 6 where cor(y,x) = 0 by construct (square grid of points)
y0 = rep(c(1,2,3,4,5),5)
x0 = c(rep(1,5),rep(2,5),rep(3,5),rep(4,5),rep(5,5))
plot(x0, y0)
Ex6 = as.data.frame(cbind(x0,y0))
zero.res = lmodel2(y0 ~ x0, data=Ex6, "relative", "relative")
print(zero.res)
op < par(mfrow = c(1,2))
plot(zero.res, "OLS")
plot(zero.res, "MA")
par(op)
