lmodel2: Model II regression
In lmodel2: Model II Regression
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
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
lm and aov.
data
A data frame containing the two variables specified in the formula.
range.y, range.x
Parametres for ranged major axis regression
(RMA). If range.y = NULL and range.x = NULL, RMA will
not be computed. If only one of them is NULL, the program will stop.
If range.y = "relative": variable y has a true zero
(relative-scale variable). If range.y = "interval": variable
y possibly includes negative values (interval-scale
variable). If range.x = "relative": variable x has a
true zero (relative-scale variable). If range.x = "interval":
variable x possibly includes negative values (interval-scale
variable)
nperm
Number of permutations for the tests. If nperm =
0, tests will not be computed.
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
(one-tailed, 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 (R-square) of the OLS regression.
P.param
2-tailed parametric P-value for the test of r and the OLS slope.
theta
Angle between the two OLS regression lines,
lm(y ~ x) and lm(x ~ y).
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
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## 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)
Example output
RMA was not requested: it will not be computed.
Model II regression
Call: lmodel2(formula = Predicted_by_model ~ Survival, data = mod2ex1,
nperm = 99)
n = 54 r = 0.8387315 r-square = 0.7034705
Parametric P-values: 2-tailed = 2.447169e-15 1-tailed = 1.223585e-15
Angle between the two OLS regression lines = 9.741174 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 0.6852956 0.6576961 33.33276 0.01
2 MA 0.4871990 0.7492103 36.84093 0.01
3 SMA 0.4115541 0.7841557 38.10197 NA
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 0.4256885 0.9449028 0.5388717 0.7765204
2 MA 0.1725753 0.7633080 0.6216569 0.8945561
3 SMA 0.1349629 0.6493905 0.6742831 0.9119318
Eigenvalues: 0.1332385 0.01090251
H statistic used for computing C.I. of MA: 0.007515993
Model II regression
Call: lmodel2(formula = Prey ~ Predators, data = mod2ex2, range.y =
"relative", range.x = "relative", nperm = 99)
n = 20 r = 0.8600787 r-square = 0.7397354
Parametric P-values: 2-tailed = 1.161748e-06 1-tailed = 5.808741e-07
Angle between the two OLS regression lines = 5.106227 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 20.02675 2.631527 69.19283 0.01
2 MA 13.05968 3.465907 73.90584 0.01
3 SMA 16.45205 3.059635 71.90073 NA
4 RMA 17.25651 2.963292 71.35239 0.01
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 12.490993 27.56251 1.858578 3.404476
2 MA 1.347422 19.76310 2.663101 4.868572
3 SMA 9.195287 22.10353 2.382810 3.928708
4 RMA 8.962997 23.84493 2.174260 3.956527
Eigenvalues: 269.8212 6.418234
H statistic used for computing C.I. of MA: 0.006120651
Model II regression
Call: lmodel2(formula = No_eggs ~ Mass, data = mod2ex3, range.y =
"relative", range.x = "relative", nperm = 99)
n = 11 r = 0.882318 r-square = 0.7784851
Parametric P-values: 2-tailed = 0.0003241737 1-tailed = 0.0001620869
Angle between the two OLS regression lines = 5.534075 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 19.766816 1.869955 61.86337 0.01
2 MA 6.656633 2.301728 66.51716 0.01
3 SMA 12.193785 2.119366 64.74023 NA
4 RMA 13.179672 2.086897 64.39718 0.01
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS -4.098376 43.63201 1.117797 2.622113
2 MA -36.540814 27.83295 1.604304 3.724398
3 SMA -14.576929 31.09957 1.496721 3.001037
4 RMA -18.475744 35.25317 1.359925 3.129441
Eigenvalues: 494.634 17.49327
H statistic used for computing C.I. of MA: 0.02161051
Model II regression
Call: lmodel2(formula = y ~ x, data = mod2ex4, range.y = "interval",
range.x = "interval", nperm = 99)
n = 100 r = 0.896898 r-square = 0.804426
Parametric P-values: 2-tailed = 1.681417e-36 1-tailed = 8.407083e-37
Angle between the two OLS regression lines = 6.218194 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 0.7713474 0.8648893 40.85618 0.01
2 MA 0.2905866 0.9602938 43.83962 0.01
3 SMA 0.2703395 0.9643118 43.95915 NA
4 RMA 0.3142417 0.9555996 43.69937 0.01
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 0.2663618 1.2763329 0.7794015 0.950377
2 MA -0.2121756 0.7477724 0.8695676 1.060064
3 SMA -0.1795065 0.6820701 0.8826059 1.053581
4 RMA -0.1848274 0.7702125 0.8651145 1.054637
Eigenvalues: 17.56697 0.9534124
H statistic used for computing C.I. of MA: 0.002438452
Model II regression
Call: lmodel2(formula = random_y ~ random_x, data = mod2ex5, range.y =
"interval", range.x = "interval", nperm = 99)
n = 100 r = -0.0837681 r-square = 0.007017094
Parametric P-values: 2-tailed = 0.4073269 1-tailed = 0.2036634
Angle between the two OLS regression lines = 80.41387 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Confidence interval = NA when the limits of the confidence interval
cannot be computed. This happens when the correlation is 0
or the C.I. includes all 360 deg. of the plane (H >= 1)
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 0.07074386 -0.08010978 -4.580171 0.2
2 MA 0.11293376 -0.60004584 -30.965688 0.2
3 SMA 0.14184407 -0.95632810 -43.721176 NA
4 RMA 0.26977945 -2.53296649 -68.456190 0.2
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS -0.1220525 0.26354020 -0.2711429 0.1109234
2 MA NA NA NA NA
3 SMA 0.1278759 0.15887844 -1.1662547 -0.7841884
4 RMA 0.0965427 -0.06826575 1.6330043 -0.3980472
Eigenvalues: 1.071318 0.8857145
H statistic used for computing C.I. of MA: 1.106884
No permutation test will be performed
Warning messages:
1: In lmodel2(y0 ~ x0, data = Ex6, "relative", "relative") :
MA: R-square = 0
2: In lmodel2(y0 ~ x0, data = Ex6, "relative", "relative") :
SMA: R-square = 0
3: In lmodel2(y0 ~ x0, data = Ex6, "relative", "relative") :
RMA: R-square = 0
Model II regression
Call: lmodel2(formula = y0 ~ x0, data = Ex6, range.y = "relative",
range.x = "relative")
n = 25 r = 0 r-square = 0
Parametric P-values: 2-tailed = 1 1-tailed = 0.5
Angle between the two OLS regression lines = 90 degrees
MA, SMA, RMA slopes = NA when the correlation is zero
Confidence interval = NA when the limits of the confidence interval
cannot be computed. This happens when the correlation is 0
or the C.I. includes all 360 deg. of the plane (H >= 1)
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 3 -9.420555e-17 -5.39758e-15 NA
2 MA NA NA NA NA
3 SMA NA NA NA NA
4 RMA NA NA NA NA
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 1.569391 4.430609 -0.4313449 0.4313449
2 MA NA NA NA NA
3 SMA NA NA NA NA
4 RMA NA NA NA NA
Eigenvalues: 2.083333 2.083333
H statistic used for computing C.I. of MA: NA
Warning message:
In plot.lmodel2(zero.res, "MA") :
R-square = 0: model and C.I. not drawn for MA, SMA or RMA
lmodel2 documentation built on May 2, 2019, 8:34 a.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
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 (one-tailed, 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 (R-square) of the OLS regression. |
P.param |
2-tailed parametric P-value 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)
|
Example output
RMA was not requested: it will not be computed.
Model II regression
Call: lmodel2(formula = Predicted_by_model ~ Survival, data = mod2ex1,
nperm = 99)
n = 54 r = 0.8387315 r-square = 0.7034705
Parametric P-values: 2-tailed = 2.447169e-15 1-tailed = 1.223585e-15
Angle between the two OLS regression lines = 9.741174 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 0.6852956 0.6576961 33.33276 0.01
2 MA 0.4871990 0.7492103 36.84093 0.01
3 SMA 0.4115541 0.7841557 38.10197 NA
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 0.4256885 0.9449028 0.5388717 0.7765204
2 MA 0.1725753 0.7633080 0.6216569 0.8945561
3 SMA 0.1349629 0.6493905 0.6742831 0.9119318
Eigenvalues: 0.1332385 0.01090251
H statistic used for computing C.I. of MA: 0.007515993
Model II regression
Call: lmodel2(formula = Prey ~ Predators, data = mod2ex2, range.y =
"relative", range.x = "relative", nperm = 99)
n = 20 r = 0.8600787 r-square = 0.7397354
Parametric P-values: 2-tailed = 1.161748e-06 1-tailed = 5.808741e-07
Angle between the two OLS regression lines = 5.106227 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 20.02675 2.631527 69.19283 0.01
2 MA 13.05968 3.465907 73.90584 0.01
3 SMA 16.45205 3.059635 71.90073 NA
4 RMA 17.25651 2.963292 71.35239 0.01
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 12.490993 27.56251 1.858578 3.404476
2 MA 1.347422 19.76310 2.663101 4.868572
3 SMA 9.195287 22.10353 2.382810 3.928708
4 RMA 8.962997 23.84493 2.174260 3.956527
Eigenvalues: 269.8212 6.418234
H statistic used for computing C.I. of MA: 0.006120651
Model II regression
Call: lmodel2(formula = No_eggs ~ Mass, data = mod2ex3, range.y =
"relative", range.x = "relative", nperm = 99)
n = 11 r = 0.882318 r-square = 0.7784851
Parametric P-values: 2-tailed = 0.0003241737 1-tailed = 0.0001620869
Angle between the two OLS regression lines = 5.534075 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 19.766816 1.869955 61.86337 0.01
2 MA 6.656633 2.301728 66.51716 0.01
3 SMA 12.193785 2.119366 64.74023 NA
4 RMA 13.179672 2.086897 64.39718 0.01
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS -4.098376 43.63201 1.117797 2.622113
2 MA -36.540814 27.83295 1.604304 3.724398
3 SMA -14.576929 31.09957 1.496721 3.001037
4 RMA -18.475744 35.25317 1.359925 3.129441
Eigenvalues: 494.634 17.49327
H statistic used for computing C.I. of MA: 0.02161051
Model II regression
Call: lmodel2(formula = y ~ x, data = mod2ex4, range.y = "interval",
range.x = "interval", nperm = 99)
n = 100 r = 0.896898 r-square = 0.804426
Parametric P-values: 2-tailed = 1.681417e-36 1-tailed = 8.407083e-37
Angle between the two OLS regression lines = 6.218194 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 0.7713474 0.8648893 40.85618 0.01
2 MA 0.2905866 0.9602938 43.83962 0.01
3 SMA 0.2703395 0.9643118 43.95915 NA
4 RMA 0.3142417 0.9555996 43.69937 0.01
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 0.2663618 1.2763329 0.7794015 0.950377
2 MA -0.2121756 0.7477724 0.8695676 1.060064
3 SMA -0.1795065 0.6820701 0.8826059 1.053581
4 RMA -0.1848274 0.7702125 0.8651145 1.054637
Eigenvalues: 17.56697 0.9534124
H statistic used for computing C.I. of MA: 0.002438452
Model II regression
Call: lmodel2(formula = random_y ~ random_x, data = mod2ex5, range.y =
"interval", range.x = "interval", nperm = 99)
n = 100 r = -0.0837681 r-square = 0.007017094
Parametric P-values: 2-tailed = 0.4073269 1-tailed = 0.2036634
Angle between the two OLS regression lines = 80.41387 degrees
Permutation tests of OLS, MA, RMA slopes: 1-tailed, tail corresponding to sign
A permutation test of r is equivalent to a permutation test of the OLS slope
P-perm for SMA = NA because the SMA slope cannot be tested
Confidence interval = NA when the limits of the confidence interval
cannot be computed. This happens when the correlation is 0
or the C.I. includes all 360 deg. of the plane (H >= 1)
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 0.07074386 -0.08010978 -4.580171 0.2
2 MA 0.11293376 -0.60004584 -30.965688 0.2
3 SMA 0.14184407 -0.95632810 -43.721176 NA
4 RMA 0.26977945 -2.53296649 -68.456190 0.2
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS -0.1220525 0.26354020 -0.2711429 0.1109234
2 MA NA NA NA NA
3 SMA 0.1278759 0.15887844 -1.1662547 -0.7841884
4 RMA 0.0965427 -0.06826575 1.6330043 -0.3980472
Eigenvalues: 1.071318 0.8857145
H statistic used for computing C.I. of MA: 1.106884
No permutation test will be performed
Warning messages:
1: In lmodel2(y0 ~ x0, data = Ex6, "relative", "relative") :
MA: R-square = 0
2: In lmodel2(y0 ~ x0, data = Ex6, "relative", "relative") :
SMA: R-square = 0
3: In lmodel2(y0 ~ x0, data = Ex6, "relative", "relative") :
RMA: R-square = 0
Model II regression
Call: lmodel2(formula = y0 ~ x0, data = Ex6, range.y = "relative",
range.x = "relative")
n = 25 r = 0 r-square = 0
Parametric P-values: 2-tailed = 1 1-tailed = 0.5
Angle between the two OLS regression lines = 90 degrees
MA, SMA, RMA slopes = NA when the correlation is zero
Confidence interval = NA when the limits of the confidence interval
cannot be computed. This happens when the correlation is 0
or the C.I. includes all 360 deg. of the plane (H >= 1)
Regression results
Method Intercept Slope Angle (degrees) P-perm (1-tailed)
1 OLS 3 -9.420555e-17 -5.39758e-15 NA
2 MA NA NA NA NA
3 SMA NA NA NA NA
4 RMA NA NA NA NA
Confidence intervals
Method 2.5%-Intercept 97.5%-Intercept 2.5%-Slope 97.5%-Slope
1 OLS 1.569391 4.430609 -0.4313449 0.4313449
2 MA NA NA NA NA
3 SMA NA NA NA NA
4 RMA NA NA NA NA
Eigenvalues: 2.083333 2.083333
H statistic used for computing C.I. of MA: NA
Warning message:
In plot.lmodel2(zero.res, "MA") :
R-square = 0: model and C.I. not drawn for MA, SMA or RMA
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