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\title{Model II regression user's guide, R edition}

\author{Pierre Legendre}

\address{D{\'e}partement de sciences biologiques, Universit{\'e} de
Montr{\'e}al, C.P. 6128, succursale Centre-ville, Montr{\'e}al,
Qu{\'e}bec H3C 3J7, Canada}
\email{Pierre.Legendre@umontreal.ca}

\date{\Sexpr{packageDate('lmodel2')} (version \Sexpr{packageVersion('lmodel2')})}

\begin{document}

\setkeys{Gin}{width=0.55\linewidth}
\SweaveOpts{strip.white=true}
<<echo=false>>=
require(lmodel2)
options(width=72)
figset <- function() par(mar=c(4,4,1,1)+.1)
options(SweaveHooks = list(fig = figset))
@


\maketitle

\tableofcontents

Function \texttt{lmodel2} computes model II simple linear regression
using the following methods: major axis (MA), standard major axis
(SMA), ordinary least squares (OLS), and ranged major axis
(RMA). Information about these methods is available, for instance, in
section 10.3.2 of \citet{Legendre-Legendre98} and in sections 14.13
and 15.7 of \citet{SokalRohlf95}\footnote{In Sokal and Rohlf
  (Biometry, 2nd edition, 1981: 551), the numerical result for MA
  regression for the example data set is wrong. The mistake has been
  corrected in the 1995 edition.}. Parametric 95\% confidence
intervals are computed for the slope and intercept parameters. A
permutation test is available to determine the significance of the
slopes of MA, OLS and RMA and also for the correlation
coefficient. This function represents an evolution of a
\textsc{Fortran} program written in 2000 and 2001.

Bartlett's three-group model II regression method, described by the
above mentioned authors, is not computed by the program because it
suffers several drawbacks. Its main handicap is that the regression
lines are not the same depending on whether the grouping (into three
groups) is made based on $x$ or $y$. The regression line is not
guaranteed to pass through the centroid of the scatter of points and
the slope estimator is not symmetric, i.e. the slope of the regression
$y = f(x)$ is not the reciprocal of the slope of the regression $x =
f(y)$.

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; see example in chapter \ref{sec:exa4}
(p. \pageref{sec:exa4}). Detailed recommendations follow.

\section{Recommendations on the use of model II regression methods}

Considering the results of simulation studies,
\citet{Legendre-Legendre98} offer the following recommendations to
ecologists who have to estimate the parameters of the functional
linear relationships between variables that are random and measured
with error (Table \ref{tab:recommend}).

\begin{table}
\label{tab:recommend}
  \caption{Application of the model II regression methods. The numbers
    in the left-hand column refer to the corresponding paragraphs in
    the text.}
\begin{Small}
\begin{tabular}{lllc}
\toprule
Par.& Method &Conditions of application& Test possible\\
\midrule
1 &OLS &Error on $y \gg$ error on $x$ &Yes\\
3 &MA& Distribution is bivariate normal & Yes \\
& &Variables are in the same physical units or dimensionless \\
& &Variance of error about the same for $x$ and $y$ \\

4 & &
Distribution is bivariate normal\\
& &Error variance on each axis proportional to variance of\\
& &corresponding variable \\

4a & RMA &
Check scatter diagram: no outlier & Yes\\
4b &SMA& Correlation $r$ is significant& No\\
5 &OLS& Distribution is not bivariate normal& Yes\\
& &Relationship between $x$ and $y$ is linear\\
6 &OLS& To compute forecasted (fitted) or predicted $y$ values& Yes\\
& &(Regression equation and confidence intervals are irrelevant) \\
7& MA& To compare observations to model predictions &Yes\\
\bottomrule
\end{tabular}
\end{Small}
\end{table}

\begin{enumerate}
\item If the magnitude of the random variation (i.e. the error
  variance\footnote{Contrary to the sample variance, the error
    variance on $x$ or $y$ cannot be estimated from the data. An
    estimate can only be made from knowledge of the way the variables
    were measured.}) on the response variable $y$ is much larger
  (i.e. more than three times) than that on the explanatory variable
  $x$, use OLS.  Otherwise, proceed as follows.
\item Check whether the data are approximately bivariate normal,
  either by looking at a scatter diagram or by performing a formal
  test of significance. If not, attempt transformations to render them
  bivariate normal. For data that are or can be made to be reasonably
  bivariate normal, consider recommendations 3 and 4. If not, see
  recommendation 5.
\item For bivariate normal data, use major axis (MA) regression if
  both variables are expressed in the same physical units
  (untransformed variables that were originally measured in the same
  units) or are dimensionless (e.g. log-transformed variables), if it
  can reasonably be assumed that the error variances of the variables
  are approximately equal.

  When no information is available on the ratio of the error variances
  and there is no reason to believe that it would differ from 1, MA
  may be used provided that the results are interpreted with
  caution. MA produces unbiased slope estimates and accurate
  confidence intervals \citep{Jolicoeur90}.

  MA may also be used with dimensionally heterogeneous variables when
  the purpose of the analysis is (1) to compare the slopes of the
  relationships between the same two variables measured under
  different conditions (e.g. at two or more sampling sites), or (2) to
  test the hypothesis that the major axis does not significantly
  differ from a value given by hypothesis (e.g. the relationship $E =
  b_1 m$ where, according to the famous equation of Einstein, $b_1 =
  c^2$, $c$ being the speed of light in vacuum).

\item For bivariate normal data, if MA cannot be used because the
  variables are not expressed in the same physical units or the error
  variances on the two axes differ, two alternative methods are
  available to estimate the parameters of the functional linear
  relationship if it can reasonably be assumed that the error variance
  on each axis is proportional to the variance of the corresponding
  variable, i.e. (the error variance of $y$ / the sample variance of
  $y$) (the error variance of $x$ / the sample variance of $x$). This
  condition is often met with counts (e.g. number of plants or
  animals) or log-transformed data \citep{McArdle88}.

\begin{enumerate}
\item Ranged major axis regression (RMA) can be used. The method is
  described below. Prior to RMA, one should check for the presence of
  outliers, using a scatter diagram of the objects.
\item Standard major axis regression (SMA) can be used. One should
  first test the significance of the correlation coefficient ($r$) to
  determine if the hypothesis of a relationship is supported.  No SMA
  regression equation should be computed when this condition is not
  met.

  This remains a less-than-ideal solution since SMA slope estimates
  cannot be tested for significance.  Confidence intervals should also
  be used with caution: simulations have shown that, as the slope
  departs from $\pm 1$, the SMA slope estimate is increasingly biased
  and the confidence interval includes the true value less and less
  often. Even when the slope is near $\pm 1$ (e.g. example \S
  \ref{sec:exa5}), the confidence interval is too narrow if $n$ is
  very small or if the correlation is weak.
\end{enumerate}

\item If the distribution is not bivariate normal and the data cannot
  be transformed to satisfy that condition (e.g. if the distribution
  possesses two or several modes), one should wonder whether the slope
  of a regression line is really an adequate model to describe the
  functional relationship between the two variables. Since the
  distribution is not bivariate normal, there seems little reason to
  apply models such as MA, SMA or RMA, which primarily describe the
  first principal component of a bivariate normal distribution. So,
  (1) if the relationship is linear, OLS is recommended to estimate
  the parameters of the regression line. The significance of the slope
  should be tested by permutation, however, because the distributional
  assumptions of parametric testing are not satisfied. (2) If a
  straight line is not an appropriate model, polynomial or nonlinear
  regression should be considered.

\item When the purpose of the study is not to estimate the parameters
  of a functional relationship, but simply to forecast or predict
  values of $y$ for given $x$'s, use OLS in all cases.  OLS is the
  only method that minimizes the squared residuals in y. The OLS
  regression line itself is meaningless. Do not use the standard error
  and confidence bands, however, unless $x$ is known to be free of
  error \citep[p.~545, Table~14.3]{SokalRohlf95}; this warning applies
  in particular to the 95\% confidence intervals computed for OLS by
  this program.
\item Observations may be compared to the predictions of a
  statistical or deterministic model (e.g. simulation model) in order
  to assess the quality of the model. If the model contains random
  variables measured with error, use MA for the comparison since
  observations and model predictions should be in the same units.

  If the model fits the data well, the slope is expected to be $1$ and
  the intercept $0$. A slope that significantly differs from $1$
  indicates a difference between observed and simulated values which
  is proportional to the observed values. For relative-scale
  variables, an intercept which significantly differs from $0$
  suggests the existence of a systematic difference between
  observations and simulations \citep{Mesple.ea96}.

\item With all methods, the confidence intervals are large when n is
  small; they become smaller as n goes up to about 60, after which
  they change much more slowly. Model II regression should ideally be
  applied to data sets containing 60 observations or more. Some of the
  examples presented below have fewer observations; they are only
  presented for illustration.
\end{enumerate}

\section{Ranged major axis regression}

Ranged major axis regression (RMA) is described in
\citep[p.~551]{Legendre-Legendre98}. It is computed as follows:

\begin{enumerate}
\item Transform the $y$ and $x$ variables into $y'$ and $x'$,
  respectively, whose range is 1. Two formulas are available for
  ranging, depending on the nature of the variables:
  \begin{itemize}
  \item For variables whose variation is expressed relative to an
    arbitrary zero (interval-scale variables, e.g. temperature in
    $^{\circ}$C), the formula for ranging is:
\begin{equation}
\label{eq:range1}
y'_i = \frac{y_i - y_{\min}}{y_{\max} - y_{\min}} \quad \text{or} \quad
x'_i = \frac{x_i - x_{\min}}{x_{\max}-x_{\min}}
\end{equation}

\item For variables whose variation is expressed relative to a true
  zero value (ratio-scale or relative-scale variables, e.g. species
  abundances, or temperature expressed in $^{\circ}$K), the recommended
  formula for ranging assumes a minimum value of $0$;
  eq. \ref{eq:range1} reduces to:
\begin{equation}
y'_i = \frac{y_i}{y_{\max}} \quad \text{or} \quad
x'_i = \frac{x_i}{x_{\max}}
\end{equation}
\end{itemize}
\item Compute MA regression between the ranged variables $y'$ and
  $x'$. Test the significance of the slope estimate by permutation if
  needed.
\item Back-transform the estimated slope, as well as its confidence
  interval limits, to the original units by multiplying them by the
  ratio of the ranges:
\begin{equation}
b_1 = b'_1 \frac{y_{\max} - y_{\min}}{x_{\max} - x_{\min}}
\end{equation}

\item Recompute the intercept $b_0$ and its confidence interval
  limits, using the original centroid ($\bar x$, $\bar y$) of the
  scatter of points and the estimates of the slope $b_1$ and its
  confidence limits:
\begin{equation}
b_0= \bar y - b_1 \bar x
\end{equation}
\end{enumerate}

The RMA slope estimator has several desirable properties when the
variables $x$ and $y$ are not expressed in the same units or when the
error variances on the two axes differ.  (1) The slope estimator
scales proportionally to the units of the two variables: the position
of the regression line in the scatter of points remains the same
irrespective of any linear change of scale of the variables. (2) The
estimator is sensitive to the covariance of the variables; this is not
the case for SMA. (3) Finally, and contrary to SMA, it is possible to
test the hypothesis that an RMA slope estimate is equal to a stated
value, in particular 0 or 1. As in MA, the test may be done either by
permutation, or by comparing the confidence interval of the slope to
the hypothetical value of interest. Thus, whenever MA regression
cannot be used because of incommensurable units or because the error
variances on the two axes differ, RMA regression can be used. There is
no reason, however, to use RMA when MA is justified.

Prior to RMA, one should check for the presence of outliers, using a
scatter diagram of the objects. RMA should not be used in the presence
of outliers because they cause important changes to the estimates of
the ranges of the variables. Outliers that are not aligned fairly well
with the dispersion ellipse of the objects may have an undesirable
influence on the slope estimate. The identification and treatment of
outliers is discussed in \citep[Chapter~13.4]{SokalRohlf95}. Outliers
may, in some cases, be eliminated from the data set, or they may be
subjected to a winsorizing procedure described by these authors.

\section{Input file}

A data frame with objects in rows and variables in columns.

\section{Output file}

The output file obtained by \texttt{print.lmodel2} contains the
following results:

\begin{enumerate}
\item The call to the function.
\item General regression statistics: number of objects ($n$),
  correlation coefficient ($r$), coefficient of determination ($r^2$)
  of the OLS regression, parametric $P$-values (2-tailed, one-tailed)
  for the test of the correlation coefficient and the OLS slope, angle
  between the two OLS regression lines, \texttt{lm(y \~\ x)} and
  \texttt{lm(x \~\ y)}.
\item 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).

\item A table with rows corresponding to the four regression
  methods. The method name is followed by the parametric 95\%
  confidence intervals (2.5 and 97.5 percentiles) for the intercept
  and slope estimates.
\item The eigenvalues of the bivariate dispersion, computed during
  major axis regression, and the $H$ statistic used for computing the
  confidence interval of the major axis slope (notation following
  \citealt{SokalRohlf95}).

  For the slopes of MA, OLS and RMA, the permutation tests are carried
  out using the slope estimates $b$ themselves as the reference
  statistics. In OLS simple linear regression, a permutation test of
  significance based on the $r$ statistic is equivalent to a
  permutation test based on the pivotal $t$-statistic associated with
  $b_{OLS}$ \citep[p.~24]{Legendre-Legendre98}. On the other hand, across
  the permutations, the slope estimate ($b_{OLS}$) differs from $r$ by a
  constant ($s_y/s_x$) since $b_{OLS} = r_{xy} s_y/s_x$, so that $b_{OLS}$ and
  $r$ are equivalent statistics for permutation testing. As a
  consequence, a permutation test of $b_{OLS}$ is equivalent to a
  permutation test carried out using the pivotal $t$-statistic
  associated with $b_{OLS}$. This is not the case in multiple linear
  regression, however, as shown by \citet{AnderLeg99}.
\end{enumerate}

If the objective is simply to assess the relationship between the two
variables under study, one can simply compute the correlation
coefficient r and test its significance. A parametric test can be used
when the assumption of binormality can safely be assumed to hold, or a
permutation test when it cannot.

For the intercept of OLS, the confidence interval is computed using
the standard formulas found in textbooks of statistics; results are
identical to those of standard statistical software. No such formula,
providing correct $\alpha$-coverage, is known for the other three
methods. In the program, the confidence intervals for the intercepts
of MA, SMA and RMA are computed by projecting the bounds of the
confidence intervals of the slopes onto the ordinate; this results in
an underestimation of these confidence intervals.

\begin{figure}
  \centering
\resizebox{\linewidth}{!}{
  \begin{tikzpicture}
    \draw[->, very thick] (0,0) -- (6,0);
    \draw[<-, very thick]  (3,3) -- (3,-3);
    \draw[->, very thick] (8,0) -- (14, 0);
    \draw[->, very thick] (11,-3) -- (11,3);
    \draw[->, thick] (6.8,-2) -- (8.6,-2);
    \draw[<-, thick] (7.7,-1.1) -- (7.7,-2.8);
    \draw[font=\footnotesize] (8.2,-1.6) node {I};
    \draw[font=\footnotesize] (7.2,-1.6) node {II};
    \draw[font=\footnotesize] (7.2,-2.4) node {III};
    \draw[font=\footnotesize] (8.2,-2.4) node {IV};
    \draw[dashed] (3,0) -- (3+3/2.75,3) node[above]{$b_{1 \inf} = 2.75$};
    \draw (3,0) -- (3-3/2.75,-3) node[at start,sloped,above]{Lower bound
    of CI};
    \draw (3,0) -- (3+3/5.67,-3) node[midway,sloped,above]{MA
      regression line} node[below]{$b_1 = -5.67$};
    \draw (3,0) -- (3+2.7/1.19, -2.7) node[midway,sloped,above]{Upper
      bound of CI} node[below]{$b_{1 \sup}=-1.19$};
    \draw (11,0) -- (11-3/5.67,3);
    \draw[dashed] (11,0) -- (11+3/5.67,-3)
    node[at start,sloped,below]{Upper bound of CI} node[below]{$b_{1 \sup}
      = -5.67$};
    \draw (11,0) -- (11+3/2.75, 3) node[near end,sloped,above]{MA
      regression line} node[right]{$b_1 = 2.75$};
    \draw (11,0) -- (11+2/0.84, 2) node[near end,sloped,above]{Lower
      bound of CI} node[right]{$b_{1 \inf} =0.84$};
    \draw[font=\large] (0,3) node{(a)};
    \draw[font=\large] (8,3) node{(b)};
  \end{tikzpicture}
}
  \caption{(a) If a MA regression line has the lower bound of its
    confidence interval (C.I.) in quadrant III, this bound has a
    positive slope ($+2.75$ in example). (b) Likewise, if a MA
    regression line has the upper bound of its confidence interval in
    quadrant II, this bound has a negative slope ($-5.67$ in example).}
  \label{fig:rma}
\end{figure}
In MA or RMA regression, the bounds of the confidence interval (C.I.)
of the slope may, on occasions, lie outside quadrants I and IV of the
plane centred on the centroid of the bivariate distribution. When the
\emph{lower bound} of the confidence interval corresponds to a line in
quadrant III (Fig. \ref{fig:rma}a), it has a positive slope; the RMA
regression line of example in chapter \ref{sec:exa5}
(p. \pageref{sec:exa5}) provides an example of this
phenomenon. Likewise, when the \emph{upper bound} of the confidence
interval corresponds to a line in quadrant II (Fig. \ref{fig:rma}b),
it has a negative slope. In other instances, the confidence interval
of the slope may occupy all 360$^{\circ}$ of the plane, which results
in it having no bounds. The bounds are then noted 0.00000; see chapter
\ref{sec:exa5} (p. \pageref{sec:exa5}).

In SMA or OLS, confidence interval bounds cannot lie outside quadrants
I and IV. In SMA, the regression line always lies at a $+45^{\circ}$
or $-45^{\circ}$ angle in the space of the standardized variables; the
SMA slope is a back-transformation of $\pm45^{\circ}$ to the units of
the original variables. In OLS, the slope is always at an angle closer
to zero than the major axis of the dispersion ellipse of the points,
i.e. it always underestimates the MA slope in absolute value.


\section{Examples}
\subsection{Surgical unit data}
\label{sec:exa1}

\subsubsection{Input data}

This example compares observations to the values forecasted by a
model. A hospital surgical unit wanted to forecast survival of
patients undergoing a particular type of liver surgery. Four
explanatory variables were measured on patients. The response variable
Y was survival time, which was $\log_{10}$-transformed. The data are
described in detail in Section 8.2 of \citet{Neter.ea96} who also
provide the original data sets. The data were divided in two groups of
54 patients. The first group was used to construct forecasting models
whereas the second group was reserved for model validation. Several
regression models were studied. One of them, which uses variables $X_3 =$
enzyme function test score and $X_4$ = liver function test score, is used
as the basis for the present example. The multiple regression equation
is the following:


\begin{equation*}
\hat Y = 1.388778 + 0.005653 X_3 + 0.139015 X_4
\end{equation*}
This equation was applied to the second data set (also 54 patients) to
produce forecasted survival times. In the present example, these
values are compared to the observed survival times. Fig. \ref{fig:ex1}
shows the scatter diagram with $\log_{10}$(observed survival time) in
abscissa and forecasted values in ordinate. The MA regression line is
shown with its 95\% confidence region. The $45^{\circ}$ line, which
would correspond to perfect forecasting, is also shown for comparison.

\subsubsection{Output file}

MA, SMA and OLS equations, 95\% C.I., and tests of significance, were
obtained with the following R commands. The RMA method, which is
optional, was not computed since MA is the only appropriate method in
this example.

<<>>=
data(mod2ex1)
Ex1.res <- lmodel2(Predicted_by_model ~ Survival, data=mod2ex1, nperm=99)
Ex1.res
@
\begin{SCfigure}
<<fig=true,echo=false,results=hide>>=
plot(Ex1.res, centro = TRUE, xlab="log(observed survival time", ylab="Forecasted")
abline(diff(colMeans(mod2ex1)), 1, lty=2)
legend("topleft", c("MA regression", "Confidence limits", "45 degree line"), col=c("red", "grey", "black"), lty=c(1,1,2))
@
\caption{Scatter diagram of the 2.5 Example 1 data showing the major
  axis (MA) regression line and its 95\% confidence region. The
  45$^{\circ}$ line is drawn for reference. The cross indicates the
  centroid of the bivariate distribution. The MA regression line
  passes through this centroid.}
\label{fig:ex1}
\end{SCfigure}

The interesting aspect of the MA regression equation in this example
is that the regression line is not parallel to the 45$^{\circ}$ line
drawn in Fig. \ref{fig:ex1}. The 45$^{\circ}$ line is not included in
the 95\% confidence interval of the MA slope, which goes from
$\tan^{-1}(0.62166) = 31.87$ to $\tan^{-1}(0.89456) = 41.81$. The
Figure shows that the forecasting equation overestimated survival
below the mean and underestimated it above the mean. The OLS
regression line, which is often (erroneously) used by researchers for
comparisons of this type, would show an even greater discrepancy
(33.3$^{\circ}$ angle) from the 45$^{\circ}$ line, compared to the MA
regression line (36.8$^{\circ}$ angle).

\subsection{Eagle rays and \emph{Macomona} bivalves}
\label{sec:exa2}

\subsubsection{Input data}

The following table presents observations at 20 sites from a study on
predator-prey relationships \citep{Hines.ea97}. $y$ is the number of
bivalves (\emph{Macomona liliana}) larger than 15 mm in size, found in
0.25 m$^2$ quadrats of sediment; $x$ is the number of sediment
disturbance pits of a predator, the eagle ray (\emph{Myliobatis
  tenuicaudatus}), found within circles of a 15 m radius around the
bivalve quadrats.


The variables $x$ and $y$ are expressed in the same physical units and
are estimated with sampling error, and their distribution is
approximately bivariate normal. The error variance is not the same for
$x$ and $y$ but, since the data are animal counts, it seems reasonable
to assume that the error variance along each axis is proportional to
the variance of the corresponding variable. The correlation is
significant: $r = 0.86$, $p < 0.001$. RMA and SMA are thus appropriate
for this data set; MA and OLS are not. Fig. \ref{fig:ex2} shows the
scatter diagram.  The various regression lines are presented to allow
their comparison.

\subsubsection{Output file}

MA, SMA, OLS and RMA regression equations, confidence intervals, and
tests of significance, were obtained with the following R commands.
That the 95\% confidence intervals of the SMA and RMA intercepts do
not include 0 may be due to different reasons: (1) the relationship
may not be perfectly linear; (2) the C.I. of the intercepts are
underestimated; (3) the predators (eagle rays) may not be attracted to
sampling locations containing few prey (bivalves).
<<>>=
data(mod2ex2)
Ex2.res = lmodel2(Prey ~ Predators, data=mod2ex2, "relative","relative",99)
Ex2.res
@
\begin{SCfigure}
<<fig=true,echo=false>>=
plot(Ex2.res, confid=FALSE, xlab="Eagle rays (predators)", ylab="Bivalves (prey)", main = "", centr=TRUE)
lines(Ex2.res, "OLS", col=1, confid=FALSE)
lines(Ex2.res, "SMA", col=3, confid=FALSE)
lines(Ex2.res, "RMA", col=4, confid=FALSE)
legend("topleft", c("OLS", "MA", "SMA", "RMA"), col=1:4, lty=1)
@
\caption{Scatter diagram of the Example 2 data (number of bivalves as
  a function of the number of eagle rays) showing the major axis (MA),
  standard major axis (SMA), ordinary 50 least-squares (OLS) and
  ranged major axis (RMA) regression lines. SMA and RMA are the
  appropriate regression lines in this example. The cross indicates
  the centroid of the bivariate distribution. The four regression
  lines pass through this centroid.  }
\label{fig:ex2}
\end{SCfigure}

[1] In this table of the output file, the rows correspond,
respectively, to the MA, OLS and RMA slopes and to the coefficient of
correlation $r$ (\texttt{Corr}). \texttt{Stat.} is the value of the
statistic being tested for significance. As explained in the
\emph{Output file section}, the statistic actually used by the program
for the test of the MA slope, in this example, is the inverse of the
$b_{MA}$ slope estimate ($1/3.46591 = 0.28852$) because the reference
value of the statistic in this permutation test must not exceed
1. One-tailed probabilities (\texttt{One-tailed p}) are computed in
the direction of the sign of the coefficient. For a one-tailed test in
the upper tail (i.e. for a coefficient with a positive sign), $p =$ (EQ
+ GT)/(Number of permutations + 1). For a test in the lower tail
(i.e. for a coefficient with a negative sign), $p =$ (LT + EQ)/(Number
of permutations + 1), where
\begin{itemize}
\item LT is the number of values under permutation that are smaller
  than the reference value;
\item EQ is the number of values under permutation that are equal to
  the reference value of the statistic, plus 1 for the reference value
  itself;
\item GT is the number of values under permutation that are greater
  than the reference value.
\end{itemize}

\subsection{Cabezon spawning}%
\label{sec:exa3}%
\subsubsection{Input data}

The following table presents data used by
\citet[Box~14.12]{SokalRohlf95} to illustrate model II regression
analysis. They concern the mass ($x$) of unspawned females of a
California fish, the cabezon (\emph{Scorpaenichthys marmoratus}), and
the number of eggs they subsequently produced ($y$). One may be
interested to estimate the functional equation relating the number of
eggs to the mass of females before spawning. The physical units of the
variables are as in the table published by \citet[p.~546]{SokalRohlf95}.

Since the variables are in different physical units and are estimated
with error, and their distribution is approximately bivariate normal,
RMA and SMA are appropriate for this example; MA is inappropriate. The
OLS regression line is meaningless; in model II regression, OLS should
only be used for forecasting or prediction. It is plotted in
Fig. \ref{fig:ex3} only to allow comparison.

The RMA and SMA regression lines are nearly indistinguishable in this
example. The slope of RMA can be tested for significance ($H0$:
$b_{RMA} = 0$), however, whereas the SMA slope cannot. The 95\%
confidence intervals of the intercepts of RMA and SMA, although
underestimated, include the value 0, as expected if a linear model
applies to the data: a female with a mass of 0 is expected to produce
no egg.

Another interesting property of RMA and SMA is that their estimates of
slope and intercept change proportionally to changes in the units of
measurement. One can easily verify that by changing the decimal places
in the Example 2 data file and recomputing the regression
equations. RMA and SMA share this property with OLS. MA regression
does not have this property; this is why it should only be used with
variables that are in the same physical units, as those of Example 1.

\subsubsection{Output file}

MA, SMA, OLS and RMA equations, 95\% C.I., and tests of significance, were
obtained with the following R commands:
<<>>=
data(mod2ex3)
Ex3.res = lmodel2(No_eggs ~ Mass, data=mod2ex3, "relative", "relative", 99)
Ex3.res
@
\begin{SCfigure}
<<fig=true,echo=false>>=
plot(Ex3.res, method="OLS", conf = FALSE, centroid=TRUE, main="", xlab = "Fish mass (x100 g)", ylab = "No. of eggs", col=1)
lines(Ex3.res, "SMA", col=2, conf = FALSE)
lines(Ex3.res, "RMA", col=3, conf = FALSE)
legend("topleft", c("OLS", "SMA", "RMA"), col=1:3, lty=1)
@
\caption{Scatter diagram of the Example 3 data (number of eggs
  produced as a function of the mass of unspawned females) with the
  ranged major axis (RMA), standard major axis (SMA) and ordinary
  least-squares (OLS) regression lines. RMA and SMA are the
  appropriate regression lines in this example. The cross indicates
  the centroid of the bivariate distribution. The three regression
  lines pass through this centroid.}
\label{fig:ex3}
\end{SCfigure}

\subsection{Highly correlated random variables}
\label{sec:exa4}

\subsubsection{Input data}

\citet{Mesple.ea96} generated a variable $X$ containing 100 values
drawn at random from a uniform distribution in the interval $[0,
10]$. They then generated two other variables, $N_1$ and $N_2$,
containing values drawn at random from a normal distribution
$\mathcal{N}(0, 1)$. These variables were combined to create two new
variables $x = (X + N_1)$ and $y = (X + N_2)$. The relationship
constructed in this way between $x$ and $y$ is perfect and should have
a slope of 1, despite the fact that there is normal error added
independently to $x$ and $y$.

\subsubsection{Output file}

The various model II regression methods were applied to this data set
with the following results, were obtained with the following R
commands:
<<>>=
data(mod2ex4)
Ex4.res = lmodel2(y ~ x, data=mod2ex4, "interval", "interval", 99)
Ex4.res
@

The noticeable aspect is that with OLS regression, the confidence
interval of the slope does not include the value 1 and the confidence
interval of the intercept does not include the value 0. The OLS slope
underestimates the real slope of the bivariate functional
relationship, which is 1 by construct in this example. This
illustrates the fact that OLS, considered as model I regression
method, is inadequate to estimate the slope of the functional
relationship between these variables. As a model II regression method,
OLS would only be appropriate to predict the values $\hat{y}$ from $x$
(point 6 in Table \ref{tab:recommend}).

With all the other model II regression methods, the confidence
intervals of the slopes include the value 1 and the confidence
intervals of the intercepts include the value 0, as expected for this
data set because of the way the data were generated.

\subsection{Uncorrelated random variables}
\label{sec:exa5}

\subsubsection{Input data}

Two vectors of 100 random data drawn from a normal distribution
$\mathcal{N}(0, 1)$ were generated. One expects to find a null
correlation with this type of data which were submitted to the model
II regression program.

\subsubsection{Output file}

MA, SMA, OLS and RMA equations, 95\% C.I., and tests of significance,
were obtained with the following R commands. Fig. \ref{fig:ex5} shows
the scatter diagram. The various regression lines are presented to
allow their comparison.

<<>>=
data(mod2ex5)
Ex5.res <- lmodel2(random_y ~ random_x, data=mod2ex5,"interval","interval",99)
Ex5.res
@
\begin{SCfigure}
<<fig=true,echo=false>>=
plot(Ex5.res, conf = FALSE, main = "", centro = TRUE)
lines(Ex5.res, "OLS", conf=F, col=1)
lines(Ex5.res, "RMA", conf=F, col=4)
lines(Ex5.res, "SMA", conf=F, col=3)
legend("topright", c("OLS","MA","SMA", "RMA"), col=1:4, lty=1)
@
\caption{ Scatter diagram of the Example 5 data (random numbers)
  showing the major axis (MA), standard major axis (SMA), ordinary
  least-squares (OLS) and ranged major axis (RMA OLS regression
  lines. The correlation coefficient is not significantly different
  from zero. The cross indicates the centroid of the bivariate
  distribution.  The four regression lines pass through this
  centroid.}
\label{fig:ex5}
\end{SCfigure}
Neither the correlation nor any of the regression coefficients are
significant; this is as expected from the way the data were
generated. Note that the slope estimates differ widely among
methods. The MA slope is $b_{MA} = -0.60005$ but its confidence
interval, noted \texttt{NA} for `Not available' or `Missing value',
covers all 360$^{\circ}$ of the plane, as stated in the comment above
the regression table. The RMA slope estimate is $b_{RMA} =
-2.53297$. OLS, which should only be used to predict the values $\hat
y$ from $x$ (point 6 in Table 1), tends to produce slopes near zero
for random data: $b_{OLS} = 0.08011$.

Since the correlation is not significant, SMA should not have been
computed. This method tends to produce slopes near $\pm1$; with the
present example, the slope is indeed near $\pm1$ ($b_{SMA} = 0.95633$)
since the standard deviations of the two variables are nearly equal
($s_x = 1.01103$, $s_y = 0.96688$). This example shows that RMA does
not necessarily produce results that are similar to SMA.

The confidence intervals of the slope and intercept of RMA provide an
example of the phenomenon of inversion of the confidence limits
described in Fig. \ref{fig:rma}.

\bibliographystyle{plainnat}
\bibliography{lmodel2}

\end{document}
