Animal Morphology: Deriving Classification Equation with Linear Discriminat Analysis (lda)

Fellow R Users:
I'm not extremely familiar with lda or R programming, but a recent
editorial review of a manuscript submission has prompted a crash
course. I am on this forum hoping I could solicit some much needed
advice for deriving a classification equation.

I have used three basic measurements in lda to predict two groups:
male and female. I have a working model, low Wilk's lambda, graphs,
coefficients, eigenvalues, etc. (see below). I adjusted the sample
analysis for Fisher's or Anderson's Iris data provided in the MASS
library for my own data.

My final and last step is simply form the classification equation.
The classification equation is simply using standardized coefficients
to classify each group- in this case male or female. A more thorough
explanation is provided:

"For cases with an equal sample size for each group the classification
function coefficient (Cj) is expressed by the following equation:

Cj = cj0+ cj1x1+ cj2x2+...+ cjpxp

where Cj is the score for the jth group, j = 1 ??? k, cjo is the
constant for the jth group, and x = raw scores of each predictor.
If W = within-group variance-covariance matrix, and M = column matrix
of means for group j, then the constant   cjo= (-1/2)CjMj" (Julia
Barfield, John Poulsen, and Aaron French 
http://userwww.sfsu.edu/~efc/classes/biol710/discrim/discriminant.htm).

I am unable to navigate this last step based on the R output I have.
I only have the linear discriminant coefficients for each predictor
that would be needed to complete this equation.

Please, if anybody is familiar or able to to help please let me know.
There is a spot in the acknowledgments for you.

All the best,
Chase Mendenhall

[Your data and output listings removed. For comments, see at end]

On 24-May-09 13:01:26, cdm wrote:

Fellow R Users:
I'm not extremely familiar with lda or R programming, but a recent
editorial review of a manuscript submission has prompted a crash
course. I am on this forum hoping I could solicit some much needed
advice for deriving a classification equation.

I have used three basic measurements in lda to predict two groups:
male and female. I have a working model, low Wilk's lambda, graphs,
coefficients, eigenvalues, etc. (see below). I adjusted the sample
analysis for Fisher's or Anderson's Iris data provided in the MASS
library for my own data.

My final and last step is simply form the classification equation.
The classification equation is simply using standardized coefficients
to classify each group- in this case male or female. A more thorough
explanation is provided:

"For cases with an equal sample size for each group the classification
function coefficient (Cj) is expressed by the following equation:

Cj = cj0+ cj1x1+ cj2x2+...+ cjpxp

where Cj is the score for the jth group, j = 1 ??? k, cjo is the
constant for the jth group, and x = raw scores of each predictor.
If W = within-group variance-covariance matrix, and M = column matrix
of means for group j, then the constant   cjo= (-1/2)CjMj" (Julia
Barfield, John Poulsen, and Aaron French 
http://userwww.sfsu.edu/~efc/classes/biol710/discrim/discriminant.htm).

I am unable to navigate this last step based on the R output I have.
I only have the linear discriminant coefficients for each predictor
that would be needed to complete this equation.

Please, if anybody is familiar or able to to help please let me know.
There is a spot in the acknowledgments for you.

All the best,
Chase Mendenhall

The first thing I did was to plot your data. This indicates in the
first place that a perfect discrimination can be obtained on the
basis of your variables WRMA_WT and WRMA_ID alone (names abbreviated
to WG, WT, ID, SEX):

  d.csv("horsesLDA.csv")
  # names(D0) # "WRMA_WG"  "WRMA_WT"  "WRMA_ID"  "WRMA_SEX"
  WG<-D0$WRMA_WG; WT<-D0$WRMA_WT;
  ID<-D0$WRMA_ID; SEX<-D0$WRMA_SEX

  ix.M<-(SEX=="M"); ix.F<-(SEX=="F")

  ## Plot WT vs ID (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")
  lines(ID,15.5-1.0*(ID))

and that there is a lot of possible variation in the discriminating
line WT = 15.5-1.0*(ID)

Also, it is apparent that the covariance between WT and ID for Females
is different from the covariance between WT and ID for Males. Hence
the assumption (of common covariance matrix in the two groups) for
standard LDA (which you have been applying) does not hold.

Given that the sexes can be perfectly discriminated within the data
on the basis of the linear discriminator (WT + ID) (and others),
the variable WG is in effect a close approximation to noise.

However, to the extent that there was a common covariance matrix
to the two groups (in all three variables WG, WT, ID), and this
was well estimated from the data, then inclusion of the third
variable WG could yield a slightly improved discriminator in that
the probability of misclassification (a rare event for such data)
could be minimised. But it would not make much difference!

However, since that assumption does not hold, this analysis would
not be valid.

If you plot WT vs WG, a common covariance is more plausible; but
there is considerable overlap for these two variables:

  plot(WG,WT)
  points(WG[ix.M],WT[ix.M],pch="+",col="blue")
  points(WG[ix.F],WT[ix.F],pch="+",col="red")

If you plot WG vs ID, there is perhaps not much overlap, but a
considerable difference in covariance between the two groups:

  plot(ID,WG)
  points(ID[ix.M],WG[ix.M],pch="+",col="blue")
  points(ID[ix.F],WG[ix.F],pch="+",col="red")

This looks better on a log scale, however:

  lWG <- log(WG) ; lWT <- log(WT) ; lID <- log(ID)
## Plot log(WG) vs log(ID) (M & F)
  plot(lID,lWG)
  points(lID[ix.M],lWG[ix.M],pch="+",col="blue")
  points(lID[ix.F],lWG[ix.F],pch="+",col="red")

and common covaroance still looks good for WG vs WT:

  ## Plot log(WT) vs log(WG) (M & F)
  plot(lWG,lWT)
  points(lWG[ix.M],lWT[ix.M],pch="+",col="blue")
  points(lWG[ix.F],lWT[ix.F],pch="+",col="red")

but there is no improvement for WG vs IG:

  ## Plot log(WT) vs log(ID) (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")

So there is no simple road to applying a routine LDA to your data.

To take account of different covariances between the two groups,
you would normally be looking at a quadratic discriminator. However,
as indicated above, the fact that a linear discriminator using
the variables ID & WT alone works so well would leave considerable
imprecision in conclusions to be drawn from its results.

Sorry this is not the straightforward answer you were hoping for
(which I confess I have not sought); it is simply a reaction to
what your data say.

Ted.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 24-May-09                                       Time: 20:07:43
------------------------------ XFMail ------------------------------

[Your data and output listings removed. For comments, see at end]

On 24-May-09 13:01:26, cdm wrote:

Fellow R Users:
I'm not extremely familiar with lda or R programming, but a recent
editorial review of a manuscript submission has prompted a crash
course. I am on this forum hoping I could solicit some much needed
advice for deriving a classification equation.

I have used three basic measurements in lda to predict two groups:
male and female. I have a working model, low Wilk's lambda, graphs,
coefficients, eigenvalues, etc. (see below). I adjusted the sample
analysis for Fisher's or Anderson's Iris data provided in the MASS
library for my own data.

My final and last step is simply form the classification equation.
The classification equation is simply using standardized coefficients
to classify each group- in this case male or female. A more thorough
explanation is provided:

"For cases with an equal sample size for each group the classification
function coefficient (Cj) is expressed by the following equation:

Cj = cj0+ cj1x1+ cj2x2+...+ cjpxp

where Cj is the score for the jth group, j = 1 ??? k, cjo is the
constant for the jth group, and x = raw scores of each predictor.
If W = within-group variance-covariance matrix, and M = column matrix
of means for group j, then the constant   cjo= (-1/2)CjMj" (Julia
Barfield, John Poulsen, and Aaron French 
http://userwww.sfsu.edu/~efc/classes/biol710/discrim/discriminant.htm).

I am unable to navigate this last step based on the R output I have.
I only have the linear discriminant coefficients for each predictor
that would be needed to complete this equation.

Please, if anybody is familiar or able to to help please let me know.
There is a spot in the acknowledgments for you.

All the best,
Chase Mendenhall

The first thing I did was to plot your data. This indicates in the
first place that a perfect discrimination can be obtained on the
basis of your variables WRMA_WT and WRMA_ID alone (names abbreviated
to WG, WT, ID, SEX):

  d.csv("horsesLDA.csv")
  # names(D0) # "WRMA_WG"  "WRMA_WT"  "WRMA_ID"  "WRMA_SEX"
  WG<-D0$WRMA_WG; WT<-D0$WRMA_WT;
  ID<-D0$WRMA_ID; SEX<-D0$WRMA_SEX

  ix.M<-(SEX=="M"); ix.F<-(SEX=="F")

  ## Plot WT vs ID (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")
  lines(ID,15.5-1.0*(ID))

and that there is a lot of possible variation in the discriminating
line WT = 15.5-1.0*(ID)

Also, it is apparent that the covariance between WT and ID for Females
is different from the covariance between WT and ID for Males. Hence
the assumption (of common covariance matrix in the two groups) for
standard LDA (which you have been applying) does not hold.

Given that the sexes can be perfectly discriminated within the data
on the basis of the linear discriminator (WT + ID) (and others),
the variable WG is in effect a close approximation to noise.

However, to the extent that there was a common covariance matrix
to the two groups (in all three variables WG, WT, ID), and this
was well estimated from the data, then inclusion of the third
variable WG could yield a slightly improved discriminator in that
the probability of misclassification (a rare event for such data)
could be minimised. But it would not make much difference!

However, since that assumption does not hold, this analysis would
not be valid.

If you plot WT vs WG, a common covariance is more plausible; but
there is considerable overlap for these two variables:

  plot(WG,WT)
  points(WG[ix.M],WT[ix.M],pch="+",col="blue")
  points(WG[ix.F],WT[ix.F],pch="+",col="red")

If you plot WG vs ID, there is perhaps not much overlap, but a
considerable difference in covariance between the two groups:

  plot(ID,WG)
  points(ID[ix.M],WG[ix.M],pch="+",col="blue")
  points(ID[ix.F],WG[ix.F],pch="+",col="red")

This looks better on a log scale, however:

  lWG <- log(WG) ; lWT <- log(WT) ; lID <- log(ID)
## Plot log(WG) vs log(ID) (M & F)
  plot(lID,lWG)
  points(lID[ix.M],lWG[ix.M],pch="+",col="blue")
  points(lID[ix.F],lWG[ix.F],pch="+",col="red")

and common covaroance still looks good for WG vs WT:

  ## Plot log(WT) vs log(WG) (M & F)
  plot(lWG,lWT)
  points(lWG[ix.M],lWT[ix.M],pch="+",col="blue")
  points(lWG[ix.F],lWT[ix.F],pch="+",col="red")

but there is no improvement for WG vs IG:

  ## Plot log(WT) vs log(ID) (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")

So there is no simple road to applying a routine LDA to your data.

To take account of different covariances between the two groups,
you would normally be looking at a quadratic discriminator. However,
as indicated above, the fact that a linear discriminator using
the variables ID & WT alone works so well would leave considerable
imprecision in conclusions to be drawn from its results.

Sorry this is not the straightforward answer you were hoping for
(which I confess I have not sought); it is simply a reaction to
what your data say.

Ted.

[Apologies -- I made an error (see at [***] near the end)]

On 24-May-09 19:07:46, Ted Harding wrote:

[Your data and output listings removed. For comments, see at end]

On 24-May-09 13:01:26, cdm wrote:

Fellow R Users:
I'm not extremely familiar with lda or R programming, but a recent
editorial review of a manuscript submission has prompted a crash
course. I am on this forum hoping I could solicit some much needed
advice for deriving a classification equation.

I have used three basic measurements in lda to predict two groups:
male and female. I have a working model, low Wilk's lambda, graphs,
coefficients, eigenvalues, etc. (see below). I adjusted the sample
analysis for Fisher's or Anderson's Iris data provided in the MASS
library for my own data.

My final and last step is simply form the classification equation.
The classification equation is simply using standardized coefficients
to classify each group- in this case male or female. A more thorough
explanation is provided:

"For cases with an equal sample size for each group the classification
function coefficient (Cj) is expressed by the following equation:

Cj = cj0+ cj1x1+ cj2x2+...+ cjpxp

where Cj is the score for the jth group, j = 1 ??? k, cjo is the
constant for the jth group, and x = raw scores of each predictor.
If W = within-group variance-covariance matrix, and M = column matrix
of means for group j, then the constant   cjo= (-1/2)CjMj" (Julia
Barfield, John Poulsen, and Aaron French 
http://userwww.sfsu.edu/~efc/classes/biol710/discrim/discriminant.htm).

I am unable to navigate this last step based on the R output I have.
I only have the linear discriminant coefficients for each predictor
that would be needed to complete this equation.

Please, if anybody is familiar or able to to help please let me know.
There is a spot in the acknowledgments for you.

All the best,
Chase Mendenhall

The first thing I did was to plot your data. This indicates in the
first place that a perfect discrimination can be obtained on the
basis of your variables WRMA_WT and WRMA_ID alone (names abbreviated
to WG, WT, ID, SEX):

  d.csv("horsesLDA.csv")
  # names(D0) # "WRMA_WG"  "WRMA_WT"  "WRMA_ID"  "WRMA_SEX"
  WG<-D0$WRMA_WG; WT<-D0$WRMA_WT;
  ID<-D0$WRMA_ID; SEX<-D0$WRMA_SEX

  ix.M<-(SEX=="M"); ix.F<-(SEX=="F")

  ## Plot WT vs ID (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")
  lines(ID,15.5-1.0*(ID))

and that there is a lot of possible variation in the discriminating
line WT = 15.5-1.0*(ID)

Also, it is apparent that the covariance between WT and ID for Females
is different from the covariance between WT and ID for Males. Hence
the assumption (of common covariance matrix in the two groups) for
standard LDA (which you have been applying) does not hold.

Given that the sexes can be perfectly discriminated within the data
on the basis of the linear discriminator (WT + ID) (and others),
the variable WG is in effect a close approximation to noise.

However, to the extent that there was a common covariance matrix
to the two groups (in all three variables WG, WT, ID), and this
was well estimated from the data, then inclusion of the third
variable WG could yield a slightly improved discriminator in that
the probability of misclassification (a rare event for such data)
could be minimised. But it would not make much difference!

However, since that assumption does not hold, this analysis would
not be valid.

If you plot WT vs WG, a common covariance is more plausible; but
there is considerable overlap for these two variables:

  plot(WG,WT)
  points(WG[ix.M],WT[ix.M],pch="+",col="blue")
  points(WG[ix.F],WT[ix.F],pch="+",col="red")

If you plot WG vs ID, there is perhaps not much overlap, but a
considerable difference in covariance between the two groups:

  plot(ID,WG)
  points(ID[ix.M],WG[ix.M],pch="+",col="blue")
  points(ID[ix.F],WG[ix.F],pch="+",col="red")

This looks better on a log scale, however:

  lWG <- log(WG) ; lWT <- log(WT) ; lID <- log(ID)
## Plot log(WG) vs log(ID) (M & F)
  plot(lID,lWG)
  points(lID[ix.M],lWG[ix.M],pch="+",col="blue")
  points(lID[ix.F],lWG[ix.F],pch="+",col="red")

and common covaroance still looks good for WG vs WT:

  ## Plot log(WT) vs log(WG) (M & F)
  plot(lWG,lWT)
  points(lWG[ix.M],lWT[ix.M],pch="+",col="blue")
  points(lWG[ix.F],lWT[ix.F],pch="+",col="red")

but there is no improvement for WG vs IG:

  ## Plot log(WT) vs log(ID) (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")

[***]
The above is incorrect! Apologies. I plotted the raw WT and ID
instead of their logs. In fact, if you do plot the logs:

  ## Plot log(WT) vs log(ID) (M & F)
  plot(lID,lWT)
  points(lID[ix.M],lWT[ix.M],pch="+",col="blue")
  points(lID[ix.F],lWT[ix.F],pch="+",col="red")

you now get what looks like much closer agreement between the
covariance cov(lID,lWT) then before. Hence, I would now suggest
that you do your limear discrimination on the logarithms of the
variables (since you also get agreement for the other pairs on
the log scale.

In fact:

[Raw]:
  [Male]:
  cov(cbind(WG,WT,ID)[ix.M,])
  #            WG         WT          ID
  # WG  2.2552465 0.11074710 -0.02202080
  # WT  0.1107471 0.33853450  0.06601287
  # ID -0.0220208 0.06601287  0.31979368

  [Female]:
  cov(cbind(WG,WT,ID)[ix.F,])
  #           WG        WT        ID
  # WG  2.4716912 0.1577307   0.6670657
  # WT  0.1577307 0.3183928   0.2973335
  # I D 0.6670657 0.2973335   2.8326520

[log]:
  [Male]:
  cov(cbind(lWG,lWT,lID)[ix.M,])
  #               lWG          lWT           lID
  # lWG  0.0006584465 0.0001813315 -0.0002133576
  # lWT  0.0001813315 0.0030368382  0.0030442356
  # lID -0.0002133576 0.0030442356  0.0693965979

  [Female]:
  cov(cbind(lWG,lWT,lID)[ix.F,])
  #              lWG          lWT         lID
  # lWG  0.0007244826 0.0002171885  0.001951343
  # lWT  0.0002171885 0.0019640076  0.003305884
  # lID  0.0019513428 0.0033058841  0.068406840

So there is no simple road to applying a routine LDA to your data.

To take account of different covariances between the two groups,
you would normally be looking at a quadratic discriminator. However,
as indicated above, the fact that a linear discriminator using
the variables ID & WT alone works so well would leave considerable
imprecision in conclusions to be drawn from its results.

Sorry this is not the straightforward answer you were hoping for
(which I confess I have not sought); it is simply a reaction to
what your data say.

Ted.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 24-May-09                                       Time: 21:49:50
------------------------------ XFMail ------------------------------

Dear Ted,
Thank you for taking the time out to help me with this analysis.
I'm seeing that I may have left out a crucial detail concerning
this analysis. The ID measurement (interpubic distance) is a new
measurement that has never been used in the field of ornithology
(to my knowledge). The objective of the paper is to demonstrate
the usefulness of ID. The paper compared ID with plumage criterion,
a categorical variable at best, but under peer-review there is a
request to use other morphological data to compare/contrast ID.
Unfortunately, wing (WG) and weight (WT) were the only measurements
taken in addition to ID in this study.

The purpose of the LDA is to demonstrate the power if ID in the
context of WG and WT. I agree that WG is a terrible metric for
discrimination, WT is good but there is significant overlap between
groups, but ID is a good discriminator on it's own (classified 97-100%
of all individuals based on 92.5% CI).

You pointed out that I am violating assumptions with LDA based on
different covariances between sexes (thank you... I never would have
caught it). I'm wondering how to proceed.

Should I:

1) Perform linear discrimination with WT and ID, and then determine a
classification equation? And, if I do how do I derive the
classification
equation (e.g. [Cj = cj0+ cjWTxWT+ cjIDxID; Cj>x= male, Cj<x=female])

2) Demonstrate that ID is important based on linear discrimanant
coefficients and structure coefficients from this WG, WT, and ID LDA;
discuss the assumption violation and argue for it's use as a
demonstration
of variable predicting power; and NOT provide a classification equation
because we already have ID ranges and it would be inappropriate.

3) Both #1 and #2 because WT and ID provide such a good discriminating
function and use the WG, WT, and ID LDA for demonstration of variable
prediction value.

4) ??? better suggestions.




THANK YOU so much for responding and all of your insight. I'm humbled
by
your R skills... that code nearly too me all day to write (little by
little
I'm learning).

Chase



Ted.Harding-2 wrote:

[Your data and output listings removed. For comments, see at end]

On 24-May-09 13:01:26, cdm wrote:

Fellow R Users:
I'm not extremely familiar with lda or R programming, but a recent
editorial review of a manuscript submission has prompted a crash
course. I am on this forum hoping I could solicit some much needed
advice for deriving a classification equation.

I have used three basic measurements in lda to predict two groups:
male and female. I have a working model, low Wilk's lambda, graphs,
coefficients, eigenvalues, etc. (see below). I adjusted the sample
analysis for Fisher's or Anderson's Iris data provided in the MASS
library for my own data.

My final and last step is simply form the classification equation.
The classification equation is simply using standardized coefficients
to classify each group- in this case male or female. A more thorough
explanation is provided:

"For cases with an equal sample size for each group the
classification
function coefficient (Cj) is expressed by the following equation:

Cj = cj0+ cj1x1+ cj2x2+...+ cjpxp

where Cj is the score for the jth group, j = 1 ?????? k, cjo is the
constant for the jth group, and x = raw scores of each predictor.
If W = within-group variance-covariance matrix, and M = column matrix
of means for group j, then the constant   cjo= (-1/2)CjMj" (Julia
Barfield, John Poulsen, and Aaron French 
http://userwww.sfsu.edu/~efc/classes/biol710/discrim/discriminant.htm)
.

I am unable to navigate this last step based on the R output I have.
I only have the linear discriminant coefficients for each predictor
that would be needed to complete this equation.

Please, if anybody is familiar or able to to help please let me know.
There is a spot in the acknowledgments for you.

All the best,
Chase Mendenhall

The first thing I did was to plot your data. This indicates in the
first place that a perfect discrimination can be obtained on the
basis of your variables WRMA_WT and WRMA_ID alone (names abbreviated
to WG, WT, ID, SEX):

  d.csv("horsesLDA.csv")
  # names(D0) # "WRMA_WG"  "WRMA_WT"  "WRMA_ID"  "WRMA_SEX"
  WG<-D0$WRMA_WG; WT<-D0$WRMA_WT;
  ID<-D0$WRMA_ID; SEX<-D0$WRMA_SEX

  ix.M<-(SEX=="M"); ix.F<-(SEX=="F")

  ## Plot WT vs ID (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")
  lines(ID,15.5-1.0*(ID))

and that there is a lot of possible variation in the discriminating
line WT = 15.5-1.0*(ID)

Also, it is apparent that the covariance between WT and ID for Females
is different from the covariance between WT and ID for Males. Hence
the assumption (of common covariance matrix in the two groups) for
standard LDA (which you have been applying) does not hold.

Given that the sexes can be perfectly discriminated within the data
on the basis of the linear discriminator (WT + ID) (and others),
the variable WG is in effect a close approximation to noise.

However, to the extent that there was a common covariance matrix
to the two groups (in all three variables WG, WT, ID), and this
was well estimated from the data, then inclusion of the third
variable WG could yield a slightly improved discriminator in that
the probability of misclassification (a rare event for such data)
could be minimised. But it would not make much difference!

However, since that assumption does not hold, this analysis would
not be valid.

If you plot WT vs WG, a common covariance is more plausible; but
there is considerable overlap for these two variables:

  plot(WG,WT)
  points(WG[ix.M],WT[ix.M],pch="+",col="blue")
  points(WG[ix.F],WT[ix.F],pch="+",col="red")

If you plot WG vs ID, there is perhaps not much overlap, but a
considerable difference in covariance between the two groups:

  plot(ID,WG)
  points(ID[ix.M],WG[ix.M],pch="+",col="blue")
  points(ID[ix.F],WG[ix.F],pch="+",col="red")

This looks better on a log scale, however:

  lWG <- log(WG) ; lWT <- log(WT) ; lID <- log(ID)
## Plot log(WG) vs log(ID) (M & F)
  plot(lID,lWG)
  points(lID[ix.M],lWG[ix.M],pch="+",col="blue")
  points(lID[ix.F],lWG[ix.F],pch="+",col="red")

and common covaroance still looks good for WG vs WT:

  ## Plot log(WT) vs log(WG) (M & F)
  plot(lWG,lWT)
  points(lWG[ix.M],lWT[ix.M],pch="+",col="blue")
  points(lWG[ix.F],lWT[ix.F],pch="+",col="red")

but there is no improvement for WG vs IG:

  ## Plot log(WT) vs log(ID) (M & F)
  plot(ID,WT,xlim=c(0,12),ylim=c(8,15))
  points(ID[ix.M],WT[ix.M],pch="+",col="blue")
  points(ID[ix.F],WT[ix.F],pch="+",col="red")

So there is no simple road to applying a routine LDA to your data.

To take account of different covariances between the two groups,
you would normally be looking at a quadratic discriminator. However,
as indicated above, the fact that a linear discriminator using
the variables ID & WT alone works so well would leave considerable
imprecision in conclusions to be drawn from its results.

Sorry this is not the straightforward answer you were hoping for
(which I confess I have not sought); it is simply a reaction to
what your data say.

Ted.

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E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
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Date: 24-May-09                                       Time: 20:07:43
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