jrhouck34

02-27-2000, 11:28 PM

To All,

Here is the summary of the replies to my questions on normalizing joint

moments and how to map the local moments. The original posting and the

responses are included.

In summary:

First question - normalization of joint moments

1. Some people suggested the reason for normalizing to both mass and

height is to render the variable demensionless

2. Other people agreed that the variability is not significantly

reduced by combining both mass and height (in adults)

3. Some people are suggesting the moments be normalized to another

variable (for example - mass times length to the 3/2 power 14.9 (0.6))

Joint Moments (not as much commentary as the normalization question)

1. Depends on the situation

2. One person wrote that the vector sum of the moments mapped into the

JCS do not vectorily sum. However, this is not necessarily the case.

Moments that sum appropriately can be calculated around the axis of the

JCS for example(non-orthogonal), however, since the axes themselves do

not intersect at a common point the calculated moments do not represent

a balance of the joint moments (See Andrews, JG, J Biomech,

17(2):155-158, 1984 he does a nice job explaining this issue).

I appreciate all the time people spent in answering the questions and

hope this posting is helpful to others.

Jeff Houck, PhD, PT

ORIGINAL POSTING

************************************************** *************

To All,

First Question - Recent papers reporting the net joint moments have

taken two approaches to decreasing variability across subjects. Some

studies (Eng and Winter, J

Biomech, 1995) divide the net joint moment by body mass. Other papers

(Berchuck, JBJS 1990, Holden and Stanhope, Gait & Posture, 7:1-6, 1998)

divide the net joint moment by % body weight * height. The reasoning

behind

the second approach is that both the ground reaction force (which is

correlated with body weight)and segment length (which is correlated with

height) are used to

estimate net joint moments.

However, a recent abstract by Sum et al(Sum et al, Gait & Posture, 7,

1998) suggests that using % body weight * height generally doesn't

decrease variability, except for the ankle dorsiflexion peak during

walking. In

addition, height explained little of the variability in the frontal

plane moments

(< 11%) and explained no additional variability once the moments had

been normalized

to body mass. This appears to suggest that normalizing to % body weight

* height

does not significantly reduce variability, and therefore, dividing net

joint

moments by body mass such as in the Eng and Winter(Eng and Winter, J

Biomech, 1995)

paper is adequate. Yet current studies appear to be dividing the net

joint

moments by % Body weight * height (Holden and Stanhope, Gait & Posture,

7:1-6, 1998). Are there other compelling arguments to make % body weight

* height the

standard over body mass?

Second question -- When moving to reporting three dimensional joint

moments there are three options when mapping the moments into the local

coordinate system. The moments can be reported in the proximal segments

coordinate system, distal segments coordinate system or mapped into the

non -

orthogonal axes of the joint coordinates used to estimate kinematics

(JCS).

Siegler and Liu (In Allard et al, Three Dimensional Analysis of Human

Locomotion, Wiley and Sons, Ltd, 1997 page 203) suggest that the moments

be mapped into the JCS. However, Andrews (Andrews, JG, J Biomech,

17(2):155-158, 1984) raised the issue that the joint center is not

common to all 3 axes when using the JCS. And therefore, "...the vector

sum [of the three joint torques] does not represent the combined turning

effect of all joint structures about a single point because the the

three axes do intersect at a common point."

Comparisons of mapping the net moments into the femoral coordinate

system and tibial coordinate system during walking suggest very close

approximations in the sagittal and frontal planes, however, the

differences are > 100 % in

the transverse plane(Pilot data). In terms of the transverse plane it

appears to make sense for interpreting muscle function to map the

moments into the local coordinate system of the distal segment. For

example the hamstrings are considered internal/external rotators because

of their relationship to the tibial long axis not the femoral long

axis.

What should the standard be for reporting local joint moments?

I will post all replies.

Thanks!

Jeff Houck, PhD, PT

Ithaca College - Rochester Campus

300 East River Road Suite 1-102

Rochester, NY 14623

jhouck@ithaca.edu

RESPONSES

************************************************** ***************

Hello Jeff,

I have a few comments and opinions about the joint moment issues you

raised. First of all, there is no RIGHT answer to any of your questions.

This is an important point that I believe is often lost. The important

thing to consider is what is most practical and logical for a given

application.

Regarding normalization, I believe that the "best" reason to use a

[bodyweight * height] scheme is to render the resulting variable

dimensionless. In many other fields, this is the common interpretation

of

"normalized". Other than that, I agree that the data suggests little

improvement in terms of reducing variability. The "best" reason to use a

[bodyweight only] scheme seems to be it's existing wide-spread use. Both

seem to work reasonably well.

Regarding joint moment reference coordinate systems: proximal, distal or

global all have certain minor advantages and disadvantages over one

another.

I would note that their is a significant problem/difficulty/source of

confusion encountered when using a JCS description of the moments.

Namely,

due to the non-orthogonal nature of the JCS, the components of the

moment

are non-unique, and the magnitude of the moment is no longer the

Euclidean

norm (sum of the squares of the components). These disadvantages

outweigh,

in my opinion, any seeming advantage of having moments expressed in a

joint

coordinate system.

One additional comment: I believe that the "moment arms" reported by

some

graphics-based software systems used to develop and analyze models of

musculoskeletal structures, are consistent with the JCS moment

description.

Thus care must be taken in using/interpreting these values. I should

note

that I am not certain of this last statement, and would greatly

appreciate

clarification from anyone who knows better than I do.

And again, there is nothing "wrong" with this description, as long as

it's

implications and assumptions are understood by the end user of the data.

Well, those are my opinions for what they're worth (and we all know what

they say about opinions). Thanks for the interesting question.

-Mike-

--

Michael Schwartz, Ph.D.

Director of Bioengineering Research

Gillette Children's Specialty Healthcare

Assistant Professor

Orthopaedic Surgery, Biomedical Engineering

University of Minnesota

Phone:(651)229-3929 Fax:(651)229-3867

Jeff,

You asked very good questions.

To your first question, mechanically it makes sense to normalize joint

moments

to weight and height. Based on my observation, normalize joint moments

to

weight and height did reduce the inter-subject variations. However, the

purpose to normalize joint moments to anthropometry measures is not only

to

reduce the inter-subject variation, but also to eliminate the effect of

anthropometry measures on joint mements to fully reveal the difference

in

movements between subjects. From this point of view, it is not a bad

thing if

normalizing joint moments weight and height does not reduce

inter-subject

variation as much as normalizing them to weight only. It certainly

should not

be a reason to say it is not a good way of normalization. Recently, we

evaluated a patient's gait in stair climbing. His problem became clear

only

after we normalized his joint moments.

To your second question, I don't think there is or should be a stardard.

In

which coordinate system the joint moment should be expressed largely

depends

on research questions.

Hope this will help.

Bing

Bing Yu, Ph.D.

Assistant Professor

Division of Physical Therapy

The University of North Carolina at Chapel Hill

Dear Jeff,

I used the data we had collected in 847 children aged 6-18 years to show

you

the results of different types of normalization applied to maximum

isometric

knee extension torque data.( I attach a histograms from Statistica as a

knee_ext_torques.doc). The variability of body mass*body height

normalization is equel to the normalization by 4 power of body height

(30

%). The variability of body mass normalization is 35 %. The

distributions

of data obtained with all types of normalization data are normal (K-S

data).

So both Steve Stanhope and David Winter are right, however Steve's

normalization is slightly better. I prefer the body height normalization

in

children [2] when the body stature is the major growth factor. In adults

the

body mass*body height normalization may work better, because the growth

is

already completed and other factors may be more important.

You can also check the following papers:

[1] Lebiedowska M, Syczewska M, Graff K, Kalinowska M. (1996)

Application

of Biomechanical Growth Models in the Quantitative Evaluation of The

Child Motor System. Disability and Rehabilitation. Vol.18, No.3 137-142.

[2] Lebiedowska M. and Polisiakiewicz A. (1997) Changes in the lower leg

moment of inertia due to child's growth J. Biomechanics. Vol .30, No.7,

723:728.

Sincerely

Maria Lebiedowska Ph.D.

Director Motion Analysis Laboratory

Southern Illinois University, School of Medicine

751 North Rutledge LL 0300

P.O. Box 19652

Springfield, Il 62794-9652

Ph. (217) 782-6556

e-mail: mlebiedowska@siumed.edu

We systematically looked at various methods to normalize/scale gait data

using

data from 10 subjects who spanned a wide range of heights and weights.

With

regards to normalizing joint moments we found the following 3 factors

statistically equivalent

mass 15.0 (0.5)

mass times length 14.5 (0.6)

mass times length to the 3/2 power 14.9 (0.6)

compared to

no scaling 29.7 (1.2)

[Each value represents the corrected pooled inter-subject variation, for

the

hip, knee and ankle moments, 3 axes, at each 2% of the gait cycle, in N

m, and

its 95% confidence value]

As you can see each of these three methods reduced the inter-subject

variation

to about 50% of its not normalized values.

Personally I prefer the third approach since it is based on some nice

physical

and mechanical assumptions. These are contained in a paper we have

written

which is currently under review.

I hope these data help.

Cheers ...

Michael Raymond Pierrynowski, Ph.D. 1 905 525-9140 x22910 (phone)

Human Movement Laboratory 1 905 522-6095 (fax)

School of Rehabilitation Science McMaster University, HSC 1J11

Hamilton, Ontario L8N 3Z5

CANADA

Jeff,

I've been looking at the development of walking patterns in children

between

the ages of 5 and 12. The height of these children varied from 100 to

160cm. It was necessary to ensure that variation in this group of

children

due to height (and weight) differences was eliminated from the analysis.

At

the ESMAC 1999 conference I was very politely but firmly informed that

analysis of gait data should be made using dimensionless quantities (Hof

AL.

Scaling gait data to body size. Gait and Posture. 1996;6:265.). Using

dimensionless quantities is the only mathematically correct way of doing

it.

Scaling moments by body weight may remove a great deal of the

variability

and the additional use of height may not reduce the variablitily

further,

but at least you then know that your variability is due to other sources

and

not height differences.

Body weight scaling of moments is 'arbitrary' scaling by body weight and

height at least has a mathematical foundation in that it leads to

dimensionless quantities.

Why are you trying to use normalisation to reduce the variability of

your

results? Differences in weight and height of subjects should be taken

into

account by using dimensionless quantities. The variability is then the

variability due to other factors such as marker placement differences,

different walking styles and so on. If you want to examine the reasons

for

differences between subjects data try examining the effects of

differences

in speed of walking. You would of course have to use a dimensionless

form

of speed.

Jeff,

I've been looking at the development of walking patterns in children

between

the ages of 5 and 12. The height of these children varied from 100 to

160cm. It was necessary to ensure that variation in this group of

children

due to height (and weight) differences was eliminated from the analysis.

At

the ESMAC 1999 conference I was very politely but firmly informed that

analysis of gait data should be made using dimensionless quantities (Hof

AL.

Scaling gait data to body size. Gait and Posture. 1996;6:265.). Using

dimensionless quantities is the only mathematically correct way of doing

it.

Scaling moments by body weight may remove a great deal of the

variability

and the additional use of height may not reduce the variablitily

further,

but at least you then know that your variability is due to other sources

and

not height differences.

Body weight scaling of moments is 'arbitrary' scaling by body weight and

height at least has a mathematical foundation in that it leads to

dimensionless quantities.

Why are you trying to use normalisation to reduce the variability of

your

results? Differences in weight and height of subjects should be taken

into

account by using dimensionless quantities. The variability is then the

variability due to other factors such as marker placement differences,

different walking styles and so on. If you want to examine the reasons

for

differences between subjects data try examining the effects of

differences

in speed of walking. You would of course have to use a dimensionless

form

of speed.

Jeff

Peak muscular force is known to be proportional to body mass (BM) raised

to

the 2/3 power. Moment arm lengths should be proportional to stature, or,

using

geometric scaling, proportional to BM raised to the 1/3 power. Moments

are

forces * Length, so the next step is to combine these (BM^2/3 * BM^1/3)

to

find BM^1. One would therefore predict then that moments resulting from

muscular force should be directly proportional to BM. At the least, this

should apply to peak moments.

Watch the units you end up with. Seems to me your resulting units, after

normalization would be something like m*m /s/s. Forgive the poor

notation

necessitated by the email medium.

Final Comment: I think we should avoid allowing statistics to overrule

physics, at least in the absence of a good physical rationale. If one

doesn't

get a significant reduction in variability using a particular

normalization

technique, I think we simply need to look deeper at the causes of

variability.

__________________________________________________

E C "Ned" Frederick, PhD Phone:+1 (603)772.4689

Exeter Research, Inc. Fax:+1 (603)658.0210

80 Haigh Road nederick@exeter-research.com

Brentwood, NH 03833, USA http://www.exeter-research.com

---------------------------------------------------------------

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

For information and archives: http://isb.ri.ccf.org/biomch-l

---------------------------------------------------------------

Here is the summary of the replies to my questions on normalizing joint

moments and how to map the local moments. The original posting and the

responses are included.

In summary:

First question - normalization of joint moments

1. Some people suggested the reason for normalizing to both mass and

height is to render the variable demensionless

2. Other people agreed that the variability is not significantly

reduced by combining both mass and height (in adults)

3. Some people are suggesting the moments be normalized to another

variable (for example - mass times length to the 3/2 power 14.9 (0.6))

Joint Moments (not as much commentary as the normalization question)

1. Depends on the situation

2. One person wrote that the vector sum of the moments mapped into the

JCS do not vectorily sum. However, this is not necessarily the case.

Moments that sum appropriately can be calculated around the axis of the

JCS for example(non-orthogonal), however, since the axes themselves do

not intersect at a common point the calculated moments do not represent

a balance of the joint moments (See Andrews, JG, J Biomech,

17(2):155-158, 1984 he does a nice job explaining this issue).

I appreciate all the time people spent in answering the questions and

hope this posting is helpful to others.

Jeff Houck, PhD, PT

ORIGINAL POSTING

************************************************** *************

To All,

First Question - Recent papers reporting the net joint moments have

taken two approaches to decreasing variability across subjects. Some

studies (Eng and Winter, J

Biomech, 1995) divide the net joint moment by body mass. Other papers

(Berchuck, JBJS 1990, Holden and Stanhope, Gait & Posture, 7:1-6, 1998)

divide the net joint moment by % body weight * height. The reasoning

behind

the second approach is that both the ground reaction force (which is

correlated with body weight)and segment length (which is correlated with

height) are used to

estimate net joint moments.

However, a recent abstract by Sum et al(Sum et al, Gait & Posture, 7,

1998) suggests that using % body weight * height generally doesn't

decrease variability, except for the ankle dorsiflexion peak during

walking. In

addition, height explained little of the variability in the frontal

plane moments

(< 11%) and explained no additional variability once the moments had

been normalized

to body mass. This appears to suggest that normalizing to % body weight

* height

does not significantly reduce variability, and therefore, dividing net

joint

moments by body mass such as in the Eng and Winter(Eng and Winter, J

Biomech, 1995)

paper is adequate. Yet current studies appear to be dividing the net

joint

moments by % Body weight * height (Holden and Stanhope, Gait & Posture,

7:1-6, 1998). Are there other compelling arguments to make % body weight

* height the

standard over body mass?

Second question -- When moving to reporting three dimensional joint

moments there are three options when mapping the moments into the local

coordinate system. The moments can be reported in the proximal segments

coordinate system, distal segments coordinate system or mapped into the

non -

orthogonal axes of the joint coordinates used to estimate kinematics

(JCS).

Siegler and Liu (In Allard et al, Three Dimensional Analysis of Human

Locomotion, Wiley and Sons, Ltd, 1997 page 203) suggest that the moments

be mapped into the JCS. However, Andrews (Andrews, JG, J Biomech,

17(2):155-158, 1984) raised the issue that the joint center is not

common to all 3 axes when using the JCS. And therefore, "...the vector

sum [of the three joint torques] does not represent the combined turning

effect of all joint structures about a single point because the the

three axes do intersect at a common point."

Comparisons of mapping the net moments into the femoral coordinate

system and tibial coordinate system during walking suggest very close

approximations in the sagittal and frontal planes, however, the

differences are > 100 % in

the transverse plane(Pilot data). In terms of the transverse plane it

appears to make sense for interpreting muscle function to map the

moments into the local coordinate system of the distal segment. For

example the hamstrings are considered internal/external rotators because

of their relationship to the tibial long axis not the femoral long

axis.

What should the standard be for reporting local joint moments?

I will post all replies.

Thanks!

Jeff Houck, PhD, PT

Ithaca College - Rochester Campus

300 East River Road Suite 1-102

Rochester, NY 14623

jhouck@ithaca.edu

RESPONSES

************************************************** ***************

Hello Jeff,

I have a few comments and opinions about the joint moment issues you

raised. First of all, there is no RIGHT answer to any of your questions.

This is an important point that I believe is often lost. The important

thing to consider is what is most practical and logical for a given

application.

Regarding normalization, I believe that the "best" reason to use a

[bodyweight * height] scheme is to render the resulting variable

dimensionless. In many other fields, this is the common interpretation

of

"normalized". Other than that, I agree that the data suggests little

improvement in terms of reducing variability. The "best" reason to use a

[bodyweight only] scheme seems to be it's existing wide-spread use. Both

seem to work reasonably well.

Regarding joint moment reference coordinate systems: proximal, distal or

global all have certain minor advantages and disadvantages over one

another.

I would note that their is a significant problem/difficulty/source of

confusion encountered when using a JCS description of the moments.

Namely,

due to the non-orthogonal nature of the JCS, the components of the

moment

are non-unique, and the magnitude of the moment is no longer the

Euclidean

norm (sum of the squares of the components). These disadvantages

outweigh,

in my opinion, any seeming advantage of having moments expressed in a

joint

coordinate system.

One additional comment: I believe that the "moment arms" reported by

some

graphics-based software systems used to develop and analyze models of

musculoskeletal structures, are consistent with the JCS moment

description.

Thus care must be taken in using/interpreting these values. I should

note

that I am not certain of this last statement, and would greatly

appreciate

clarification from anyone who knows better than I do.

And again, there is nothing "wrong" with this description, as long as

it's

implications and assumptions are understood by the end user of the data.

Well, those are my opinions for what they're worth (and we all know what

they say about opinions). Thanks for the interesting question.

-Mike-

--

Michael Schwartz, Ph.D.

Director of Bioengineering Research

Gillette Children's Specialty Healthcare

Assistant Professor

Orthopaedic Surgery, Biomedical Engineering

University of Minnesota

Phone:(651)229-3929 Fax:(651)229-3867

Jeff,

You asked very good questions.

To your first question, mechanically it makes sense to normalize joint

moments

to weight and height. Based on my observation, normalize joint moments

to

weight and height did reduce the inter-subject variations. However, the

purpose to normalize joint moments to anthropometry measures is not only

to

reduce the inter-subject variation, but also to eliminate the effect of

anthropometry measures on joint mements to fully reveal the difference

in

movements between subjects. From this point of view, it is not a bad

thing if

normalizing joint moments weight and height does not reduce

inter-subject

variation as much as normalizing them to weight only. It certainly

should not

be a reason to say it is not a good way of normalization. Recently, we

evaluated a patient's gait in stair climbing. His problem became clear

only

after we normalized his joint moments.

To your second question, I don't think there is or should be a stardard.

In

which coordinate system the joint moment should be expressed largely

depends

on research questions.

Hope this will help.

Bing

Bing Yu, Ph.D.

Assistant Professor

Division of Physical Therapy

The University of North Carolina at Chapel Hill

Dear Jeff,

I used the data we had collected in 847 children aged 6-18 years to show

you

the results of different types of normalization applied to maximum

isometric

knee extension torque data.( I attach a histograms from Statistica as a

knee_ext_torques.doc). The variability of body mass*body height

normalization is equel to the normalization by 4 power of body height

(30

%). The variability of body mass normalization is 35 %. The

distributions

of data obtained with all types of normalization data are normal (K-S

data).

So both Steve Stanhope and David Winter are right, however Steve's

normalization is slightly better. I prefer the body height normalization

in

children [2] when the body stature is the major growth factor. In adults

the

body mass*body height normalization may work better, because the growth

is

already completed and other factors may be more important.

You can also check the following papers:

[1] Lebiedowska M, Syczewska M, Graff K, Kalinowska M. (1996)

Application

of Biomechanical Growth Models in the Quantitative Evaluation of The

Child Motor System. Disability and Rehabilitation. Vol.18, No.3 137-142.

[2] Lebiedowska M. and Polisiakiewicz A. (1997) Changes in the lower leg

moment of inertia due to child's growth J. Biomechanics. Vol .30, No.7,

723:728.

Sincerely

Maria Lebiedowska Ph.D.

Director Motion Analysis Laboratory

Southern Illinois University, School of Medicine

751 North Rutledge LL 0300

P.O. Box 19652

Springfield, Il 62794-9652

Ph. (217) 782-6556

e-mail: mlebiedowska@siumed.edu

We systematically looked at various methods to normalize/scale gait data

using

data from 10 subjects who spanned a wide range of heights and weights.

With

regards to normalizing joint moments we found the following 3 factors

statistically equivalent

mass 15.0 (0.5)

mass times length 14.5 (0.6)

mass times length to the 3/2 power 14.9 (0.6)

compared to

no scaling 29.7 (1.2)

[Each value represents the corrected pooled inter-subject variation, for

the

hip, knee and ankle moments, 3 axes, at each 2% of the gait cycle, in N

m, and

its 95% confidence value]

As you can see each of these three methods reduced the inter-subject

variation

to about 50% of its not normalized values.

Personally I prefer the third approach since it is based on some nice

physical

and mechanical assumptions. These are contained in a paper we have

written

which is currently under review.

I hope these data help.

Cheers ...

Michael Raymond Pierrynowski, Ph.D. 1 905 525-9140 x22910 (phone)

Human Movement Laboratory 1 905 522-6095 (fax)

School of Rehabilitation Science McMaster University, HSC 1J11

Hamilton, Ontario L8N 3Z5

CANADA

Jeff,

I've been looking at the development of walking patterns in children

between

the ages of 5 and 12. The height of these children varied from 100 to

160cm. It was necessary to ensure that variation in this group of

children

due to height (and weight) differences was eliminated from the analysis.

At

the ESMAC 1999 conference I was very politely but firmly informed that

analysis of gait data should be made using dimensionless quantities (Hof

AL.

Scaling gait data to body size. Gait and Posture. 1996;6:265.). Using

dimensionless quantities is the only mathematically correct way of doing

it.

Scaling moments by body weight may remove a great deal of the

variability

and the additional use of height may not reduce the variablitily

further,

but at least you then know that your variability is due to other sources

and

not height differences.

Body weight scaling of moments is 'arbitrary' scaling by body weight and

height at least has a mathematical foundation in that it leads to

dimensionless quantities.

Why are you trying to use normalisation to reduce the variability of

your

results? Differences in weight and height of subjects should be taken

into

account by using dimensionless quantities. The variability is then the

variability due to other factors such as marker placement differences,

different walking styles and so on. If you want to examine the reasons

for

differences between subjects data try examining the effects of

differences

in speed of walking. You would of course have to use a dimensionless

form

of speed.

Jeff,

I've been looking at the development of walking patterns in children

between

the ages of 5 and 12. The height of these children varied from 100 to

160cm. It was necessary to ensure that variation in this group of

children

due to height (and weight) differences was eliminated from the analysis.

At

the ESMAC 1999 conference I was very politely but firmly informed that

analysis of gait data should be made using dimensionless quantities (Hof

AL.

Scaling gait data to body size. Gait and Posture. 1996;6:265.). Using

dimensionless quantities is the only mathematically correct way of doing

it.

Scaling moments by body weight may remove a great deal of the

variability

and the additional use of height may not reduce the variablitily

further,

but at least you then know that your variability is due to other sources

and

not height differences.

Body weight scaling of moments is 'arbitrary' scaling by body weight and

height at least has a mathematical foundation in that it leads to

dimensionless quantities.

Why are you trying to use normalisation to reduce the variability of

your

results? Differences in weight and height of subjects should be taken

into

account by using dimensionless quantities. The variability is then the

variability due to other factors such as marker placement differences,

different walking styles and so on. If you want to examine the reasons

for

differences between subjects data try examining the effects of

differences

in speed of walking. You would of course have to use a dimensionless

form

of speed.

Jeff

Peak muscular force is known to be proportional to body mass (BM) raised

to

the 2/3 power. Moment arm lengths should be proportional to stature, or,

using

geometric scaling, proportional to BM raised to the 1/3 power. Moments

are

forces * Length, so the next step is to combine these (BM^2/3 * BM^1/3)

to

find BM^1. One would therefore predict then that moments resulting from

muscular force should be directly proportional to BM. At the least, this

should apply to peak moments.

Watch the units you end up with. Seems to me your resulting units, after

normalization would be something like m*m /s/s. Forgive the poor

notation

necessitated by the email medium.

Final Comment: I think we should avoid allowing statistics to overrule

physics, at least in the absence of a good physical rationale. If one

doesn't

get a significant reduction in variability using a particular

normalization

technique, I think we simply need to look deeper at the causes of

variability.

__________________________________________________

E C "Ned" Frederick, PhD Phone:+1 (603)772.4689

Exeter Research, Inc. Fax:+1 (603)658.0210

80 Haigh Road nederick@exeter-research.com

Brentwood, NH 03833, USA http://www.exeter-research.com

---------------------------------------------------------------

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

For information and archives: http://isb.ri.ccf.org/biomch-l

---------------------------------------------------------------