Linear Model with curve fitting parameter?

-----Original Message-----
From: stephen sefick [mailto:ssefick at gmail.com]
Sent: April-04-11 2:49 PM
To: Steven McKinney
Subject: Re: [R] Linear Model with curve fitting parameter?

Steven:

I am really sorry for my confusion.  I hope this now makes sense.

b0 == y intercept == y-intercept == (intercept) fit by lm

a <- 1:10
b <- 1:10

summary(lm(a~b))
#to show what I was calling b0

So...

################################################
manning

Q = K*A*(R^b2)*(S^b3)

log(Q) = log(K)+log(A)+(b2*log(R))+(b3*log(S))

################################################
dingman
Q = K*(A^b1)*(R^b2)*(S^b3*log(S))

log(Q) = log(K)+(b1*log(A))+(b2*log(R))+(b3*(log(S))^2)

################################################

Bjerklie

Q = K*(A^b1)*(R^b2)*(S^b3)

log(Q) = log(K)+(b1*log(A))+(b2*log(R))*(b3*log(S))

################################################





On Mon, Apr 4, 2011 at 2:58 PM, Steven McKinney <smckinney at bccrc.ca> wrote:

-----Original Message-----
From: stephen sefick [mailto:ssefick at gmail.com]
Sent: April-03-11 5:35 PM
To: Steven McKinney
Cc: R help
Subject: Re: [R] Linear Model with curve fitting parameter?

Steven:

You are exactly right sorry I was confused.


#######################################################
so log(y-intercept)+log(K) is a constant called b0 (is this right?)

Doesn't look right to me based on the information you've provided.
I don't see anything labeled "y" in your previous emails, so I'm
not clear on what y is and how it relates to the original model
you described

  > >> I have a model Q=K*A*(R^r)*(S^s)
  > >>
  > >> A, R, and S are data I have and K is a curve fitting parameter.

If the model is

  Q=K*A*(R^r)*(S^s)

then

  log(Q) = log(K) + log(A) + r*log(R) + s*log(S)

Rearranging yields

  log(Q) - log(A) = log(K) + r*log(R) + s*log(S)

Let  Z = log(Q) - log(A) = log(Q/A)

so

  Z = log(K) + r*log(R) + s*log(S)

and a linear model fit of

  Z ~ log(R) + log(S)

will yield parameter estimates for the linear equation

  E(Z) = B0 + B1*log(R) + B2*log(S)

(E(Z) = expected value of Z)

so B0 estimate is an estimate of log(K)
  B1 estimate is an estimate of r
  B2 estimate is an estimate of s

and these are the only parameters you described in the original model.

lm(log(Q)~log(A)+log(R)+log(S)-1)

is fitting the model

log(Q)=a*log(A)+r*log(R)+s*log(S) (no beta 0)

and

lm(log(Q)~log(A)+log(R)+log(S))


is fitting the model

log(Q)=b0+a*log(A)+r*log(R)+s*log(S)

K has disappeared from these equations so these model fits do
not correspond to the model originally described.  Now a b0
appears, and is used in models below.  I think changing notation
is also adding confusion.  What are "y" and "intercept" you
discuss above, in relation to your original notation?

######################################################

These are the models I am trying to fit and if I have reasoned
correctly above then I should be able to fit the below models
similarly.

You will be able to fit models appropriately once you have a
clearly defined system of notation that allows you to map between
the proposed data model, the parameters in that model, and the
corresponding regression equations.

Once you have consistent notation, you will be able to see
if you can express your model as a linear regression, or
if not, what kind of non-linear regression you will need to
do to get estimates for the parameters in your model.

Best

Steve McKinney

manning
log(Q)=log(b0)+log(K)+log(A)+r*log(R)+s*log(S)

dingman
log(Q)=log(b0)+log(K)+a*log(A)+r*log(R)+s*(log(S))^2

bjerklie
log(Q)=log(b0)+log(K)+a*log(A)+r*log(R)+s*log(S)

#######################################################

Thank you for all of your help!

Stephen

On Fri, Apr 1, 2011 at 2:44 PM, Steven McKinney <smckinney at bccrc.ca> wrote:

-----Original Message-----
From: stephen sefick [mailto:ssefick at gmail.com]
Sent: April-01-11 5:44 AM
To: Steven McKinney
Cc: R help
Subject: Re: [R] Linear Model with curve fitting parameter?

Setting Z=Q-A would be the incorrect dimensions.  I could Z=Q/A.

I suspect this is confusion about what Q is.  I was presuming that
the Q in this following formula was log(Q) with Q from the original data.

I have taken the log of the data that I have and this is the model
formula without the K part

lm(Q~offset(A)+R+S, data=x)

If the model is

  Q=K*A*(R^r)*(S^s)

then

  log(Q) = log(K) + log(A) + r*log(R) + s*log(S)

Rearranging yields

  log(Q) - log(A) = log(K) + r*log(R) + s*log(S)

so what I labeled 'Z' below is

  Z = log(Q) - log(A) = log(Q/A)

so

  Z = log(K) + r*log(R) + s*log(S)

and a linear model fit of

  Z ~ log(R) + log(S)

will yield parameter estimates for the linear equation

  E(Z) = B0 + B1*log(R) + B2*log(S)

(E(Z) = expected value of Z)

so B0 estimate is an estimate of log(K)
  B1 estimate is an estimate of r
  B2 estimate is an estimate of s

More details and careful notation will eventually lead
to a reasonable description and analysis strategy.


Best

Steve McKinney

Is fitting a nls model the same as fitting an ols?  These data are
hydraulic data from ~47 sites.  To access predictive ability I am
removing one site fitting a new model and then accessing the fit with
a myriad of model assessment criteria.  I should get the same answer
with ols vs nls?  Thank you for all of your help.

Stephen

On Thu, Mar 31, 2011 at 8:34 PM, Steven McKinney <smckinney at bccrc.ca> wrote:

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of stephen

sefick

Sent: March-31-11 3:38 PM
To: R help
Subject: [R] Linear Model with curve fitting parameter?

I have a model Q=K*A*(R^r)*(S^s)

A, R, and S are data I have and K is a curve fitting parameter.  I
have linearized as

log(Q)=log(K)+log(A)+r*log(R)+s*log(S)

I have taken the log of the data that I have and this is the model
formula without the K part

lm(Q~offset(A)+R+S, data=x)

What is the formula that I should use?

Let Z = Q - A for your logged data.

Fitting lm(Z ~ R + S, data = x) should yield
intercept parameter estimate = estimate for log(K)
R coefficient parameter estimate = estimate for r
S coefficient parameter estimate = estimate for s



Steven McKinney

Statistician
Molecular Oncology and Breast Cancer Program
British Columbia Cancer Research Centre

Thanks for all of your help.  I can provide a subset of data if necessary.



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Stephen Sefick

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Stephen Sefick

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Stephen Sefick

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Stephen Sefick