NonLinear and Smooth Regression: An Introductory Example V

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Non-Linear and Smooth Regression An Introductory
Example (VR 8.1)

• Rommel Vives
• Lin Zhang
• Statistics 6601 Project, Fall 2004

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Objective

• To present a primer on non-linear regression
• Start by comparing non-linear regression to
  linear regression.

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Similarities and Differences of Methodology

• Similarities
• Each fits a mathematical model to data.
• Both are sometimes called least squares methods
  since the procedures use sum-of-squares as a
  measure of goodness-of-fit.
• Differences
• In the non-linear regression model, at least one
  parameter is related to the explanatory
  variable(s) in a non-linear fashion.
• There is usually no closed form expression for
  the least squares parameter estimates with
  non-linear regression.
• Least squares estimates of the parameters are,
  therefore, derived from an iterative process
  using numerical methods.

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Advantages and Disadvantages

• Linear Model
• Linear Least Squared Method provided the best fit
  model.
• Linear Model has closed form, moderate computing.
• The estimated parameters always have clear
  meanings.
• Too perfect, then not realistic.
• Non-Linear Model
• More flexibility, more realistic?
• Without closed form. Need better understanding
  of the meaning of parameters.
• Mass computation requirement. Impossible in era
  of Classical Model.

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Why Choose a Non-Linear Model?
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Fitting Data with Non-Linear Regression

• Choose a model
• Estimate initial values
• Constrain a parameter
• Decide on a weighting scheme
• Handle replicate values, if any

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Choose a Model

• Functions that fit the data well using the least
  squares criterion may not be meaningful in the
  context of the experiment. That is, the
  parameters may not be interpretable for such
  functions. Choose one where the parameters and
  their estimates are scientifically meaningful, so
  that results may be of use in the data analysis.

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Estimate Initial Values

• Initial values are ballpark estimates of the
  least squares values of the parameters and are
  needed to start the iterative process of deriving
  them. Understand the model, the meaning of all
  the parameters and look at the graph of the data
  to estimate the initial values.

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A Least Squares Surface (Two Parameters) with a
Local Minimum
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Constrain a Parameter

• It is not necessary in some cases to fit each
  parameter in the model. For those not fitted, a
  constant value is assigned.

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Decide on a Weighting Scheme

• Weigh data points equally
• Apply weights to data points if, for example, the
  average distance of the points from the curve
  varies with the response variable, but relative
  distance remains constant.

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Handle Replicate Values

• If, for one set of values of the independent
  variables, replicate values of the response
  variable are collected, treat replicates as
  separate points if they are independent.
  Otherwise, use the average of the replicates.

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An Introductory Example

• Data on Weight Loss
• pertains to an obese male patient, age 48,
  height 193 cm (64) with a large body frame
• consists of the variables weight (in kilograms
  as measured under standard conditions) and days
  (time since start of weight reduction program)

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R Code

• gt library(MASS)
• gt attach(wtloss)
• gt oldpar lt- par(marc(5.1,4.1,4.1,4.1))
• gt plot(Days,Weight,type"p",ylab"Weight (kg)")
• gt Wt.lbs lt- pretty(range(Weight2.205))
• gt axis(side4,atWt.lbs/2.205,labWt.lbs,srt90)
• gt mtext("Weight (lb)",side4,line3)
• gt fm lt- lm(WeightDays)
• gt abline(fm,col'red') Fit a simple regression
  of Weight on Days.
• gt lrf lt- loess(WeightDays)
• gt lines(Days,fitted(lrf),col'blue') Fit a
  LOWESS curve of Weight on Days.

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Model

• Y b0 b12-t/q ??
• where only ? is considered random
• b0 is the ultimate lean weight, or
  asymptote
• b1 is the total amount to be lost and
• q is the time taken to lose half the
  amount remaining to be lost

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References

• The GraphPad Guide to Nonlinear Regression
• Engineering Statistics Handbook

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The End