Nonlinear least squares regression

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Nonlinear least squares regression

• Some models cannot be made linear in parameters
• E.g., theta-logistic stock-recruitment model

• Solution nonlinear least squares (NLS)
• All other assumptions from OLS (normal residuals,
  etc.) still apply
• Conceptually the same find the values of
  parameters that minimize the sum of squared
  residuals

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Nonlinear least squares regression

• Numerically intensive cant be done by hand
• As number of parameters gets large say above
  10 or so can take substantial time even on
  modern computers

• Not guaranteed to get the right answer
• Computer algorithms find local minima
  sometimes there are more than one
• Need to provide pretty good initial guesses for
  the parameter values
• Base guesses on
• scientific information
• If only a few nonlinear parameters, try fixing
  them at a few values and using OLS to estimate
  the rest choose the best overall combination

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Example stock-recruitment relationship for
Alewife
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Force c0, theta1
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Another stock-recruitment model

• Beverton-Holt model
• No overcompensation
• Competition for space rather than food
• Always some recruitment, no matter how large the
  spawner population

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A few issues

• Some heteroskedasticity caused by positivity
  constraints
• Measured on different scale from Ricker model
  cant directly compare

• Try a modified model

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Calculating AIC

• Often, AIC is not reported by software you have
  to calculate it yourself
• L is log-likelihood
• p is number of parameters
• If log-likelihood is not reported, and you are
  doing least-squares regression, then use this
  formula (n is number of data points)

When making comparisons, must have same dependent
variable and sample size for each model
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Comparing the models
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Dangers of nonlinear regression

• You can get similarly good fits with very
  different looking functions
• Often OK for interpolating

• Different models will produce very different
  predictions when extrapolating