Nonlinear least squares regression

1. 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

2. Nonlinear least squares regression Numerically intensive: can�t 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

3. Example: stock-recruitment relationship for Alewife

6. Force c=0, theta=1

7. 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

9. A few issues Some heteroskedasticity caused by positivity constraints Measured on different scale from Ricker model � can�t directly compare Try a modified model:

11. 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):

12. Comparing the models

13. 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