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Unformatted text preview: NONLINEAR MODELS I So far the models we have studied this semester have been linear in the sense that our model for the mean has been a linear function of the parameters. I We have assumed E ( y ) = X I f ( X i , ) = X i is said to be linear in the parameters of because X i = X i 1 1 + X i 2 2 + ... + X ip p is a linear combination of 1 , 2 ,..., p . I f ( X i , ) = X i is linear in even if the predictor variables, the X s are nonlinear functions of other variables. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 1 / 40 I For example, if X i 1 = 1 X i 2 = Amount of fertilizer applied to plot i X i 3 = ( Amount of fetrtilizer applied to plot i ) 2 X i 4 = log ( Concentration of fungicide on plot i ) I f ( X i , ) = X i = X i 1 1 + X i 2 2 + X i 3 3 + X i 4 4 = 1 + fert i 2 + fert 2 i 3 + log ( ( fung ) i ) 4 is still linear in the parameters 1 , 2 , 3 , 4 . I Now, we consider nonlinear models for the mean E ( y i ) . I These are models where f ( X i , ) cannot be written as a linear combination of 1 , 2 ,.., p I Small digression: What about models that can be transformed to be linear in the parameters? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 2 / 40 linearizing a nonlinear model I Example: MichaelisMenton enzyme kinetics model v s = v m S S + K m I S is concentration of substrate, v s is reaction rate at S v m is maximum reaction rate, K m is enzyme affinity= S at which v s = v m / 2 I Function is mathematically equivalent to: I LineweaverBurke: 1 v s = 1 v m + K m v m 1 S Linear regression of Y = 1 / v s on X = 1 / S I HanesWoolf: S v s = K m v m + 1 v m S Linear regression of Y = S / v s on X = S I Both are linear regressions c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 3 / 40 20 40 60 80 100 0.5 1.0 1.5 2.0 Substrate conc Reaction velocity c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 4 / 40 I However, the estimators of v m and K m derived from each model are not the same I Illustrate numerically: LS estimates from each model Model 1 v m v m K m K m nonlin 2.05 9.12 LB 0.377 5.64 1 / 2.65 1 / 14.96 HW: 4.74 0.482 1 / 1 2.07 / 1 9.83 I Why? I Because the statistical model adds a specification of variability to the mathematical model, e.g. v i = v m S i S i + K m + i , i ( , 2 ) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 5 / 40 I And v i = v m S i S i + K m + i , i ( , 2 1 ) (1) I is not the same as 1 v 1 = 1 v m + K m v m 1 S i + i , i ( , 2 2 ) (2) I If you work out all the details, (2) is equivalent to (1) with unequal variances I The statistical models for MM, LB, and HW are different I Estimates differ because I Different variance models I Leverage of specific observations is not the same c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 6 / 40 linearizing a nonlinear model: 2nd example...
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 Spring '08
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