25Nonlin - NONLINEAR MODELS For example if Xi1 = 1 Xi2 =...

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NONLINEAR MODELS 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. We have assumed E ( y ) = X β f ( X i , β ) = X i β is said to be linear in the parameters of β because X i β = X i 1 β 1 + X i 2 β 2 + . . . + X ip β p is a linear combination of β 1 , β 2 , ..., β p . f ( X i , β ) = X i β is linear in β 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 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 ) f ( X i , β ) = X i β = X i 1 β 1 + X i 2 β 2 + X i 3 β 3 + X i 4 β 4 = β 1 + fert i β 2 + fert 2 i β 3 + log ( ( fung ) i ) β 4 is still linear in the parameters β 1 , β 2 , β 3 , β 4 . Now, we consider nonlinear models for the mean E ( y i ) . These are models where f ( X i , β ) cannot be written as a linear combination of β 1 , β 2 , .., β p 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 non-linear model Example: Michaelis-Menton enzyme kinetics model v s = v m S S + K m 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 Function is mathematically equivalent to: Lineweaver-Burke: 1 v s = 1 v m + K m v m 1 S Linear regression of Y = 1 / v s on X = 1 / S Hanes-Woolf: S v s = K m v m + 1 v m S Linear regression of Y = S / v s on X = S Both are linear regressions c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 3 / 40 0 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
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However, the estimators of v m and K m derived from each model are not the same Illustrate numerically: LS estimates from each model Model ˆ β 0 ˆ β 1 v m ˆ v m K m ˆ K m nonlin 2.05 9.12 L-B 0.377 5.64 1 0 2.65 β 1 0 14.96 H-W: 4.74 0.482 1 1 2.07 β 0 1 9.83 Why? 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 + ε i , ε i ( 0 , σ 2 ) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 5 / 40 And v i = v m S i S i + K m + ε i , ε i ( 0 , σ 2 1 ) (1) is not the same as 1 v 1 = 1 v m + K m v m 1 S i + i , i ( 0 , σ 2 2 ) (2) If you work out all the details, (2) is equivalent to (1) with unequal variances The statistical models for MM, L-B, and H-W are different Estimates differ because Different variance models Leverage of specific observations is not the same c 2011 Dept. Statistics (Iowa State University) Stat 511 section 25 6 / 40 linearizing a non-linear model: 2nd example Exponential growth model Y i = β 0 e β 1 T i Nonlinear form, constant variance: Y i = β 0 e β 1 T i + ε i , ε i ( 0 , σ 2 1 ) Linearized form, constant variance, normal dist.: Y i = log Y i = log β 0 + rT i + i , i N ( 0 , σ 2 2 ) Statistically equivalent to Y i = β 0 e β 1 T i × e i , i N ( 0 , σ 2 2 ) i.e., errors are multiplicative log normal with constant lognormal
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