25Nonlin

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern