25Nonlin - NONLINEARMODELS Wehaveassumed E y)= X β f X i...

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Unformatted text preview: NONLINEARMODELS Sofarthemodelswehavestudiedthissemesterhavebeenlinear inthesensethatourmodelforthemeanhasbeenalinear functionoftheparameters. Wehaveassumed E ( y )= X β f ( X i , β )= X i β issaidtobelinearintheparametersof β because X i β = X i 1 β 1 + X i 2 β 2 + ... + X ip β p isalinearcombinationof β 1 , β 2 ,..., β p . f ( X i , β )= X i β islinearin β evenifthepredictorvariables,the X s arenonlinearfunctionsofothervariables. c 2011Dept.Statistics(IowaStateUniversity) Stat511section25 1/40 Forexample,if X i 1 = 1 X i 2 = Amountoffertilizerappliedtoplot i X i 3 =( Amountoffetrtilizerappliedtoploti ) 2 X i 4 = log ( Concentrationoffungicideonploti ) 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 isstilllinearinthe parameters β 1 , β 2 , β 3 , β 4 . Now,weconsidernonlinearmodelsforthemean E ( y i ) . Thesearemodelswhere f ( X i , β ) cannotbewrittenasalinear combinationof β 1 , β 2 ,.., β p Smalldigression:Whataboutmodelsthatcanbetransformedto belinearintheparameters? c 2011Dept.Statistics(IowaStateUniversity) Stat511section25 2/40 linearizinganon-linearmodel Example:Michaelis-Mentonenzymekineticsmodel v s = v m S S + K m S isconcentrationofsubstrate, v s isreactionrateat S v m ismaximumreactionrate, K m isenzymeaffinity= S atwhich v s = v m / 2 Functionismathematicallyequivalentto: I Lineweaver-Burke: 1 v s = 1 v m + K m v m 1 S Linearregressionof Y = 1 / v s on X = 1 / S I Hanes-Woolf: S v s = K m v m + 1 v m S Linearregressionof Y = S / v s on X = S I Botharelinearregressions c 2011Dept.Statistics(IowaStateUniversity) Stat511section25 3/40 ● ● ● ● ● ● ● ● ● ● ● ● 2 4 6 8 1 . 5 1 . 1 . 5 2 . Substrate conc R e a c t io n v e lo c i t y c 2011Dept.Statistics(IowaStateUniversity) Stat511section25 4/40 However,theestimatorsof v m and K m derivedfromeachmodel arenotthesame Illustratenumerically:LSestimatesfromeachmodel Model ˆ β 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 /β 2.65 β 1 /β 14.96 H-W: 4.74 0.482 1 /β 1 2.07 β /β 1 9.83 Why? Becausethestatisticalmodeladdsaspecificationofvariabilityto themathematicalmodel,e.g. v i = v m S i S i + K m + ε i , ε i ∼ ( ,σ 2 ) c 2011Dept.Statistics(IowaStateUniversity) Stat511section25 5/40 And v i = v m S i S i + K m + ε i , ε i ∼ ( ,σ 2 1 ) (1) isnotthesameas 1 v 1 = 1 v m + K m v m 1 S i + i , i ∼ ( ,σ 2 2 ) (2) Ifyouworkoutallthedetails,(2)isequivalentto(1)withunequal variances The statistical modelsforMM,L-B,andH-Waredifferent Estimatesdifferbecause I Differentvariancemodels I Leverageofspecificobservationsisnotthesame c 2011Dept.Statistics(IowaStateUniversity) Stat511section25 6/40 linearizinganon-linearmodel:2ndexample Exponentialgrowthmodel Y i = β e β 1 T i Nonlinearform,constantvariance: Y i = β e β 1 T i + ε i , ε i ∼ ( ,σ 2 1 ) Linearizedform,constantvariance,normaldist.: Y ∗ i = log Y i = log β + rT i + i ,...
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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25Nonlin - NONLINEARMODELS Wehaveassumed E y)= X β f X i...

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