EXST7034 - Regression Techniques Page 1Transformations - your textbook takes a rather simplistic approach, does not getinto modelling aspects or interpretation Why transform? Data appears curved, LOF shows linear fit not adequate. Is thevariance homogeneous? (Probably not).MAJOR CONSIDERATION - do you have any reason to suspect any particularcurved, functional relationship? Your textbook mentions theoreticalconsiderations late as a “comment". Transformations of X, used to fit curvature when variance is homogeneous andnormaliXYi a) log(X ) or ln(X ) or X 333ÈiXYi b) X or e 32X3iXYi c) or e 1X-X33EXST7034 - Regression Techniques Page 2Transformations of Y, used to fit curvature when variance is NOT homogeneousand NOT normalDo you ever EXPECT to see nonhomogeneous variance? If you use these modelsyou do.iXYi a) log(Y ) or ln(Y ) or Y or 333È1Y3iXYiiXYi The transformation of X and Y can also be done simultaneously. Generally therewould be some rhyme and reason to this. Don't use a square roottransformation of Y and a log transformation of X.EXST7034 - Regression Techniques Page 3Box-Cox transformations : Choose the power of Y which minimizes the regressionof Y on X.- when = 2, then the transformation is Y-2 = 1 Y = 0.5 YÈ = 0 log(Y) = 0.5 1YÈ = 1 1Y = 2 1Y2Note that since the Y values are transformed, the total SS and other SS are ofdiffering magnitudes. The best way to compare is using a standardizedvalues of some for. The recommended form is W K (Y 1) for 0 and K (log Y) for 0œ Á œ"#/--- where K Y and K#3 "3œ"8œœ”•#"8"#1K--This approach is intended primarily as an , not as a way toaid in model selectionfind the perfect model.In my judgment, theoretical considerations take precedence.CURVILINEAR AND NONLINEAR REGRESSION We will examine a number of Curvilinear techniques. POLYNOMIALS willcome later. We will also look at NONLINEAR techniques later, particularly to see how theydiffer from curvilinear techniques. Nonlinear techniques are iterative. There is no simple, unique least squaressolution, so different values of the parameters to be estimated are triedand adjusted until the best combination is found.EXST7034 - Regression Techniques Page 4CURVILINEAR MODELS 1) Y = b X e log log model3!33b"this model is used to fit meristic or morphometric relationships.they are linearized by taking logarithms ln(Y ) = ln(b ) b ln(X ) ln(e )3!"3 3 which is actually a simple linear regressionLog - log Models0102030405060708090100051015202530EXST7034 - Regression Techniques Page 5There are 4 models commonly used for morphometric relationships Linear, “Direct proportion", Log-log and PolynomialsThe correct model can often be determined by hypothesis testing. a) Linear: if H : = 0 is not rejected then use Direct Proportion!!" b) Log-log: if H : = 1 is not rejected then use Direct Proportion!"" c) Polynomial: if H : = 0 is not rejected then use Linear!#" d) Polynomial: if H : = 0 is not rejected then use Direct Proportion!""However, determination between Linear and Log-log cannot be made byhypothesis testing. Use other CONSIDERATIONS a) Is the variance homogeneous? Yes Linear; No Log-logpp b) Should the line go through the origin? No Linear; Yes Log-logpp c) Does the line appear curved? No Linear; Yes Log-logppEXST7034 - Regression Techniques Page 61) LENGTH - WEIGHT RELATIONSHIP a) always use the model W = L3! 33"%"" Log(W ) = Log( ) + * Log(L ) + Log( )3!"33"" % b) in SAS INPUT ... LT WT ...; LWT = LOG(WT); LLT = LOG(LT); PROC GLM: MODEL LWT = LLT; c) statistically all assumptions apply except, (a) multiplicative (non-homogeneous) error is implied for raw data because of theoriginal model (b) Ricker (1973) discusses the fact that length is probably not measured withouterrorEXST7034 - Regression Techniques Page 72) Y = b e e = N e e3! 3 !3bX rt3 this models exponential growth (or decay => mortality) used for modeling short term growth ( ) or mortality ( ) linearize by logs ln(Y ) = ln(b) bX ln(e )Ê333Exponential growth and decay curves010020030040050060070080090010000 5 10 15 20 25 30 3) Y = b b e3! " 31X3 used for some recruitment models hyperbolic model where b is the asymptote!Hyperbolic models-50510152003691215EXST7034 - Regression Techniques Page 84) Y = b X e e3!3 3bX"3 This model is the Ricker Stock Recruit Model, it is linearized (best) by dividingthrough by X and then logs ln = ln(b ) bX e’“YX33!33EXST7034 - Regression Techniques Page 9NOTES on curvilinear models from transformations Those curvilinear models shown above areSIMPLE LINEAR REGRESSIONS AFTER TRANSFORMATION The assumptions are made for the transformed models, not for the untransformedversion, and the assumptions are the same as for any simple linearregression. The tests of hypothesis and other diagnostics should be done on the transformedversion of the model. Confidence intervals are done on the transformed variables. Predicted values canbe obtained from the transformed version and untransformed, or directlyfrom the reconstructed original model. However, confidence limitsshould obtained from the transformed, linear model and detransformed.EXST7034 - Regression Techniques Page 10Examples of Linear and Curvilinear ModelsLINEAR MODELS - Y = b bX e LINEAR3! "3 3 Y = b bX bX e LINEAR3 ! " "3 # #3 3 Y = b bX bX bX X e LINEAR (interaction)3 "3#3"3#33  the cross product of independent variables, when used as an independent variable,for the interaction or the two variables. Since the value of both is known,this is also just another term in a multiple regression and still linear.NONLINEAR MODELS PARAMETER Y = b X e cannot take logarithms with addition sign3! 3ib" Y = b X e log of addition, even for error term3! 33b" Y = b e e this is model fitted by NLIN, not GLM3! 3bX"3 Y = L 1 e von Bertalanffy growth curve3_‘-k(t-t )! Y = L e Gompertz growth curve3!L(1e)_-kt Y = Logistic growth curve3L1+ e__!!Š‹LLL-kt Y = b b X b X e with and UNKNOWN POWER3! "3 # 33 P is a NONLINEAR model Y = b + bX bX bX e CURVILINEAR (polynomial)3!"3 # $ 33$32 However, if