05a_CurvilinearTrans(1)

05a_CurvilinearTrans - Curvilinear Regression Statistical Techniques II EXST7015 Curvilinear Regression Curvilinear Regression(continued We will

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Statistical Techniques II EXST7015 Curvilinear Regression Curvilinear Regression As the name implies, these are regressions that fit curves. However, the regressions we will discuss are also linear models, so most of the techniques and SAS procedures we have discussed will still be relevant. Curvilinear Regression (continued) We will discuss two basic types of curvilinear model. The first are models that are not linear, but that can be "linearized" by transformation. These models are referred to as "intrinsically linear", because after transformation they are linear, often SLR. Later we will cover polynomial regressions. These are an extraordinarily flexible family of curves that will fit almost anything. Unfortunately, they rarely have a good, interpretation of the parameter estimates. Curvilinear Regression (continued) Intrinsically linear models These are models that contain some transformed variable, logarithms, inverses, square roots, sines, etc. We will concentrate on logarithms, since these models are some of the most useful. What is the effect of taking a logarithm of a dependent or independent variable? For example, instead of Y i =b 0 +b 1 X i +e i , fit log(Y i )=b 0 +b 1 X i +e i Curvilinear Regression (continued) If we fit log(Y i ) = b 0 + b 1 X i + e i Then the original model, before we took logarithms, must have been Y i =b' 0 exp b1Xi e i Where " exp " is the base of the natural logarithm (2.718281828) This model is called the "Exponential Growth model" if b 1 is positive, or the exponential decay model if it is not. It is used in the biological sciences to fit exponential growth (+b 1 ) or mortality (-b 1 ). Curvilinear Regression (continued) Exponential model Y i b 0 exp b 1 X i e i = Exponential growth and decay 0 5 10 15 20 25 30 35 01 02 03 0 Blue b 0 =34 b 1 =-0.0953 e b1 =0.909 Red b 0 =2 b 1 =+0.0953 e b1 =1.1 05s-Slr-Curvilinear-Trees 1-6
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Curvilinear Regression (continued) Other examples of curvilinear models. log(Y i = b 0 X i b1 e i ) produces log(Y i )=b 0 +b 1 log(X i ) + log(e i ) This model is used to fit many things, including morphometric data, A model with an inverse (1/X i ) will fit a "hyperbola", with it's asymptote. Y i = b 0 + b 1 (1/X i ) + e i Curvilinear Regression (continued) Power model Y i b 0 X i b 1 e i = 0 5 10 15 20 25 30 0 5 10 15 20 25 30 b 1 =negative b 1 =0 b 1 >1 b 1 =1 0<b 1 <1 b0, b1 29, -1 19,0 4, 0.5 1,1 0.03, 2 Curvilinear Regression (continued) Hyperbolic model: Y i = b 0 + b 1 (1/X i ) + e i note that b 0 fits the asymptote Hyperbolic curves 0 5 10 15 20 25 01 0 2 0 3 0 4 0 Yhat = 10 + 10(1/X i ) Yhat = 10 - 10(1/X i ) Curvilinear Regression (continued) These are a few of many possible curvilinear regressions. Models including power terms, exponents, logarithms, inverses, roots, and trigonometric functions fit may be curvilinear.
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This note was uploaded on 12/29/2011 for the course EXST 7087 taught by Professor Wang,j during the Fall '08 term at LSU.

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05a_CurvilinearTrans - Curvilinear Regression Statistical Techniques II EXST7015 Curvilinear Regression Curvilinear Regression(continued We will

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