Statistical Techniques IIEXST7015Curvilinear RegressionCurvilinear RegressionAs 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 modelsThese 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 Yi=b0+b1Xi+ei, fit log(Yi)=b0+b1Xi+ei Curvilinear Regression(continued)If we fit log(Yi) = b0+ b1Xi+ ei Then the original model, before we took logarithms, must have been Yi=b'0expb1Xiei Where "exp" is the base of the natural logarithm (2.718281828) This model is called the "Exponential Growth model" if b1is positive, or the exponential decay model if it is not. It is used in the biological sciences to fit exponential growth (+b1) or mortality (-b1). Curvilinear Regression(continued)Exponential model Yib0expb1Xiei=Exponential growth and decay051015202530350102030Blue b0=34 b1=-0.0953 eb1=0.909Red b0=2 b1=+0.0953 eb1=1.105s-Slr-Curvilinear-Trees 1-6Curvilinear Regression(continued)Other examples of curvilinear models. log(Yi= b0Xib1ei) produces log(Yi)=b0+b1log(Xi) + log(ei)This model is used to fit many things, including morphometric data, A model with an inverse (1/Xi) will fit a "hyperbola", with it's asymptote. Yi= b0+ b1(1/Xi) + ei Curvilinear Regression(continued)Power model Yib0Xib1ei=0510152025300 5 10 15 20 25 30b1=negativeb1=0b1>1b1=10<b1<1b0, b1 29, -1 19,0 4, 0.5 1,1 0.03, 2Curvilinear Regression(continued)Hyperbolic model: Yi= b0+ b1(1/Xi) + ei note that b0fits the asymptote Hyperbolic curves0510152025010203040Yhat = 10 + 10(1/Xi)Yhat = 10 - 10(1/Xi)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. Curvilinear Regression(continued)However, not all are curves can be fitted by linear models with transformations. Some are nonlinear, and require nonlinear curve fitting techniques. For example, Yi= b0Xib1eiis curvilinear Yi= b0Xib1+eiis nonlinear Yi= b0+ b1Xi+ b2X2i+ eiis linear (polynomial) Yi= b0+ b1Xi+ b2Xib3+ eiis nonlinear Curvilinear Regression(continued)Note that Yi= b0Xib1eihas an error multiplied by Xi. This is interesting because when the error is multiplied by the independent variable, the variance about the regression line should appear to increase as Xiincreases. The log transformation (of Yi) should remove this nonhomogeneous variance. This is not true for the log transformation of Xi. 05s-Slr-Curvilinear-Trees 7-12Curvilinear Residual PatternsTransformations of Yi, like log transformations, will affect homogeneity of variance. The raw data should actually appear nonhomogeneous. iXYiiXYiCurvilinear Residual Patterns (continued)Transformations of Xiwill not. iXYiiXYiCurvilinear Residual Patterns (continued)Polynomials assume homogeneous variance and will not adjust variance. iXYiCurvilinear Regression ExamplesAir speed example. A small example from The Science of Flight by Peter P. Wagener, Am Sci, volume 74,(3),May-June 1986, page 274. The author fitted a quadratic model to this data (we will later). However, many examples of technological development over time follow an "exponential" model. So, we will fit an exponential model to this example. Curvilinear Regression ExamplesAir speed example. I digitized the following data from a graph. Like the author, I omit values after 1963 (speed changes little once jet age reached). YEAR SPEED AIRCRAFT1926 108 Ford 5-AT1932 150 247D1935 179 DC-31939 200 307 Strat1941 204 DC-41942 292 L-7491946 304 DC-61947 283 Convair 21947 292 377 strat1950 308 DC-6B1952 354 DC-71954 304 Viscount1951 458 Comet1958 404 L188A Ele1957 550 707/DC-81964 500 BAC1-11-21963 571 727Curvilinear Example 1 (continued)The exponential is a logical and interesting model for this data. Dependent Variable: LOGSPEED Sum of MeanSource DF Squares Square F Value Pr>FModel 1 3.11456238 3.11456238 145.18 0.0001Error 15 0.32179173 0.02145278Corrected Total 16 3.43635410 R-Square C.V. Root MSE LOGSPEED Mean 0.906357 2.579479 0.146468 5.678188Source DF Type I SS Mean Square F Value Pr > FYR 1 3.11456238 3.11456238 145.18 0.0001Source DF Type III SS Mean Square F Value Pr > FYR 1 3.11456238 3.11456238 145.18 0.0001 T for H0: Pr > |T| Std Error ofParameter Estimate Parameter=0 EstimateINTERCEPT 4.750697794 56.04 0.0001 0.08477711YR 0.041602463 12.05 0.0001 0.0034527305s-Slr-Curvilinear-Trees 13-18Curvilinear Example 1 (continued)For exponential models the slope is interpreted as a "proportional" or percentage increase per X variable unit. To find the percentage value per Xiunit, assess EXP(b1) = exp(0.0416) = 1.0425. So there was an average annual increase in speed of 4.25%. Curvilinear Example 1 (continued)Note that I adjusted years (YR = YEAR - 1925) so that 1925 is year zero. Otherwise the zero value would be 1 BC. Another use of exponential models is to calculate doubling times or half-life values. Yi= b0at time = Xi= 0, so how long does it take to get to speed=2b0? Set 2b0=b0expb1X', 2=expb1X', log(2)=b1X', log(2)/b1= X', X' = 0.693/0.0416 = 16.67. So speed doubled every 16.67 years. Curvilinear Example 1 (continued)Exponential models. What can I say? Good fit, Few d.f. (basically a SLR), clear