Statistical Techniques IIEXST7015Curvilinear Regression11a_CurvilinearPoly 1Curvilinear 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. 11a_CurvilinearPoly 2Curvilinear Regression (continued)We will discuss two basic types of curvilinear model. The first are 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.The second 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. 11a_CurvilinearPoly 3Polynomial RegressionPolynomial regressions are multiple regressions that use power terms of the Xivariable to fit curves. As long as the value of the power is known, the model is linear. Only a single Xiis needed (though more can be used). The assumptions are the same as for any other multiple regression. 11a_CurvilinearPoly 4Polynomial Regression (continued)Polynomial regressions are of the formYi= b0+ b1Xi+ b2X2i+ b3X3i+ ... + bkXki+ ei The simplest in this family of models is the "linear", which is just a simple linear regression. Polynomials proceed, Quadratic Yi= b0+ b1Xi+ b2X2i+ eiCubic Yi= b0+ b1Xi+ b2X2i+ b3X3i+ ei Quartic Yi= b0+ b1Xi+ b2X2i+ b3X3i+ b4X4i+ ei Quintic, etc. 11a_CurvilinearPoly 5Polynomial Regression (continued)The quadratic fits a simple parabolic curve. Either concave or convex, depending on the sign on the regression coefficient.XYXY11a_CurvilinearPoly 6Polynomial Regression (continued)The cubic fits parabolic curves with an inflection. The inflection does not always occur within the range of the data. XYXYInflection11a_CurvilinearPoly 7Polynomial Regression (continued)The quartic polynomial adds another inflection, and another peak or valley (maximum or minimum point). These are not usually symmetric. XYXY11a_CurvilinearPoly 8Polynomial Regression (continued)The same pattern continues for larger models. XY11a_CurvilinearPoly 9Polynomial Regression (continued)What good are polynomials? They will fit anything. In fact, if no two X values are repeated, then a large enough polynomial will go through every observation.A SLR exactly fits 2 points A quadratic polynomial will exactly fit 3 pointsA cubic will pass through each of 4 points For n points, n-1 polynomial terms will pass through every point. 11a_CurvilinearPoly 10Polynomial Regression (continued)Sounds like a good thing? Only if you want to fit random scatter. How would you interpret the graph below? XYNote11a_CurvilinearPoly 11Polynomial Regression (continued)Recall the air speed example from The Science of Flight by Peter P. Wagener, Am Sci, volume 74,(3),May-June 1986, page 274.We previously fitted an exponential growth curve with good results. Air speed example. I digitized the following data from a graph and omitted values after 1963. 11a_CurvilinearPoly 12Polynomial Reg. - Example 1YEAR 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 727Year of Airplane introduction01002003004005006007001920 1930 1940 1950 1960Air speed (mph)11a_CurvilinearPoly 13Polynomial Reg. - Example 1 (continued)We will now proceed to fit a polynomial model (quadratic) to the data. This was the model chosen by the author. 11a_CurvilinearPoly 14Polynomial Reg. - Example 1 (continued)The SAS statements are,PROC GLM DATA=ONE; MODEL SPEED = YEAR YEAR*YEAR;RUN;Note that I used Year*Year to fit YEAR squared. You can do this in GLM, but not in PROC REG. The GLM output follows. 11a_CurvilinearPoly 15Polynomial Reg. - Example 1 (continued)PROC GLM on airspeed example. Dependent Variable: SPEED Sum ofSource DF Squares Mean Square F Value Pr>FModel 2 405441.6302 202720.8151 74.75 <.0001Error 17 46104.9198 2712.0541Corrected Total 19 451546.5500R-Square Coeff Var Root MSE SPEED Mean0.897896 14.56099 52.07739 357.6500Source DF Type I SS Mean Square F Value Pr > FYEAR 1 404010.2946 404010.2946 148.97 <.0001YEAR*YEAR 1 1431.3356 1431.3356 0.53 0.4774 11a_CurvilinearPoly 16Polynomial Reg. - Example 1 (continued)There is clearly a linear increasing trend over time (P>F)<0.0001. However, the additional term for quadratic curvature is not significant. There is not apparently any significant curvature in this example. At least not of a parabolic shape. 11a_CurvilinearPoly 17Poly. Reg. - Ex 1 (continued)Airspeed example with linear (blue) and quadratic (red) models fitted. 01002003004005006001920 1930 1940 1950 1960Year of Airplane introductionAir speed (mph)11a_CurvilinearPoly 18Polynomial Reg. - Example 1 (continued)There is not a good test between the two models (exponential fitted earlier and quadratic/linear here). However, the exponential fitted well, adjusted for possible nonhomogeneous variance and was readily interpretable. The quadratic is not justified, and would reduce to a SLR. Also simple and interpretable. 11a_CurvilinearPoly 19Polynomial Reg. - Example 210 K Race Results - Vermont. Separate race results for 527 Women & 963 MenHypothesize that fastest runners will be neither the oldest nor the youngest. This can be fitted with a polynomial. Scatter plots for the two sexes, and the regression were run in SAS (below). 11a_CurvilinearPoly 20Polynomial Reg. - Example 2 (continued)Scatter plot Sex=Fsex=F Plot of TIME*Age. Legend: A = 1 obs, B = 2 obs, etc. TIME | | | A A A A B A | A A A A A A A A A A A A | B A A B B A A A A B A A A B A 280 + A A A A A A B A A B A A A A A | A A A A A