EXST 7015 Fall 2014 Lab 08: Polynomial Regression

Transcription

1 EXST 7015 Fall 2014 Lab 08: Polynomial Regression OBJECTIVES Polynomial regression is a statistical modeling technique to fit the curvilinear data that either shows a maximum or a minimum in the curve, or that could show a max or min if you extrapolated the curve beyond your data. The ability to determine a minimum or maximum point based on the experimental data is a useful application of polynomials. The simple polynomial regressions are multiple regression that use power terms of the independent variable (X i ) with the form of Y=β 0 + β 1 X i + β 2 X i β k X i k + e i. Notice the subtle difference from multiple linear regression model, here the numbers 2, 3,,k represent the powers of the same variable. For data that are shaped like a parabola, you probably won't need more than a quadratic model. If the curve trends up again at one end, you'll need a cubic model. Curves with multiple kinks need even higher-order terms. It is obvious that multi collinearity is unavoidable issue in Polynomial regression because the model terms are related to each other. Use of sequentially adjusted type I SS is the solution as presented as the following. Several hypotheses are tested during polynomial regression which is fitted successively starting with the liner term (a first order polynomial). The first null hypothesis, then, is that a quadratic equation does not fit the data significantly better than a linear equation; the next null hypothesis may be that a cubic equation does not fit the data significantly better than a quadratic equation, and so on. Therefore, the sequentially adjusted type I SS should be used when one attempts to test whether a polynomial model is as good as the one with a higher order term. If a particular higher order term is significant, all terms of lower order should be assumed significant and retained in the regression model. There is also a null hypothesis for each equation that says that it does not fit the data significantly better than a horizontal line; in other words, that there is no relationship between the X and Y variables. It should be noticed that the fully adjusted regression coefficients are still used to fit the polynomial regression, which usually leads to no practical explanation of regression coefficients. Further, extrapolation outside the range of the fitted experimental data is untenable, and should not be attempted. This is because the shape of the regression function, as predicted by the model, may not at all accurately represent reality outside the range of the experimental data. The assumptions for performing polynomial regression are similar to those of ordinary regression. The assumptions of normality and homogeneity can be evaluated by customary diagnostic technique (Shapiro-Wilk, residual plots). Residual and influence statistics still work with these regression models.

2 LABORATORY INSTRUCTIONS Housekeeping Statements dm 'log; clear; output; clear'; options nodate nocenter pageno = 1 ls=78 ps=53; title1 'EXST7015 lab 8, Name, Section#'; ods rtf file = 'c:/temp/lab8.rtf'; ods html file = 'c:/temp/lab8.html'; Data Set: The data set that we use contains the per capita state and local public expenditures and associated state demographic and economic characteristics for 48 states during the year of Detailed information can be found at The variables in the dataset are: EX: Per capita state and local public expenditures ($) ECAB: Economic ability index, in which income, retail sales, and the value of output (manufactures, mineral, and agricultural) per capita are equally weighted. MET: Percentage of population living in standard metropolitan areas GROW: Percent change in population, YOUNG: Percent of population aged 5-19 years OLD: Percent of population over 65 years of age WEST: Western state (1) or not (0) In this lab, we will use MET, Percentage of population living in standard metropolitan area, as the independent variable and the expenditure (EX) as the dependent variable. Other variables are dropped from the dataset. data expenditure; input ex ecab met grow young old west state$; drop ecab grow young old west; cards; ME NH VT MA RI CT NY NJ PA DE MD VA MI OH IN IL WI WV KY

3 TE NC SC GA FL AL MS MN IA MO ND SD NB KS LA AR OK TX NM AZ MT ID WY CO UT WA OR NV CA ; Proc print data=expenditure; proc gplot data=expenditure; plot ex*met; title3 "Plot of raw data"; symbol1 c=red v=dot; Proc gplot: It can produce high-resolution graphic plot, and improve the appearance of the plot by defining plot symbols. C=red specifies the color of the symbol. V=dot would place a dot for each data point. Take time to exam the plot, can you spot a maximum or minimum point? Fitting a Polynomial Regression Model with a cubic term of MET: proc glm data=expenditure; title2 "Polynomial regression model with cubic term"; model ex = met met*met met*met*met / ss1 ss2; output out=outdata1 p=yhat1 r=resid1; proc plot data=outdata1;

4 title3 'residual plot'; plot resid1*yhat1; proc univariate data=outdata1 normal; title3 'univariate procedure on residuals'; var resid1; Notice that PROC GLM is used for polynomial regression rather than PROC REG. While they are essentially the same for the analysis, there is one difference. When using PROC GLM, one can enter the higher order terms directly at the model statement, does not need to create new variables in the data step for each of them, which on the contrary has to be done before PROC REG works properly. Carefully exam the F-tests of parameter estimates. Keep in mind that the sequentially adjusted type I sums of squares are used for polynomial regression. Fitting a Polynomial Regression Model with a Quadratic term of MET: proc glm data=expenditure; title2 "Polynomial regression model with cubic term"; model ex = met met*met / ss1 ; output out=outdata2 p=yhat2 r=resid2; proc plot data=outdata2; title2 'residual plot'; plot resid2*yhat2; proc univariate data=outdata2 normal; title3 'univariate procedure on residuals'; var resid2; ods rtf close; quit;

5 LAB ASSIGNMENT 1. Describe the trend in the scatter plot of the raw data: what is the relationship between variables EX and MET? 2. Fit a polynomial regression model with cubic term of MET. When you decide whether the cubic term and quadratic term should be included in the model, do you use the Type I SS or Type II SS? Why? 3. Is the cubic effect significant? How about quadratic and linear effects? 4. Based on your answers to the above questions and the SAS output, which polynomial model do you consider the best? Write down the polynomial model with the estimated coefficient values. Do you keep the linear term in the model? Why? 5. Now assume that there is a state where 100 percent of its residents live in standard metropolitan areas. Use the best model to predict the per capita public expenditure of this state. Is there any problem in doing so?