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Topic 9: Remedies

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Outline Review diagnostics for residuals Discuss remedies –Nonlinear relationship –Nonconstant variance –Non-Normal distribution –Outliers

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Diagnostics for residuals Look at residuals to find serious violations of the model assumptions –nonlinear relationship –nonconstant variance –non-Normal errors presence of outliers a strongly skewed distribution

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Recommendations for checking assumptions Plot Y vs X (is it a linear relationship?) Look at distribution of residuals Plot residuals vs X, time, or any other potential explanatory variable Use the i=sm## in symbol statement to get smoothed curves

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Plots of Residuals Plot residuals vs –Time (order) –X or predicted value (b 0 +b 1 X) Look for –nonrandom patterns –outliers (unusual observations)

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Residuals vs Order Pattern in plot suggests dependent errors / lack of indep Pattern usually a linear or quadratic trend and/or cyclical If you are interested read KNNL pgs

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Residuals vs X Can look for –nonconstant variance –nonlinear relationship –outliers –somewhat address Normality of residuals

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Tests for Normality H 0 : data are an i.i.d. sample from a Normal population H a : data are not an i.i.d. sample from a Normal population KNNL (p 115) suggest a correlation test that requires a table look-up

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Tests for Normality We have several choices for a significance testing procedure Proc univariate with the normal option provides four proc univariate normal; Shapiro-Wilk is a common choice

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Tests for Normality TestStatisticp Value Shapiro-WilkW Pr < W Kolmogorov-SmirnovD Pr > D> Cramer-von MisesW-Sq Pr > W-Sq> Anderson-DarlingA-Sq Pr > A-Sq> All P-values > 0.05…Do not reject H0

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Other tests for model assumptions Durbin-Watson test for serially correlated errors (KNNL p 114) Modified Levene test for homogeneity of variance (KNNL p ) Breusch-Pagan test for homogeneity of variance (KNNL p 118) For SAS commands see topic9.sas

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Plots vs significance test Plots are more likely to suggest a remedy Significance tests results are very dependent on the sample size; with sufficiently large samples we can reject most null hypotheses

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Default graphics with SAS 9.3 proc reg data=toluca; model hours=lotsize; id lotsize; run;

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Will discuss these diagnostics more in multiple regression Provides rule of thumb limits Questionable observation (30,273)

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Additional summaries Rstudent: Studentized residual…almost all should be between ± 2 Leverage: “Distance” of X from center…helps determine outlying X values in multivariable setting…outlying X values may be influential Cooks’D: Influence of i th case on all predicted values

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Lack of fit When we have repeat observations at different values of X, we can do a significance test for nonlinearity Browse through KNNL Section 3.7 Details of approach discussed when we get to KNNL 17.9, p 762 Basic idea is to compare two models Gplot with a smooth is a better (i.e., simpler) approach

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SAS code and output proc reg data=toluca; model hours=lotsize / lackfit; run; Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Lack of Fit Pure Error Corrected Total

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Nonlinear relationships We can model many nonlinear relationships with linear models, some have several explanatory variables (i.e., multiple linear regression) –Y = β 0 + β 1 X + β 2 X 2 + e (quadratic) –Y = β 0 + β 1 log(X) + e

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Nonlinear Relationships Sometimes can transform a nonlinear equation into a linear equation Consider Y = β 0 exp(β 1 X) + e Can form linear model using log log(Y) = log(β 0 ) + β 1 X + log(e) Note that we have changed our assumption about the error

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Nonlinear Relationship We can perform a nonlinear regression analysis KNNL Chapter 13 SAS PROC NLIN

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Nonconstant variance Sometimes we model the way in which the error variance changes –may be linearly related to X We can then use a weighted analysis KNNL 11.1 Use a weight statement in PROC REG

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Non-Normal errors Transformations often help Use a procedure that allows different distributions for the error term –SAS PROC GENMOD

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Generalized Linear Model Possible distributions of Y: –Binomial (Y/N or percentage data) –Poisson (Count data) –Gamma (exponential) –Inverse gaussian –Negative binomial –Multinomial Specify a link function for E(Y)

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Ladder of Reexpression (transformations) p Transformation is x p

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Circle of Transformations X up, Y up X up, Y down X down, Y up X down, Y down X Y

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Box-Cox Transformations Also called power transformations These transformations adjust for non-Normality and nonconstant variance Y´ = Y or Y´ = (Y - 1)/ In the second form, the limit as approaches zero is the (natural) log

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Important Special Cases = 1, Y´ = Y 1, no transformation =.5, Y´ = Y 1/2, square root = -.5, Y´ = Y -1/2, one over square root = -1, Y´ = Y -1 = 1/Y, inverse = 0, Y´ = (natural) log of Y

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Box-Cox Details We can estimate by including it as a parameter in a non-linear model Y = β 0 + β 1 X + e and using the method of maximum likelihood Details are in KNNL p SAS code is in boxcox.sas

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Box-Cox Solution Standardized transformed Y is –K 1 (Y - 1) if ≠ 0 –K 2 log(Y) if = 0 where K 2 = ( Y i ) 1/n (the geometric mean) and K 1 = 1/ ( K 2 -1 ) Run regressions with X as explanatory variable estimated minimizes SSE

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data a1; input age plasma cards; ; Example

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Box Cox Procedure *Procedure that will automatically find the Box-Cox transformation; proc transreg data=a1; model boxcox(plasma)=identity(age); run;

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Transformation Information for BoxCox(plasma) Lambda R-Square Log Like * < * < - Best Lambda * - Confidence Interval + - Convenient Lambda

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* The first part of the program gets the geometric mean; data a2; set a1; lplasma=log(plasma); proc univariate data=a2 noprint; var lplasma; output out=a3 mean=meanl;

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data a4; set a2; if _n_ eq 1 then set a3; keep age yl l; k2=exp(meanl); do l = -1.0 to 1.0 by.1; k1=1/(l*k2**(l-1)); yl=k1*(plasma**l -1); if abs(l) < 1E-8 then yl=k2*log(plasma); output; end;

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proc sort data=a4 out=a4; by l; proc reg data=a4 noprint outest=a5; model yl=age; by l; data a5; set a5; n=25; p=2; sse=(n-p)*(_rmse_)**2; proc print data=a5; var l sse;

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Obs l sse

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symbol1 v=none i=join; proc gplot data=a5; plot sse*l; run;

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data a1; set a1; tplasma = plasma**(-.5); tage = (age+.5)**(-.5); symbol1 v=circle i=sm50; proc gplot; plot tplasma*age; proc sort; by tage; proc gplot; plot tplasma*tage; run;

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Background Reading Sections describe significance tests for assumptions (read it if you are interested). Box-Cox transformation is in nknw132.sas Read sections 4.1, 4.2, 4.4, 4.5, and 4.6

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