1 / 45

Diagnosing and Remedying Model Assumptions for Residual Analysis in Statistical Models

This guide explores essential diagnostics for assessing residuals in statistical models. It emphasizes identifying serious violations of model assumptions such as nonlinear relationships, nonconstant variance, non-normal distribution, and outliers. Recommendations include plotting residuals against explanatory variables and testing for normality, as well as discussing approaches for obtaining smoothed curves through SAS procedures. The document also highlights transformation techniques (e.g., Box-Cox) to address non-normality and variance issues, ensuring accurate and reliable statistical inferences.

asasia
Télécharger la présentation

Diagnosing and Remedying Model Assumptions for Residual Analysis in Statistical Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Topic 9: Remedies

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

  3. 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

  4. 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

  5. Plots of Residuals • Plot residuals vs • Time (order) • X or predicted value (b0+b1X) • Look for • nonrandom patterns • outliers (unusual observations)

  6. 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 108-110

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

  8. Tests for Normality • H0: data are an i.i.d. sample from a Normal population • Ha: data are notan i.i.d. sample from a Normal population • KNNL (p 115) suggest a correlation test that requires a table look-up

  9. 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

  10. All P-values > 0.05…Do not reject H0

  11. Other tests for model assumptions • Durbin-Watson test for serially correlated errors (KNNL p 114) • Modified Levene test for homogeneity of variance (KNNL p 116-118) • Breusch-Pagan test for homogeneity of variance (KNNL p 118) • For SAS commands see topic9.sas

  12. 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

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

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

  15. 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 ith case on all predicted values

  16. 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

  17. SAS code and output proc reg data=toluca; model hours=lotsize / lackfit; run;

  18. Nonlinear relationships • We can model many nonlinear relationships with linear models, some have several explanatory variables (i.e., multiple linear regression) • Y = β0 + β1X+ β2X2+ e(quadratic) • Y = β0 + β1log(X)+ e

  19. Nonlinear Relationships • Sometimes can transform a nonlinear equation into a linear equation • Consider Y = β0exp(β1X)+ e • Can form linear model using log log(Y) = log(β0) + β1X + log(e) • Note that we have changed our assumption about the error

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

  21. 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

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

  23. 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)

  24. Ladder of Reexpression(transformations) 1.5 p Transformation is xp 1.0 0.5 0.0 -0.5 -1.0

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

  26. 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

  27. Important Special Cases •  = 1, Y´ = Y1, no transformation •  = .5, Y´ = Y1/2, square root •  = -.5, Y´ = Y-1/2, one over square root •  = -1, Y´ = Y-1 = 1/Y, inverse •  = 0, Y´ = (natural) log of Y

  28. Box-Cox Details • We can estimate  by including it as a parameter in a non-linear model • Y = β0 + β1X+ e and using the method of maximum likelihood • Details are in KNNL p 134-137 • SAS code is in boxcox.sas

  29. Box-Cox Solution • Standardized transformed Y is • K1(Y - 1) if  ≠ 0 • K2log(Y) if  = 0 where K2 = ( Yi)1/n (the geometric mean) and K1 = 1/ ( K2 -1) • Run regressions with X as explanatory variable • estimated  minimizes SSE

  30. Example data a1; input age plasma @@; cards; 0 13.44 0 12.84 0 11.91 0 20.09 0 15.60 1 10.11 1 11.38 1 10.28 1 8.96 1 8.59 2 9.83 2 9.00 2 8.65 2 7.85 2 8.88 3 7.94 3 6.01 3 5.14 3 6.90 3 6.77 4 4.86 4 5.10 4 5.67 4 5.75 4 6.23 ;

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

  32. Transformation Information for BoxCox(plasma) Lambda R-Square Log Like -2.50 0.76 -17.0444 -2.00 0.80 -12.3665 -1.50 0.83 -8.1127 -1.00 0.86 -4.8523 * -0.50 0.87 -3.5523 < 0.00 + 0.85 -5.0754 * 0.50 0.82 -9.2925 1.00 0.75 -15.2625 1.50 0.67 -22.1378 2.00 0.59 -29.4720 2.50 0.50 -37.0844 < - Best Lambda * - Confidence Interval + - Convenient Lambda

  33. *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;

  34. 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;

  35. 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;

  36. Obs l sse 1 -1.0 33.9089 2 -0.9 32.7044 3 -0.8 31.7645 4 -0.7 31.0907 5 -0.6 30.6868 6 -0.5 30.5596 7 -0.4 30.7186 8 -0.3 31.1763 9 -0.2 31.9487 10 -0.1 33.0552

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

  38. 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;

  39. Background Reading • Sections 3.4 - 3.7 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

More Related