Lecture 4. Checking Model Adequacy

Transcription

1 Lecture 4. Checking Model Adequacy Montgomery: 3-4, Page 1

2 Model Checking and Diagnostics Model Assumptions 1 Model is correct 2 Independent observations 3 Errors normally distributed 4 Constant variance y ij = (y.. +(y i. y.. )) + (y ij y i. ) y ij = ŷ ij + ˆɛ ij observed = predicted + residual Note that the predicted response at treatment i is ŷ ij =ȳ i. Diagnostics use predicted responses and residuals. Page 2

3 Diagnostics Normality Histogram of residuals Normal probability plot / QQ plot Shapiro-Wilk Test Constant Variance Plot ˆɛ ij vs ŷ ij (residual plot) Bartlett s or Levene s Test Independence Plot ˆɛ ij vs time/space Plot ˆɛ ij vs variable of interest Outliers Page 3

4 Constant Variance In some experiments, error variance (σ 2 i ) depends on the mean response E(y ij )=µ i = µ + τ i. So the constant variance assumption is violated. Size of error (residual) depends on mean response (predicted value) Residual plot Plot ˆɛ ij vs ŷ ij Is the range constant for different levels of ŷ ij More formal tests: Bartlett s Test Modified Levene s Test. Page 4

5 Bartlett s Test H 0 : σ 2 1 = σ 2 2 =...= σ 2 a Test statistic: χ 2 0 = q c where q =(N a)log 10 S 2 p a i=1 (n i 1)log 10 S 2 i c =1+ 1 3(a 1) ( a i=1 (n i 1) 1 (N a) 1 ) and S 2 i is the sample variance of the ith population and S2 p is the pooled sample variance. Decision Rule: reject H 0 when χ 2 0 >χ 2 α,a 1. Remark: sensitive to normality assumption. Page 5

6 Modified Levene s Test For each fixed i, calculate the median m i of y i1,y i2,...,y ini. Compute the absolute deviation of observation from sample median:[ d ij = y ij m i for i =1, 2,...,aand j =1, 2,...,n i, Apply ANOVA to the deviations: d ij Use the usual ANOVA F -statistic for testing H 0 : σ1 2 =...= σa 2. Page 6

7 options ls=80 ps=65; title1 Diagnostics Example ; data one; infile c:\saswork\data\tensile.dat ; input percent strength time; proc glm data=one; class percent; model strength=percent; means percent / hovtest=bartlett hovtest=levene; output out=diag p=pred r=res; proc sort; by pred; symbol1 v=circle i=sm50; title1 Residual Plot ; proc gplot; plot res*pred/frame; run; proc univariate data=diag normal noprint; var res; qqplot res / normal (L=1 mu=est sigma=est); histogram res / normal; run; Page 7

8 run; proc sort; by time; symbol1 v=circle i=sm75; title1 Plot of residuals vs time ; proc gplot; plot res*time / vref=0 vaxis=-6 to 6 by 1; run; symbol1 v=circle i=sm50; title1 Plot of residuals vs time ; proc gplot; plot res*time / vref=0 vaxis=-6 to 6 by 1; run; Page 8

9 Diagnostics Example Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Levene s Test for Homogeneity of strength Variance ANOVA of Squared Deviations from Group Means Sum of Mean Source DF Squares Square F Value Pr > F percent Error Bartlett s Test for Homogeneity of strength Variance Source DF Chi-Square Pr > ChiSq percent Page 9

10 Non-constant Variance: Impact and Remedy Does not affect F-test dramatically when experiment is balanced Why concern? Comparison of treatments depends on MSE Incorrect intervals and comparison results Variance-Stabilizing Transformations Common transformations x, log(x), 1/x, arcsin( x), and 1/ x Box-Cox transformations 1. approximate the relationship σ i = θµ β i X 1 β 2. use maximum likelihood principle Distribution often more normal after transformation, then the transformation is Page 10

11 Ideas for Finding Proper Transformations Consider response Y with mean E(Y )=µ and variance Var(Y )=σ 2. That σ 2 depends on µ leads to nonconstant variances for different µ. Let f be a transformation and Ỹ = f(y ); What is the mean and variance of Ỹ? Approximate f(y ) by a linear function (Delta Method): f(y ) f(µ)+(y µ)f (µ) Mean µ = E(Ỹ )=E(f(Y )) E(f(µ)) + E((Y µ)f (µ)) = f(µ) Variance σ 2 = Var(Ỹ ) [f (µ)] 2 Var(Y )=[f (µ)] 2 σ 2 f is a good transformation if σ 2 does not depend on µ anymore. So, Ỹ has constant variance for different f(µ). Page 11

12 Transformations Suppose σ 2 is a function of µ, that is σ 2 = g(µ) Want to find transformation f such that Var(Ỹ ) does not depend on µ. Ỹ = f(y ) has constant variance: Have shown Var(Ỹ ) [f (µ)] 2 σ 2 [f (µ)] 2 g(µ) Want to choose f such that [f (µ)] 2 g(µ) c Examples g(µ) =µ (Poisson) f(x) = 1 µ dµ f(x) = X g(µ) =µ(1 µ) (Binomial) f(x) = 1 dµ f(x) = arcsin( X) µ(1 µ) g(µ) =µ 2β (Box-Cox) f(x) = µ β dµ f(x) =X 1 β g(µ) =µ 2 (Box-Cox) f(x) = 1 µ dµ f(x) = log X Page 12

13 Identify Box-Cox Transformation Using Data: Approximate Method From the previous slide, if σ = θµ β, the transformation is Y 1 β β 1; f(y )= log Y β =1 So it is crucial to estimate β based on data y ij, i =1,...,a. We have logσ i = logθ + βlogµ i Let s i and ȳ i. be the sample standard deviations and means. Because ˆσ i = s i and ˆµ i =ȳ i., approximately, where i =1,...,a. logs i = constant + βlogȳ i., We can plot logs i against logȳ i., fit a straight line and use the slope to estimate β. Page 13

14 Identify Box-Cox Transformation: Formal Method 1. For a fixed λ, perform analysis of variance on y ij (λ) = y λ ij 1 λẏ λ 1 λ 0 ẏ log y ij λ =0 whereẏ = a n i y ij i=1 j=1 1/N. 2. Step 1 generates a transformed data y ij (λ). Apply ANOVA to the new data and obtain SS E. Because SS E depends on λ, it is denoted by SS E (λ). Repeat 1 and 2 for various λ in an interval, e.g., [-2,2], and record SS E (λ) 3 Find λ 0 which minimizes SS E (λ) and pick up a meaningful λ in the neighborhood of λ 0. Denote it again by λ. 4 The transformation is: ỹ ij = y λ 0 ij if λ 0 0; ỹ ij = log y ij if λ 0 =0. Page 14

15 An Example: boxcox.dat trt response Page 15

16 Approximate Method: trans.sas options nocenter ps=65 ls=80; title1 Increasing Variance Example ; data one; infile c:\saswork\data\boxcox.dat ; input trt resp; proc glm data=one; class trt; model resp=trt; output out=diag p=pred r=res; title1 Residual Plot ; symbol1 v=circle i=none; proc gplot data=diag; plot res*pred /frame; proc univariate data=one noprint; var resp; by trt; output out=two mean=mu std=sigma; data three; set two; logmu = log(mu); logsig = log(sigma); proc reg; model logsig = logmu; title1 Mean vs Std Dev ; symbol1 v=circle i=rl; proc gplot; plot logsig*logmu / regeqn; run; Page 16

17 Residual Plot Page 17

18 Plot of logs i vs logµ i Page 18

19 Formal Method: trans1.sas options ls=80 ps=65 nocenter; title1 Box-Cox Example ; data one; infile c:\saswork\data\boxcox.dat ; input trt resp; logresp = log(resp); proc univariate data=one noprint; var logresp; output out=two mean=mlogresp; data three; set one; if _n_ eq 1 then set two; ydot = exp(mlogresp); do l=-1.0 to 1.0 by.25; den = l*ydot**(l-1); if abs(l) eq 0 then den = 1; yl=(resp**l -1)/den; if abs(l) < then yl=ydot*log(resp); output; end; Page 19

20 keep trt yl l; proc sort data=three out=three; by l; proc glm data=three noprint outstat=four; class trt; model yl=trt; by l; data five; set four; if _SOURCE_ eq ERROR ; keep l SS; proc print data=five; run; symbol1 v=circle i=sm50; proc gplot; plot SS*l; run; Page 20

21 SS E (λ) and λ OBS L SS OBS L SS Page 21

22 Plot of SS E (λ) vs λ Page 22

23 Using Proc Transreg proc transreg data=one; model boxcox(y/lambda=-2.0 to 2.0 by 0.1)=class(trt); run; The TRANSREG Procedure Transformation Information for BoxCox(y) Lambda R-Square Log Like : : : * * < * * < Best Lambda : : : * Confidence Interval Convenient Lambda Page 23

24 Nonnormality trt nitrogen Page 24

25 Test ---Statistic p Value----- Shapiro-Wilk W Pr < W Page 25

26 Kruskal-Wallis Test: a Nonparametric alternative a treatments, H 0 : a treatments are not different. Rank the observations y ij in ascending order Replace each observation by its rank R ij (assign average for tied observations) Test statistic H = 1 S 2 [ a i=1 where S 2 = 1 N 1 R 2 i. n i N(N+1)2 4 [ a i=1 Decision Rule: reject H 0 if H>χ 2 α,a 1. ] χ 2 a 1 ni j=1 R2 ij N(N+1)2 4 Let F 0 be the F -test statistic in ANOVA based on R ij. Then ] F 0 = H/(a 1) (N 1 H)/(N a) Page 26

27 options nocenter ps=65 ls=80; data new; input strain nitrogen cards; ; proc npar1way; class strain; var nitrogen; run; Page 27

28 The NPAR1WAY Procedure Analysis of Variance for Variable nitrogen Classified by Variable strain strain N Mean Source DF Sum of Squares Mean Square F Value Pr > F Among Within Page 28

29 The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable nitrogen Classified by Variable strain Sum of Expected Std Dev Mean strain N Scores Under H0 Under H0 Score Average scores were used for ties. Kruskal-Wallis Test Chi-Square DF 5 Pr > Chi-Square Page 29