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Statistics 514: Model Adequacy Lecture 4. Checking Model Assumptions: Diagnostics and Remedies Montgomery: 3-4, 15-1.1 Page 1

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Statistics 514: Model Adequacy Model Assumptions 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 = ˆ y ij + ˆ ij observed = predicted + residual Note that the predicted response at treatment i is ˆ y ij = ¯ y i. Diagnostics use predicted responses and residuals. Page 2
Statistics 514: Model Adequacy Diagnostics Normality Histogram of residuals Normal probability plot / QQ plot (refer to Lecture 3) Shapiro-Wilk Test (refer to Lecture 3) Constant Variance Plot ˆ ij vs ˆ y ij (residual plot) Bartlett’s or Levene’s Test Independence Plot ˆ ij vs time/space (refer to Lecture 3) Plot ˆ ij vs variable of interest Outliers Page 3

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Statistics 514: Model Adequacy 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 ˆ ij vs ˆ y ij Is the range constant for different levels of ˆ y ij More formal tests: Bartlett’s Test Modified Levene’s Test. Page 4
Statistics 514: Model Adequacy Bartlett’s Test Uses sample variances as estimates of population variances H 0 : σ 2 1 = σ 2 2 = . . . = σ 2 a Test statistic: χ 2 0 = 2 . 3026 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 ) S 2 i = n i j =1 ( y ij - ¯ y i. ) 2 n i - 1 ( sample variance at treatment i ) S 2 p = a i =1 ( n i - 1) S 2 i N - a = MS E (pooled variance) Decision Rule: reject H 0 when χ 2 0 > χ 2 α,a - 1 . Remark: sensitive to normality assumption. Page 5

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Statistics 514: Model Adequacy Modified Levene’s Test Use mean absolution deviations as estimates of population variances For each fixed i , calculate the median (Modified Levene) m i of y i 1 , y i 2 , . . . , y in i . Compute the absolute deviation of observation from sample median: d ij = | y ij - m i | for i = 1 , 2 , . . . , a and j = 1 , 2 , . . . , n i , Apply ANOVA to the deviations: d ij Use the usual ANOVA F -statistic for testing H 0 : σ 2 1 = . . . = σ 2 a . Page 6
Statistics 514: Model Adequacy 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 hovtest=bf; 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

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