Analy Of Variance
Analy Of Variance STA 106
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This 17 page Class Notes was uploaded by Carmen Mayer on Tuesday September 8, 2015. The Class Notes belongs to STA 106 at University of California - Davis taught by Jie Peng in Fall. Since its upload, it has received 57 views. For similar materials see /class/191910/sta-106-university-of-california-davis in Statistics at University of California - Davis.
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Date Created: 09/08/15
Stat 106 V nter 2007 Stat 106 Analysis of Variance Lecture 28 Final Review Mar 14 2007 Stat 106 V nter 2007 Introduction of Designs I 0 Experimental studies 0 Observational studies 0 Randomization CRD 0 Association vs causation 0 Basic concepts factors treatments response variable experimental units and sample size Stat 106 V nter 2007 SingleFactor Studies I KJZM TiEij7 i1 v7n j17 7ni39 Fixed effects model ANOVA model I balanced 717 E n and unbalanced studies Model assumptions ANOVA decomposition SSTO SSTR l SSE F test Analysis of factor effects point estimation and interval estimation Power calculation and planning of sample size Random effects model ANOVA model II for balanced studies Model assumptions ANOVA decomposition the only difference from the fixed effects model is the expected mean squares Stat 106 V nter 2007 Estimation of overall mean u 7 gt LL and M STR 827 gt Var7 nr Estimation of variance components 02 by MSE gtllt 0i2by MSTR MSEn 0 th f ttht Wuse eac a MSTR a2na2 MSE N 02 Fr 1nT r gtllt Satterthwaite procedure for linear combinations of the mean squares Fgtilt Predication of the random effects BLUP A 7162 BLUPm 2 Wm Y 039 039 p Stat 106 V nter 2007 Table 1 ANOVA Table for Single Factor Studies Source of Sum of Degrees of MS EMS Variation Squares SS Freedomdf 7 7 2 2 2 Between SSTR 7 I 1 nZYZ Y r 1 MSTR 039 no MUI 7 2 E HIM M02 treatments T n 039 T I ErrorWithin SSE E E Z Yij Yi2 nT r MSE a2 i1 j1 treatments T n Total SSTO E E Z Yij Y2 nT 1 i1 j1 For m E n Stat 106 Writer 2007 I TwoFactor Studies Yijk MO i jo ij ijka 13 17H397a9j 17quot39vb k lama nz39j o Fixed effects models ANOVA model I Model assumptions Interactions interaction plots curves parallel or not Balanced case mj E n gtllt ANOVA decomposition SSTO SSTR l SSE and SSTR SSA l SSB l SSAB Three F tests gtllt Analysis of factor effects i without interactions based on factor level means ii with interactions based on treatment means gtllt One case pertreatment n 1 additive model use MSAB in place of MSE as the estimator of 02 Tukey s test for additivity Stat 106 V nter 2007 Unbalanced case gtllt Imbalance destroys the orthogonality of the ANOVA decomposition gtllt Regression formulation of the model indicator variables gtllt F Test compare a larger model full model to a smaller model reduced model under H0 SSER SSEFl de dfpl SSEFdfp which follows Fde dfp dfp under H0 Analysis of factor effects same as the balanced case F 96 0 Random effects model and mixed effects model ANOVA model II and III for balanced case only Model assumptions ANOVA decompositions are the same as the fixed effects model the only difference is the expected mean squares F tests based on the expected mean squares Stat 106 V nter 2007 Estimation of variance components point estimation and approximate Cl Satterthwaite procedure Table 2 Estimation of Variance Components for ANOVA Models II and Ill Variance components Random ANOVA model Mixed ANOVA Model A and B random AfixedBrandom a Si MSAZMSAB NA 02 S2 7 MSB RJSAB S2 7 MSB MSE na na 2 2 i MSAB MSE 2 i MSAB MSE a s 7 s i 043 043 n a n Estimation of fixed effects in the mixed effects model compared to the fixed effects model use MSAB in place of MSE 647 7 7 H7 and A 02n0 EMSAB Varoz m m Stat 106 V nter 2007 Table 3 ANOVA Table for Balanced TwoFactor ANOVA Models I II III Sourceof SS clf MS Variation FactorA SSA an71H 72 a1 MSA i FactorB SSB na2 j 2 b1 MSB 3 ABinteractions SSAB n E 7 71 73 72 a 1b1 MSAB iyj a b n i E E E 2 i SS E Error SSE 7 ka Y1 n1ab MSE 7 n1ab i1j1k1 a b n Total SSTO E E E Yijk 72 nab1 i1j1k1 SSAB a 1b 1 Stat 106 V nter 2007 Table 4 Expected Mean Squares for Balanced TwoFactor ANOVA Models I II II Mean df Fixed ANOVA Model Random ANOVA Model Mixed ANOVA Model Square A and B fixed A and B random A fixed B random 0 a E a E a 1 1 2 11 2 2 2 2 11 2 MSA 0 1 039 nb a1 039 nbo39oznaoz 039 nb a1 no39a b 2 Z j 2 91 2 2 2 2 2 MSB b 1 039 na b1 039 naa n0a 039 7100 a b 2 2 EM MSAB a 1b 1 02 71L 02 7102 02 7102 a 1b 1 a a MSE n 1ab 02 02 2 Stat 106 Writer 2007 Randomized Complete Block Designs RCBDI Yzj Hpz39739j ij7 i1 7nbj17 77 o The idea of blocking reduce experimental variation 0 As special a case of twofactor studies with n 1 o Fixed block effects ANOVA decomposition F tests Analysis of treatment effects Tukey s test for additivity Planning of number of blocks 0 Random block effects and fixed treatment effects Two models ANOVA decomposition F test for treatment effects and analysis of Stat 106 V nter 2007 treatment effects are all the same as the fixed effects model Table 5 ANOVA Table for Randomized Complete Block Design Block Effects Fixed or Random Source of 88 MS Variation Blocks SSBL TE i l2 nb 1 MSBL fig 139 Treatments SSTRnb E YU Y2 77 1 MSTR S1le j Error SSBLTR E Yij YZ YjYt2 nb 1r 1 MSBLTR igtj SSBLTR h 17 1 Total nb 7v SSTO E E Yij V2 i1 j1 nb39r l Stat 106 Writer 2007 Analysis of Covariance ANCOVA I 0 Introducing covariates into the model reduce error variance 0 Compare with blocking blocking is usually preferred though not always possible Regression formulation Indicator variables Design matrix X Least square estimate of the regression coefficients and its variancecovariance matrix MSE gtllt XTX1 F test to compare two models full vs reduced Test for parallel slopes Onefactor and twofactor ANCOVA Stat 106 Writer 2007 Multiple Comparison Procedures I Pairwise comparison contrast and linear combination of means Tukey s procedure applicable for pairwise comparisons 1 T EQ1057N7df Scheffe s procedure applicable for finite or infinite number of contrasts S N 1F1 aN 1df Bonferroni s procedure applicable for predetermined finite number of linear combinations For g inferences 04 B t 1 d In all three procedures df is the degrees of freedom of the mean square which is used as the estimator of the error variance 02 N is either the Stat 106 V nter 2007 number of factor levels or the number of treatments depending on which one is of interest Stat 106 Writer 2007 Model Diagnostic and Remedial Measures I 0 General model assumptions normality equal variance and independence of errorterms 0 Diagnostic tools Residual e Y Y observed valuefitted value Residual plots i residual vs fitted value plot examine equal variance ii normal probability plot examine normality Tests for equal variance i Hartley test max 83 min 39 ii BrownForsythe test F test for equal means on the absolute deviations dij Ym where Y is the sample median of the ith treatment group Interaction plots Stat 106 V nter 2007 Symbolic scatter plot in ANCOVA for parallel slopes Remedial measures Transformations to achieve equal variance and normality variance stabilizing transformations BoxCox procedure Nonparametric rank F test Transformations to reduce interactions
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