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Unformatted text preview: Chapter 8 1Way Analysis of Variance  Completely Randomized Design Comparing t > 2 Groups  Numeric Responses • Extension of Methods used to Compare 2 Groups • Independent Samples and Paired Data Designs • Normal and nonnormal data distributions D ata D esig n N o rm al N o n  n o rm al In d ep en d en t S am p les (C R D ) F T est 1 W ay A N O V A K ru sk al W allis T est P aired D ata (R B D ) F T est 2 W ay A N O V A F ried m an ’s T est Completely Randomized Design (CRD) • Controlled Experiments  Subjects assigned at random to one of the t treatments to be compared • Observational Studies  Subjects are sampled from t existing groups • Statistical model y ij is measurement from the j th subject from group i : ij i ij i ij y ε μ ε α μ + = + + = where μ is the overall mean, α i is the effect of treatment i , ε ij is a random error, and μ i is the population mean for group i 1Way ANOVA for Normal Data (CRD) • For each group obtain the mean, standard deviation, and sample size: 1 ) ( 2 . . = = ∑ ∑ i j i ij i i j ij i n y y s n y y • Obtain the overall mean and sample size N y N y n y n y n n N i j ij t t t ∑ ∑ = + + = + + = . . 1 1 .. 1 ... ... Analysis of Variance  Sums of Squares • Total Variation 1 ) ( 1 1 2 .. = = ∑ ∑ = = N df y y TSS Total k i n j ij i • Between Group (Sample) Variation ∑ ∑ ∑ = = = = = = t i n j t i T i i i i t df y y n y y SST 1 1 1 2 .. . 2 .. . 1 ) ( ) ( • Within Group (Sample) Variation E T Total E t i i i t i n j i ij df df df SSE SST TSS t N df s n y y SSE i + = + = = = = ∑ ∑ ∑ = = = 1 2 1 1 2 . ) 1 ( ) ( Analysis of Variance Table and FTest Source of Variation Sum of Squares Degrres of Freedom Mean Square F Treatments SST t1 MST=SST/(t 1) F=MST/MSE Error SSE Nt MSE=SSE/(Nt) Total TSS N 1 • Assumption: All distributions normal with common variance • H : No differences among Group Means ( α 1 = ⋅ ⋅ ⋅ = α t =0) • H A : Group means are not all equal (Not all α i are 0) ) ( : ) 9 ( : . . : . . , 1 , obs t N t obs obs F F P val P Table F F R R MSE MST F S T ≥ ≥ = α Expected Mean Squares • Model: y ij = μ + α i + ε ij with ε ij ~ N (0, σ 2 ), Σα i = 0: 1 ) ( ) ( true), is ( otherwise 1 ) ( ) ( true, is : When ) 1 ( 1 1 ) ( ) ( 1 ) ( ) ( 1 2 1 2 2 1 2 2 1 2 2 2 = = = = + = + = ⇒ + = = ∑ ∑ ∑ = = = MSE E MST E H MSE E MST E H t n t n MSE E MST E t n MST E MSE E a t t i i i t i i i t i i i α α σ α σ α σ α σ σ Expected Mean Squares • 3 Factors effect magnitude of Fstatistic (for fixed t ) – True group effects ( α 1 ,…, α t ) – Group sample sizes ( n 1 ,…, n...
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This note was uploaded on 06/04/2011 for the course STA 6166 taught by Professor Staff during the Fall '08 term at University of Florida.
 Fall '08
 Staff
 Variance

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