Chapter8 - Chapter 8 1-Way Analysis of Variance Completely Randomized Design Comparing t> 2 Groups Numeric Responses • Extension of Methods

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Unformatted text preview: Chapter 8 1-Way 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 non-normal 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 1-Way 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 F-Test Source of Variation Sum of Squares Degrres of Freedom Mean Square F Treatments SST t-1 MST=SST/(t- 1) F=MST/MSE Error SSE N-t MSE=SSE/(N-t) 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 F-statistic (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.

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Chapter8 - Chapter 8 1-Way Analysis of Variance Completely Randomized Design Comparing t> 2 Groups Numeric Responses • Extension of Methods

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