# meank - Comparing k> 2 Groups Numeric Responses •...

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Unformatted text preview: Comparing k > 2 Groups - Numeric Responses • Extension of Methods used to Compare 2 Groups • Parallel Groups and Crossover Designs • Normal and non-normal data structures D a ta D e s ig n N o r m a l N o n - n o r m a l P a r a lle l G r o u p s ( C R D ) F - T e s t 1 - W a y A N O V A K r u s k a l- W a llis T e s t C r o s s o v e r ( R B D ) F - T e s t 2 - W a y A N O V A F r ie d m a n ’ s T e s t Parallel Groups - Completely Randomized Design (CRD) • Controlled Experiments - Subjects assigned at random to one of the k treatments to be compared • Observational Studies - Subjects are sampled from k existing groups • Statistical model Y ij is a 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 k k k ∑ ∑ = + + = + + = 1 1 1 Analysis of Variance - Sums of Squares • Total Variation 1 ) ( 1 1 2- =- = ∑ ∑ = = n df y y TotalSS Total k i n j ij i • Between Group Variation ∑ ∑ ∑ = = =- =- =- = k i n j k i T i i i i k df y y n y y SST 1 1 1 2 2 1 ) ( ) ( • Within Group Variation E T Total E k i i i k i n j i ij df df df SSE SST TotalSS k 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 k-1 MST=SST/(k-1) F=MST/MSE Error SSE n-k MSE=SSE/(n-k) Total Total SS n- 1 • H : No differences among Group Means ( α 1 = ⋅ ⋅ ⋅ = α k =0) • H A : Group means are not all equal (Not all α i are 0) ) ( : ) 4 . ( : . . : . . , 1 , obs k n k obs obs F F P val P A Table F F R R MSE MST F S T ≥- ≥ =-- α Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy • Three Conditions (Treatments): – Music and Self-sedation ( i = 1) – Self-Sedation Only ( i = 2) – Music alone ( i = 3) • Outcomes – Patient satisfaction score (all 3 conditions) – Amount of self-controlled dose (conditions 1 and 2) Source: Lee, et al (2002) Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy • Summary Statistics and Sums of Squares Calculations: Trt ( i ) n i Mean Std. Dev. 1 55 7.8 2.1 2 55 6.8 2.3 3 55 7.4 2.3 Total 165 overall mean=7.33--- 164 162 2 75 . 840 46 . 809 29 . 31 162 3 165 46 . 809 ) 3 . 2 )( 1 55 ( ) 3 . 2 )( 1 55 ( ) 1 . 2 )( 1 55 ( 2 1 3 29 . 31 ) 33 . 7 4 . 7 ( 55 ) 33 . 7 8 . 6 ( 55 ) 33 . 7 8 . 7 ( 55 2 2 2 2 2 2 = + = = + = =- = =- +- +- = =- = =- +- +- = Total E T df TotalSS df SSE df SST Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy • Analysis of Variance and F-Test for Treatment effects Source of Variation Sum of Squares...
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## This note was uploaded on 07/28/2011 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.

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meank - Comparing k> 2 Groups Numeric Responses •...

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