Why ANOVA? Analysis of Variance (ANOVA) One-Way ANOVA F-Test. One-Way ANOVA F-Test. One-Way ANOVA F-Test. Completely Randomized Design

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1 Why? () Eample: Heart performance core for 3 group of ubject, Non-moer, Moderate moer, 3Heavy moer 3 Comparing More Than Mean Average When comparing three independent random ample with two-ample t-tet, three t-tet would be needed. (Group, Group ), (Group, Group 3), (Group, Group 3) Each tet uffer a Type I Error rate at α level. If three tet were ued imultaneouly, the Type I Error will be inflated. One approach to tet the ignificant difference between everal mean uing one ingle tet without inflating Type I Error i. 3. Tet the Equality of or More () Population Mean (μ μ μ ). Variable One Categorical Independent Variable or More () Treatment Level or Group One Quantitative Dependent Variable 3. Ued to Analyze Completely Randomized Eperimental Deign Completely Randomized Deign Homogeneou Subject or Eperimental Unit Are Aigned Randomly to Treatment

2 Aumption. Randomne & Independence of Error Independent Random Sample are Drawn. Normality Population are Normally Ditributed 3. Homogeneity of Variance (σ σ σ ) Population have Equal Variance Hypothee H 0 : μ μ μ 3... μ All Population Mean are Equal No Treatment Effect H Α : Not All μ j Are Equal At Leat one pair of Pop. Mean i Different Ha Treatment Effect Not imply μ μ... μ f() f() μ μ μ 3 μ μ μ 3 8 Why i it called? Eample: HP for three group of ubject Cae I Cae II CASE I CASE II CASE...0 CASE Average GROUPID GROUPID Cae Group Group Group 3 Why Variance? Cae.0..0 Pop Pop Pop 3 Pop Pop Pop High BETWEEN variability B σ B High WITHIN variability W σ W

3 One-Way Baic Idea. If Treatment Variation I Significantly Greater Than Random Variation then Mean Are Not Liely Equal B W. Compare Type of Variation to Tet Equality of Mean Baed on Ratio of Variance Heart Performance Eample Step: Hypothei: H 0 : μ μ μ 3 (All mean are equal.) H A : μ i μ j for at leat one pair of (i, j), i, j,, 3. (Not all mean are equal.) 3 Notation ij i i : the j-th element from the i-th group : the i-th group mean : the i-th group tandard deviation : the overall ample mean n : the total ample ize (n + n + + n ) 3.90,.08.0,.00.00,.08 Group : Group : Group 3: n n n n Etimation of Variance Between Group Variance n ( ) + n( ) n ( ) B (.9.) + (..) + (.).0 3 Within Group Variance.0..0 ( n ) ) W + ( n ) ( n n.0..0 ( ) ( ) ( ) Tet Statitic Critical Value Tet Statitic B F ~ F (df, df) ditribution (under H 0 ) W B i Mean Square Between (MSB ) W i Mean Square for Error (MSW ) Degree of Freedom df (Numerator Degree of Freedom) df n (Denominator Degree of Freedom) # Population, Group, or Level n Sample Size If there no ignificant difference between mean, then F B W would be a mall value. 0 Do Not Reject H 0 Reject H 0 F α (, n ) Only reject H 0 if having large F! α F 8 3

4 Deciion Rule & Concluion Step: B.0 Tet Statitic: F. Step: 3 W Deciion Rule: Critical Value Approach: Reject null hypothei, if F > F.0,, (Table, page A-.) p-value Approach: Reject null hypothei if p -value i le than.0. Step: Concluion: The tet tatitic F. > 3.89, and ince F.00,,.9 <., o p -value < 0.00 < 0.0, null hypothei i rejected. There i tatitically ignificant difference between group mean. 9 Birth Weight Eample Eample: The birth weight of an infant ha been hypotheized to be aociated with the moing tatu of the mother during the firt trimeter of pregnancy. The mother are divided into four group according to moing habit, and the ample of birth weight in pound within each group i given a follow: 0 Birth Weight Eample (Obervational Study) Eample uing ample variance Group : (Mother i a nonmoer) Group : (Mother i an e-moer but not during the pregnancy) Group 3: (Mother i a current moer and moe le than pac per day) Group : (Mother i a current moer and moe more than pac per day) Hypothei: H 0 : μ μ μ 3 μ H a : μ i μ j for at leat one pair of (i, j), i, j,, 3,. Tet Statitic: B F ~ F-ditribution (df 3, df 0) W Over all mean, n n n 3 n Mean Square for F Tet Between group variability: B ( ) + ( ) + ( 3 ) + ( ) Within group variability:.3 Deciion Rule & Concluion B.3.08 Tet Statitic: F W Deciion Rule: (Table, page A-3.) Critical Value Approach: Reject null hypothei, if F > F.0, 3, p-value Approach: Reject null hypothei, if p -value i le than.0. ( ) + ( ) + ( ) 3 + ( ) W.09, where n + + +, i a good etimate of σ. W Concluion: The tet tatitic F > F.0, 3, 0 3.0, and p -value i le than.00 ince F.00, 3, < 9.088, o p-value < 0.0, the null hypothei i rejected. There i ignificant difference between group mean. 3

5 weight SPSS Output Error Bar Chart SPSS Error Bar Chart p-value 9 8 Between Group Within Group Sum of Square df Mean Square F Sig % CI Birth Weight w B N Non moer E-moer Smoer < Smoer > Smoer Categorie Error Bar Chart 8.00 ] ] What in the Table?.00 ] ] Nonmoer E-moer Smoe < Smoe > tatu 8 One-Way Partition Variation Repone, SS What in the Table? Variation due to treatment variation Variation due to random ampling Repone, SS Between 3 Sum of Square Between Sum of Square Treatment Sum of Square Within Sum of Square Error Repone, SS Within 3 9 Group Group Group 3 30

6 Variation ( ) ( ) ( ) ( ) ( ) ( ) n ni ( ij ) TSS K i j TSS (..) + + (..) 8.9 Treatment Variation ( ) + n ( ) + + n ( ) SSB K n n i ( i ) i SSB (.8.) + (..) +. Between Group Sum of Square df Mean Square F Sig Between Group Sum of Square df Mean Square F Sig Within Group Within Group Between Group Within Group Random (Error) Variation ( ) ( ) ( ) n n i ( ij i ) ( ni ) i SSW K i i j i SSW (..8) + + (..8).8 Sum of Square df Mean Square F Sig Source of Variation Treatment (Between ample) Error (Within ample) One-Way Summary Table Sum of Square SSB (.) SSW (.8) Degree of Freedom - (3) n - (0) TSS n - (8.9) (3) Mean Square (Variance) MSB (.3) MSW (.09) * MSBSSB/( - ); MSWSSW/(n - ); TSSSSB+SSW F MSB MSW (9.088) 3 SPSS Output p-value What if the aumption are not atified? Between Group Within Group Sum of Square df Mean Square F Sig Try a nonparametric method: Krual-Walli Tet

7 Pot Hoc Analyi Multiple Comparion SPSS output table from Bonferroni and Tuey -b option. Multiple Comparion Dependent Variable: Bonferroni What hould we do if we found ignificant difference between population mean? (I) SMOKEST Non moer E-moer (J) SMOKEST E-moer Smoer < Smoer > Non moer Smoer < Smoer > 9% Confidence Interval Mean Difference (I-J) Std. Error Sig. Lower Bound Upper Bound * * * E Smoer < Non moer -.8* E-moer Smoer > Smoer > Non moer -.99* E-moer -.39* E-03 3 Smoer < *. The mean difference i ignificant at the.0 level. 38 Tuey B Error in Multiple Comparion Procedure WEIGHT Subet for alpha.0 STATUS N Tuey B a,b Smoe > pac.8 Smoe < pac.0000 E-moer.000 Nonmoer.8 Mean for group in homogeneou ubet are diplayed. a. Ue Harmonic Mean Sample Size.9. b. The group ize are unequal. The harmonic mean of the group ize i ued. Type I error level are not guaranteed. Individual error rate: The probability that a comparion of mean will be falely declared ignificant in an eperiment. (α Ι ) Family-wie error rate: The probability that at leat a pair of mean will be falely declared ignificant in an eperiment that mae m comparion. ( ( α Ι ) m ) 39 0 Bonferroni Adjutment Method In Multiple Comparion uing Confidence Interval Etimate for difference of two mean with w a the pooled etimate of common variance for multiple comparion and with Bonferroni correction, α i the corrected confidence level, and α i the corrected level of ignificance when maing c comparion. * α Since ( ( α ) c ) cα α α c For having 3 pair, a the eample, the probability of falely reject at leat one pair of mean i about 0.0, if α.0/3 0.0, and o ( (.0) 3 ) 0.0. Bonferroni Adjutment Method! Number of pair to be compared i c!( )! α So, α*. Number of pair to be compared In confidence interval etimation, α * would be the corrected confidence level. ± t α* + w n n

8 Confidence Interval Eample: (from the previou problem about moing mother) For comparing Smoe < & Smoe > α 0.0, α* α/c 0.0/{!/(![-]!)}.0/.008, w.09, degree of freedom 0 t α*/, df t.00,0.9, [If not adjuted, α 0.0, t 0.0, 0.08.] ± (.0,.) Thi interval doe contain zero. It implie the difference between the mean of Smoe < & Smoe > group i inignificant. 3 Two-ample t -Tet Eample: (from the previou problem about moing mother) For comparing Smoe < & Smoe > α 0.0, α* α/c 0.0/{!/(!!)}.0/.008, Critical value: t α*/ t.00.9, (d.f. 0) w t Since 0.39 >.3, o p-value > 0. >.008. It implie the difference between the mean of Smoe < & Smoe > group i inignificant. Dependent Variable: Bonferroni (I) SMOKEST Non moer E-moer Smoer < Smoer > (J) SMOKEST E-moer Smoer < Smoer > Non moer Smoer < Smoer > Non moer E-moer Smoer > Non moer E-moer Smoer < *. The mean difference i ignificant at the.0 level. Multiple Comparion Mean 9% Confidence Interval Difference (I-J) Std. Error Sig. Lower Bound Upper Bound * * * E * * * E (.0,.) Multiple Comparion Procedure Planned Comparion Unplanned Comparion (Pot hoc) Multiple Comparion Procedure Fiher-Hayter* Tuey HSD* Bonferroni procedure* Scheffe procedure* (ue linear contrat) Dunnett procedure (compare with control group) Fiher Leat Significance Difference procedure Student-Newman-Keul procedure (Mod. Tuey ) Duncan procedure (Multiple range) * Strongly protected family-wie error Multiple Comparion Procedure in SPSS The Bonferroni and Tuey' honetly ignificant difference tet are commonly ued multiple comparion tet. The Bonferroni tet, baed on Student' t tatitic, adjut the oberved ignificance level for the fact that multiple comparion are made. Sida' t tet alo adjut the ignificance level and provide tighter bound than the Bonferroni tet. Tuey' honetly ignificant difference tet ue the Studentized range tatitic to mae all pairwie comparion between group and et the eperimentwie error rate to the error rate for the collection for all pairwie comparion. When teting a large number of pair of mean, Tuey' honetly ignificant difference tet i more powerful than the Bonferroni tet. For a mall number of pair, Bonferroni i more powerful. (From SPSS Help) 8 8