©2003 Thomson/South-Western 1 Chapter 11 – Analysis of Variance Slides prepared by Jeff Heyl, Lincoln University ©2003 South-Western/Thomson Learning™ Introduction to Business Statistics, 6e Kvanli, Pavur, Keeling

©2003 Thomson/South-Western 2 Analysis of Variance  Analysis of Variance (ANOVA) determines if a factor has a significant effect on the variable being measured  Examine variation within samples and variation between samples

©2003 Thomson/South-Western 3 Measuring Variation  SS(factor) measures between-sample variation [SS(between)]  SS(error) measures within-sample variation [SS(within)]  SS(total) measures the total variation in the sample [SS(factor)] [SS(error)]

©2003 Thomson/South-Western 4 Determining Sum of Squares SS(factor) = + - T2T2nnT2T2nnn T12T12n1n1T12T12n1n1 T22T22n2n2T22T22n2n2 SS(total) = ∑x 2 - = ∑x 2 - (∑x) 2 n T2T2nnT2T2nnn SS(error) = ∑x or T12T12n1n1T12T12n1n1 T22T22n2n2T22T22n2n2 SS(error) = SS(total) - SS(factor)

©2003 Thomson/South-Western 5 ANOVA Test for H o : µ 1 = µ 2 Versus H a : µ 1 ≠ µ 2 MS(factor) = = SS(factor) df for factor SS(factor)1 MS(error) = = SS(error) n 1 + n SS(error) df for error F = = estimated population variance based on the variation among the sample means estimated population variance based on the variation within each of the samples MS(factor)MS(error)

©2003 Thomson/South-Western 6 Defining the Rejection Region Figure 11.1 F

©2003 Thomson/South-Western 7 p-Values for Battery Lifetime Example Figure 11.2 t curve with 8 df 4.20 t p-value= 2 (shaded area) =.0030 F curve with 1 and 8 df F p-value= shaded area =.0030

©2003 Thomson/South-Western 8 Dot Array Diagram Figure 11.4 | BBBBBAAAAA Number of cartons | BBBBBAAAAA Number of cartons Figure 11.3

©2003 Thomson/South-Western 9 Assumptions  The replicates are obtained independently and randomly from each of the populations  The replicates from each population follow a (approximate) normal distribution  The normal populations all have a common variance

©2003 Thomson/South-Western 10 Deriving the Sum of Squares SS(factor) = T12T12n1n1T12T12n1n1 T22T22n2n2T22T22n2n2 Tk2Tk2nknkTk2Tk2nknk T2T2nnT2T2nnn SS(total) = ∑x 2 - T2T2nnT2T2nnn SS(error) = ∑x T12T12n1n1T12T12n1n1 T22T22n2n2T22T22n2n2 Tk2Tk2nknkTk2Tk2nknk = SS(total) - SS(factor)

©2003 Thomson/South-Western 11 The ANOVA Table SourcedfSSMSF Factork - 1SS(factor)MS(factor)MS(factor) Errorn - 2SS(error)MS(error)MS(error) Totaln - 1SS(total) SS(factor) k - 1 MS(factor) = SS(error) n - k MS(error) = MS(factor)MS(error) F =F =F =F =

©2003 Thomson/South-Western 12 Test for Equal Variances H o :  1 2 =  2 2 = … =  k 2 H a : at least 2 variances are unequal reject H o if H > H Table A.14 H =H =H =H = maximum s 2 minimum s 2

©2003 Thomson/South-Western 13 Confidence Intervals in One-Factor ANOVA X i - t  /2,n-k s p to X i + t  /2,n-k s p 1 11nini11nini1 11nini11nini where k= number of populations (levels) n i = number of replicates in the ith sample n= total number of observations s p = MS(error)

©2003 Thomson/South-Western 14 Confidence Intervals in One-Factor ANOVA The (1 -  ) 100% confidence interval for µ i - µ j is (X i - X j ) - t  /2,n-k s p + to (X i - X j ) + t  /2,n-k s p nini11nini1 11njnj11njnj 1 11nini11nini1 11njnj11njnj

©2003 Thomson/South-Western 15 Multiple Comparisons Procedure The multiple comparisons procedure compares all possible pairs of means in such a way that the probability of making one or more Type 1 errors is  Tukey’s Test Q = maximum (X i ) - minimum (X i ) MS(error)/n r

©2003 Thomson/South-Western 16 Multiple Comparisons Procedure 1.Find Q ,k,v using Table A.16 4.If two means differ by more than D, the conclusion is that the corresponding population means are unequal 2.Determine D = Q ,k,v D = Q ,k,v MS(error) n r 3.Place the sample means in order, from smallest to largest

©2003 Thomson/South-Western 17 Plot of Group Means Group Group Means Figure 11.5 Nylon Breaking Strength

©2003 Thomson/South-Western 18 Figure 11.6 Nylon Breaking Strength

©2003 Thomson/South-Western 19 Figure 11.7 Nylon Breaking Strength

©2003 Thomson/South-Western 20 Figure 11.7 Nylon Breaking Strength

©2003 Thomson/South-Western 21 One-Factor ANOVA Procedure 1.The replicates are obtained independently and randomly from each of the populations 2.The observations from each population follow (approximately) a normal distribution 3.The populations all have a common variance Requirements

©2003 Thomson/South-Western 22 One-Factor ANOVA Procedure H o : µ 1 = µ 2 = … = µ k H a : not all µ’s are equal Hypotheses SourcedfSSMSF Factork - 1SS(factor)MS(factor) Errorn - 2SS(error)MS(error) Totaln - 1SS(total) MS(factor)MS(error) reject H o if F * > F ,k-1,n-k

©2003 Thomson/South-Western 23 Completely Randomized Design  Replicates are obtained in a completely random manner from each population Null hypothesis is H o : µ 1 = µ 2 =... = µ n

©2003 Thomson/South-Western 24 Randomized Block Design The samples are not independent, the data are grouped (blocked) by another variable The difference between the randomized block design and the completely randomized design is that here we use a blocking strategy rather than independent samples to obtain a more precise test for examining differences in the factor level means

©2003 Thomson/South-Western 25 Randomized Block Design SS(factor) = [T T T k 2 ] - 1b T 2 bk where k= number of factor levels in the design b= number of blocks in the design n= number of observations = bk T 1, T 2,..., T k represent the totals for the k factor levels S 1, S 2,..., S b are the totals for the b blocks T= T 1 + T T k = S 1 + S S b = total of all observations

©2003 Thomson/South-Western 26 Randomized Block Design SS(blocks) = [S S S b 2 ] - 1k T 2 bk SS(total) = ∑x 2 - T 2 bk SS(error) + SS(total) - SS(factor) - SS(blocks) df for factor= k - 1 df for blocks= b - 1 df for error= (k - 1)(b - 1) df for total= bk - 1

©2003 Thomson/South-Western 27 Randomized Block Design SourcedfSSMSF Factork - 1SS(factor)MS(factor)F 1 Blocksb - 1SS(blocks)MS(blocks)F 2 Error(k - 1)(b - 1)SS(error)MS(error) Totalbk - 1SS(total) F1 =F1 =F1 =F1 =MS(factor)MS(error) F2 =F2 =F2 =F2 =MS(blocks)MS(error)

©2003 Thomson/South-Western 28 Factor Hypothesis Test H o : µ 1 = µ 2 = … = µ k H a : not all µ’s are equal reject H o if F * > F ,k-1,(k-1)(b-1) F1 =F1 =F1 =F1 =MS(factor)MS(error)

©2003 Thomson/South-Western 29 Block Hypothesis Test H o : µ 1 = µ 2 = … = µ b H a : not all µ’s are equal reject H o if F * > F ,b-1,(k-1)(b-1) F2 =F2 =F2 =F2 =MS(blocks)MS(error)

©2003 Thomson/South-Western 30 Hardness Test Data Analysis Figure 11.10

©2003 Thomson/South-Western 31 Hardness Test Data Analysis Figure 11.11

©2003 Thomson/South-Western 32 Confidence Interval Difference Between Two Means Randomized Block (1-  ) 100% confidence interval (1-  ) 100% confidence interval (X i - X j ) - t  /2,df s + to (X i - X j ) + t  /2,df s + 1b1b 1b1b

©2003 Thomson/South-Western 33 Dental Claim Data Analysis Figure 11.12

©2003 Thomson/South-Western 34 Multiple Comparisons Procedure: Randomized Block |X i - X j | > D D = Q ,k,(k-1)(b-1) MS(error)b

©2003 Thomson/South-Western 35 Machine Choice Example Figure 11.13

©2003 Thomson/South-Western 36 Two-Way Factorial Design SingleMarried MaleLowHigh FemaleHighLow Figure 11.14

©2003 Thomson/South-Western 37 Two-Way Factorial Design Figure b 1xxx 2xxx... axxx Factor B Factor A

©2003 Thomson/South-Western 38 Two-Way Factorial Design Figure BTotals 1T 1 2T 2... aT a S 1 S 2 S b Factor A Factor B x, x (total = R 11 ) x, x (total = R 21 ) x, x (total = R 12 ) x, x (total = R a1 ) x, x (total = R a2 ) x, x (total = R 22 ) x, x (total = R 1b ) x, x (total = R 1b ) x, x (total = R ab ) Totals

©2003 Thomson/South-Western 39 Two-Way Factorial Design Factor A: SSA = [T T T a 2 ] - 1br T 2 abr 1r abr Interaction: SSAB = [∑R 2 ] - SSA - SSB - Factor B: SSB = [S S S a 2 ] - T 2 abr 1ar abr Total: SS(total) = ∑x 2 -

©2003 Thomson/South-Western 40 Two-Way Factorial Design SS(error) = SS(total) - SSA - SSB - SSAB MS(error) = SS(error) ab(r - 1) MSA = SSA a - 1 MSB = SSB b - 1 MSAB = SSAB (a - 1)(b - 1)

©2003 Thomson/South-Western 41 Two-Way Factorial Design SourcedfSSMSF Factor Aa - 1SSAMSAF 1 Factor Bb - 1SSBMSBF 2 Interaction(a - 1)(b - 1)SSABMSABF 3 Errorab(r - 1)SS(error)MS(error) Totalabr - 1SS(total)

©2003 Thomson/South-Western 42 Hypothesis Test - Factor A H o : Factor A is not significant (µ M = µ F ) H a : Factor A is significant (µ M ≠ µ F ) reject H o,A if F 1 > F ,v1,v2 F1 =F1 =F1 =F1 =MSAMS(factor)

©2003 Thomson/South-Western 43 Hypothesis Test - Factor B H o : Factor B is not significant (µ 1 = µ 2 = µ 3 ) H a : Factor B is significant (not all µ’s are equal) reject H o,B if F 2 > F ,v1,v2 F2 =F2 =F2 =F2 =MSBMS(error)

©2003 Thomson/South-Western 44 Hypothesis Test - Interaction H o : Interaction is not significant H a : Interaction is significant reject H o,AB if F 2 > F ,v1,v2 F3 =F3 =F3 =F3 =MSABMS(error)

©2003 Thomson/South-Western 45 Multiple Comparisons Procedure: Two-Way Factorial Design D = Q ,k,v D = Q ,k,v MS(error) n r v = df for error n r = number of replicates in each sample

©2003 Thomson/South-Western 46 Interaction Effect Figure – – – – – – – – – – | Category 1 | Category 2 | Category 3 | Category 4 Male Female Employee classification Annual amount claimed on dental insurance A

©2003 Thomson/South-Western 47 Interaction Effect Figure – – – – – – – – – – | Category 1 | Category 2 | Category 3 | Category 4 Male Female Employee classification Annual amount claimed on dental insurance B

©2003 Thomson/South-Western 48 Gender Factor Analysis Figure 11.18

©2003 Thomson/South-Western 49 Gender Factor Analysis Figure 11.19