1. CHAPTER 3 Analysis of Variance (ANOVA) PART 3 = TWO-WAY ANOVA WITH REPLICATION (FACTORIAL EXPERIMENT) MADAM SITI AISYAH ZAKARIA EQT 271 SEM 2 2014/2015

2. Two-Way ANOVA with Replication (Factorial Experiment) Factorial experiments and their corresponding ANOVA computations are valuable designs when simultaneous conclusions about two or more factors are required. Replication means an independent repeat of each factor combination.(interaction) Purpose: Examines (1) the effect of Factor A on the dependent variable, y ; (2) the effect of Factor B on the dependent variable, y ; along with (3) the effects of the interactions between different levels of the two factors on the dependent variable, y.

3. Effects model for factorial experiment: Factorial Experiment, cont.

4. Two-Factor Factorial Experiment Three Sets of Hypothesis: i. Factor A Effect: H 0 :  1 =  2 =... =  a =0 (there is no effect on Factor A) H 1 : at least one  i  0 (there is an effect on Factor A) ii. Factor B Effect: H 0 :  1 =  2 =... =  b =0 (there is no effect on Factor B) H 1 : at least one  j ≠ 0 (there is an effect on Factor B) iii. Interaction Effect: H 0 : (   ) ij = 0 for all i,j (there is no interaction AB effect) H 1 : at least one (   ) ij  0 (there exist an interaction AB effect)

5. Two-Factor Factorial Experiment, cont. Format for data: Data appear in a grid, each cell having two or more entries. The number of values in each cell is constant across the grid and represents r, the number of replications within each cell. Calculations: – Sum of squares total (SST) = sum of squared differences between each individual data value (regardless of group membership) minus the grand mean,, across all data... total variation in the data (not variance).

6. Two-Factor Factorial Experiment, cont. Calculations, cont.: – Sum of squares Factor A (SSA) = sum of squared differences between each group mean for Factor A and the grand mean, balanced by sample size... between-factor-groups variation (not variance). – Sum of squares Factor B (SSB) = sum of squared differences between each group mean for Factor B and the grand mean, balanced by sample size... between-factor-groups variation (not variance).

7. Two-Factor Factorial Experiment, cont. Calculations, cont.: – Sum of squares Error (SSE) = sum of squared differences between individual values and their cell mean... within-groups variation (not variance). – Sum of squares Interaction: SSAB = SST – SSA – SSB – SSE

8. Two-Factor Factorial Experiment, cont. Calculations, cont.: – Mean Square Factor A (MSA) = SSA/( a – 1), where a = the number of levels of Factor A. – Mean Square Factor B (MSB) = SSB/( b – 1), where b = the number of levels of Factor B. – Mean Square Interaction (MSAB) = SSAB/( a – 1)( b – 1). – Mean Square Error (MSE) = SSE/ ab ( r – 1), where ab ( r – 1) = the degrees of freedom on error.

9. Two-Factor Factorial Experiment, cont. Calculations - F -Ratios: – F -Ratio, Factor A = MSA/MSE, where numerator degrees of freedom are a – 1 and denominator degrees of freedom are ab ( r – 1). This F -ratio is the test statistic for the hypothesis that the Factor A group means are equal. To reject the null hypothesis means that at least one Factor A group had a different effect on the dependent variable than the rest. – F -Ratio, Factor B = MSB/MSE, where numerator degrees of freedom are b – 1 and denominator degrees of freedom are ab ( r – 1). This F -ratio is the test statistic for the hypothesis that the Factor B group means are equal. To reject the null hypothesis means that at least one Factor B group had a different effect on the dependent variable than the rest.

10. Two-Factor Factorial Experiment, cont. Calculations - F -Ratios: – F -Ratio, Interaction = MSAB/MSE, where numerator degrees of freedom are ( a – 1)( b – 1) and denominator degrees of freedom are ab ( r – 1). This F -ratio is the test statistic for the hypothesis that Factors A and B operate independently. To reject the null hypothesis means that there is some relationship where levels of Factor A operate differently with different levels of Factor B. – If F -Ratio> F  or p-value< , reject H 0 at the  level.

11. Two-Factor Factorial Experiment, cont. Two-Factor ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square Fp -Value Factor A SSAa-1 Factor B SSBb-1 Interaction SSAB(a-1)(b-1) Error SSEab(r-1) Total SSTabr-1

12. A survey was conducted of hourly wages for a sample of workers in two industries at three locations in Ohio. Part of the purpose of the survey was to determine if differences exist in both industry type and location. The sample data are shown on the next slide. Example 4.4 : State of Ohio Wage Survey Two-Factor Factorial Experiment,

13. Two-Factor Factorial Experiment- An Example IndustryCincinnatiClevelandColumbus I $12.10$11.80$12.90 11.80 11.20 12.70 12.10 12.00 12.20 II 12.40 12.60 13.00 12.50 12.00 12.10 12.00 12.50 12.70 Factors: Factor A: Industry Type (2 levels) Factor B: Location (3 levels) Replications: Each experimental condition is repeated 3 times

14. Two-Factor Factorial Experiment- An Example 1. Hypothesis for this analysis: i. Factor A Effect: H 0 :  1 =  2 =0 H 1 : at least one  i  0 ii. Factor B Effect: H 0 :  1 =  2 =  3 =0 H 1 : at least one  j ≠ 0 iii. Interaction Effect: H 0 : (   ) ij = 0 for all i,j H 1 : at least one (   ) ij  0

15. Two-Factor Factorial Experiment- An Example Result: Two-Way ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square Fp -Value Factor A0.501 4.190.06 Factor B1.1220.564.690.03 Interaction0.3720.191.550.25 Error1.43120.12 Total3.4217 2. Test Statistics : (F test) i.SSA – MSA v. SSAB = SST – SSA – SSB – SSE ii.SSB – MSB vi. MSAB iii. SSE – MSE vii. F test (A, B & AB) iv. SST viii. ANOVA TABLE

16. 5. Conclusions Using the Critical Value Approach Two-Factor Factorial Experiment-An Example Interaction is not significant. Interaction: F  = 3.89 Mean wages differ by location. Locations: F  = 3.89 Mean wages do not differ by industry type. Industries: F  = 4.75 3. F (alfa) value Industries: F = 4.19 < F  = 4.75 - Do not Reject H 0 Locations: F = 4.69 > F  = 3.89 - Reject H 0 Interaction: F = 1.55 < F  = 3.89 - Do not Reject H 0 4. Rejection region : (Draw picture)

17. 5. Conclusions Using the p -Value Approach Two-Factor Factorial Experiment-An Example Interaction is not significant. Interaction: p -value =.25 >  =.05 Mean wages differ by location. Locations: p -value =.03 <  =.05 Mean wages do not differ by industry type. Industries: p -value =.06 >  =.05

18. EXERCISE TEXT BOOK PAGE 153 ( Example 9.6) PAGE 157 ( 1 & 2) 5 IMPORTANT STEP: 1.HYPOTHESIS TESTING 2.TEST STATISTIC – F TEST 3.F (alfa) – VALUE (CRITICAL VALUE) 4.REJECTION REGION 5.CONCLUSION 2 step 1.SSA – MSA 2.SSB – MSB 3.SST 4.SSE - MSE 5.SSAB = SST – SSA – SSB - SSE 6.F TEST SSA= MSA/MSE 7.F TEST SSB = MSB/MSE 8.F TEST SSAB = MSAB/MSE 9.BUILD ANOVA TABLE

19. FACTOR A FACTOR B abc Total mean (factor B) d X1 X2 A X3 X4 B X5 X6 C e X7 X8 D X9 X10 E X11 X12 F Total mean (factor A ) Grand mean a = no. of Factor A b = no. of Factor B

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