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ANOVA-UP-march-20-2024

ANOVA- Lecture of Engr. Charlton Inao to ME 172 (Experimentation in Engineering) UP Diliman

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ANOVA-UP-march-20-2024

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  1. ANOVA: Analysis of VarianceME 172 Engr. Charlton Inao Senior Lecturer Department of Mechanical Engineering ,UP DILIMAN

  2. Page 53 Probability & Inference PTA, 05-2004 ANALYSIS OF VARIANCE

  3. Page 54 • A powerful analysis tool which is based on the ratio of variances within an experiment; as such it makes use of the F-distribution Test or F-test • Used to test the differences in means (for groups or variables) for statistical significance by analyzing the variance, that is, by partitioning the total variance into its components: (1) variability due to true random error (difference of means within the group), and (2) variability due to the treatment (differences of means between the group) Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) Total Variance Variability Due to Error Variability Due to the Treatment

  4. Page 55 • Used to determine whether differences among three or more sample means represent random fluctuations of data or truly significant differences among the populations from which they were taken • Types of variables used in ANOVA: • Independent variable: variables that are manipulated or controlled (treatment); also called factors • Dependent variable: variables that are measured; also called responses • Assumptions in using ANOVA: 1) independent samples 2) normally distributed populations with equal variances These assumptions need to be satisfied before even proceeding with the analysis. Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA)

  5. Page 56 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) A) ANOVA (One-Way Classification): SSA Legend: a = treatment n = replicates where:

  6. Page 57 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) • Significant treatment effect exists if the variation due to treatment is larger than the variation due to error: if SSE >> SSA, the factor does not matter; SSE << SSA, the factor does matter  Basic Test: 1) Treatment Effect: H0: i = 0 for all i Ha: At least one i is not zero if variation due to treatment is larger than the variation due to error, then there is enough evidence to conclude the presence of treatment effect

  7. Page 58 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) Test Criteria: If Fratio > Ftab, reject H0 Note: What happens to the formula (a x n) if replicate size r is not the same in all treatments? (a x n): na1 + na2 + ... Simply add the count of replicate in each treatment

  8. Page 59 Example 1: For a one-factor experiment with the results given below, perform an analysis of variance using an  = 0.05. Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA)

  9. Page 60 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) B.1) ANOVA (Two-Way Classification Without Interaction):  set of observations are classified according to two criteria at once  partitioning of total variation in the response is based on the contribution of the levels of two factors and experimental error  a rectangular array of data where the columns represent one criterion of classification and the rows represent a second criterion (see table in next page)

  10. Page 61 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) The ANOVA Table SSA abn-1 Legend: a = factor A b = factor B n = replicates

  11. Page 62 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) where: Note: If replicate = 1, collapse the last ‘.’ or ‘k’ in each Y. Remove n (for n=1) and .

  12. Page 63 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA)  Basic Tests: 1) Treatment Effect: H01: i = 0 for all i Ha1: At least one i is not zero if variation due to treatment is larger than the variation due to error, then there is enough evidence to conclude the presence of treatment effect 2) Homogeneity of the experimental materials (blocks): H02: j = 0 for all j Ha2: At least one j is not zero if variation due to block is larger than the variation due to error, then there is enough evidence to confirm that the materials are indeed heterogeneous

  13. Page 64 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) Example 2: Perform a statistical analysis for the unreplicated experimental data below involving two input factors. Use  = 0.05.

  14. Page 65 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) B.2) ANOVA (Two-Way Classification with Interaction):  partitioning of total variation in the response based on the contribution of the levels of the 1st factor, the 2nd factor, interaction of the two factors and experimental error The ANOVA Table

  15. Page 66 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) where:

  16. Page 67 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA)  Basic Tests: 1) Effect of the 1st Factor: H01: i = 0 for all i Ha1: At least one i is not zero if variation due to the 1st factor is larger than the variation due to error, then there is enough evidence to conclude that the factor affects the response 2) Effect of the 2nd Factor: H02: j = 0 for all j Ha2: At least one j is not zero if variation due to the 2nd factor is larger than the variation due to error, then there is enough evidence to conclude that the factor affects the response 3) Presence of Interaction: H03: ()ij = 0 for all (i,j) Ha3: At least one ()ij is not zero if variation due to interaction is larger than the variation due to error, then the effect of the two factors cannot be segregated from each other. Hence, no directional hypothesis can be made on the two factors. In which case, simple effects are analyzed using the CRD approach.

  17. Page 68 Probability & Inference PTA, 05-2004 Analysis of Variance (ANOVA) Example 3: Consider the following response observations from a two-factor experiment with two stimuli A and B. Perform a statistical analysis using an  = 0.01.

  18. Page 69 Probability & Inference PTA, 05-2004 Appendix Statistical Tables • Binomial Probability Sums • Poisson Probability Sums • Z-Distribution (Normal Probability) • t-Distribution (Normal Probability) • Chi-Squared Distribution • F-Distribution

  19. Page 70 Probability & Inference PTA, 05-2004 References 1. Understanding Statistics, 3rd Edition By Mendenhall and Ott 2. Introduction to the Theory of Statistics, 3rd Edition By Mood, Graybill and Boes 3. Probability and Statistics for Engineers and Scientists, 3rd Edition, By Walpole and Myers 4. Introduction to Statistics By Walpole 5. Elementary Statistics for Basic Education By Melencio Deauna 6. Basic Statistical Methods, 5th Edition By Downie and Heath 7. Statistics II Manual By Motorola University

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