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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 8 Analysis of Variance. Flow Rate Experiment. MGH Fig 6.1. Uncontrolled Error. Assignable Cause (Factor Changes). Flow Rate Experiment. What is an appropriate statistical comparison

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Statistical Analysis

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  1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 8 Analysis of Variance

  2. Flow Rate Experiment MGH Fig 6.1

  3. Uncontrolled Error Assignable Cause (Factor Changes) Flow Rate Experiment What is an appropriate statistical comparison of the filter means? 0.35 0.30 Average Flow Rate 0.25 0.20 A B C D Filter Type

  4. Does not account for multiple comparisons What is an appropriate statistical comparison of the diet means?

  5. 5 Comparisons, Some averages used more than once (e.g., N/R50)

  6. Analysis of Variance for Single-Factor Experiments Model yij = m + ai + eij i = 1, ..., a; j = 1, ..., ri Total Sum of Squares Total Adjusted Sum of Squares Corrected Sum of Squares (Numerator of the Sample Variance)

  7. Analysis of Variance for Single-Factor Experiments Model yij = m + ai + eij i = 1, ..., a; j = 1, ..., ri Total Sum of Squares Goal Partition TSS into Components Associated with Assignable Causes: Controllable Factors and Measured Covariates Experimental Error: Uncontrolled Variation, Measurement Error, Unknown Systematic Causes

  8. Analysis of Variance for Single-Factor Experiments

  9. Analysis of Variance for Single-Factor Experiments Show

  10. Analysis of Variance for Single-Factor Experiments

  11. Estimation Estimating Factor Effects Model yij = m + ai + eij i = 1, ..., a; j = 1, ..., ri Assumption Parameter Constraint E(eij ) = 0

  12. Analysis of Variance for Single-Factor Experiments Main Effect Sum of Squares: SSA Main Effects: SSA : Sum of Squares attributable to variation in the effects of Factor A

  13. Is a pooled estimate of the error variance correct, or just ad-hoc?

  14. Analysis of Variance for Single-Factor Experiments Error Sum of Squares: SSE Residuals: SSE : Sum of Squares attributable to uncontrolled variation

  15. Analysis of Variance for Single-Factor Experiments Error Sum of Squares: SSE Factor Levels: i = 1, 2, ... , a Sample Variances:

  16. Analysis of Variance for Single-Factor Experiments Error Sum of Squares: SSE Factor Levels: i = 1, 2, ... , a Sample Variances: Pooled Variance Estimate:

  17. Degrees of Freedom Total Sum of Squares Constraint Show Degrees of Freedom n -1 = ar - 1

  18. Degrees of Freedom Main Effect Sum of Squares Constraint Show Degrees of Freedom a -1

  19. Degrees of Freedom Error Sum of Squares Constraints Show Degrees of Freedom n – a = a(r – 1)

  20. Analysis of Variance Table

  21. Analysis of Variance for the Flow Rate Data Assumptions ? Conclusions ?

  22. Individual confidence intervals and tests are not appropriate unless SIMULTANEOUS significance levels or confidence levels are used (Multiple Comparisons)

  23. Viscosity of a Chemical Process Two Factors Replicate

  24. Uncontrolled experimental error Assignable causes: two factor main effects and their interaction Viscosity of a Chemical Process 160 150 Viscosity 140 130 15% 20 lb/hr 25% 20 lb/hr 15% 30 lb/hr 25% 30 lb/hr Reactant Concentration / Flow Rate

  25. Interaction Main Effects Viscosity of a Chemical Process Average Viscosity

  26. Viscosity of a Chemical Process :Main Effect for Concentration 160 Main Effect 150 Viscosity 140 130 15% 20 lb/hr 25% 20 lb/hr 15% 30 lb/hr 25% 30 lb/hr Reactant Concentration / Flow Rate

  27. Viscosity of a Chemical Process :Main Effect for Flow Rate 160 Main Effect 150 Viscosity 140 130 15% 20 lb/hr 25% 20 lb/hr 15% 30 lb/hr 25% 30 lb/hr Reactant Concentration / Flow Rate

  28. Viscosity of a Chemical Process :Flow Rate & Concentration Interaction 160 20 lb/hr 150 30 lb/hr Viscosity 140 Interaction ? 130 15% 25% Reactant Concentration

  29. Analysis of Variance for Multi-Factor Experiments Model yijk = m + ai + bj + (ab)ij + eijk Total Sum of Squares Balanced Design Goal Partition TSS into components associated with Assignable Causes: main effects for Factors A &B, interaction between Factors A & B Experimental Error: uncontrolled variation, measurement error, unknown systematic causes

  30. Analysis of Variance for Multi-Factor Experiments

  31. Analysis of Variance for Multi-Factor Experiments

  32. Analysis of Variance for Multi-Factor Experiments Show

  33. Analysis of Variance for Multi-Factor Experiments Show

  34. Analysis of Variance for Multi-Factor Experiments Show

  35. Analysis of Variance for Multi-Factor Experiments Don’t memorize the formulas, understand what they measure

  36. Analysis of Variance Table Understand the degrees of freedom

  37. Viscosity Data Conclusions ?

  38. Sums of Squares:Connections to Model Parameters

  39. Unbalanced Experiments(including rij = 0) Calculation formulas are not correct “Sums of Squares” in computer-generated ANOVA Tables are NOT sums of squares (can be negative) usually are not additive; need not equal the usual calculation formula values

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