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

2 Flow Rate Experiment MGH Fig 6.1

3 Flow Rate Experiment ABCD Average Flow Rate Filter Type 0.35 Assignable Cause (Factor Changes) UncontrolledError What is an appropriate statistical comparison of the filter means?

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

6 5 Comparisons, Some averages used more than once (e.g., N/R50)

7 Analysis of Variance for Single-Factor Experiments Total Sum of Squares Model y ij =  +  i + e ij i = 1,..., a; j = 1,..., r i Total Adjusted Sum of Squares Corrected Sum of Squares (Numerator of the Sample Variance)

8 Analysis of Variance for Single-Factor Experiments Total Sum of Squares Model y ij =  +  i + e ij i = 1,..., a; j = 1,..., r i Goal Partition TSS into Components Associated with Assignable Causes: Controllable Factors and Measured Covariates Experimental Error: Uncontrolled Variation, Measurement Error, Unknown Systematic Causes

9 Analysis of Variance for Single-Factor Experiments

10 Analysis of Variance for Single-Factor Experiments Show

11 Analysis of Variance for Single-Factor Experiments

12 Estimating Factor Effects Model y ij =  +  i + e ij i = 1,..., a; j = 1,..., r i Estimation Assumption E(e ij ) = 0 Parameter Constraint

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

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

15 Analysis of Variance for Single-Factor Experiments Error Sum of Squares: SS E Residuals: SS E : Sum of Squares attributable to uncontrolled variation Sum of Squares attributable to uncontrolled variation

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

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

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

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

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

21 Analysis of Variance Table

22 Analysis of Variance for the Flow Rate Data Assumptions ? Conclusions ? Assumptions ? Conclusions ?

23 Individual confidence intervals and tests are not appropriate unlessSIMULTANEOUSsignificance levels or confidence levels are used (Multiple Comparisons) (Multiple Comparisons)

24 Viscosity of a Chemical Process Replicate Two Factors

25 Viscosity of a Chemical Process Viscosity 15% 20 lb/hr 25% 20 lb/hr 15% 30 lb/hr 25% 30 lb/hr Reactant Concentration / Flow Rate Assignable causes: two factor main effects and their interaction Uncontrolledexperimentalerror

26 Viscosity of a Chemical Process Average Viscosity Main Effects Interaction

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

28 Viscosity of a Chemical Process : Main Effect for Flow Rate Viscosity 15% 20 lb/hr 25% 20 lb/hr 15% 30 lb/hr 25% 30 lb/hr Reactant Concentration / Flow Rate Main Effect

29 Viscosity of a Chemical Process : Flow Rate & Concentration Interaction Viscosity 15%25% Reactant Concentration 20 lb/hr 30 lb/hr Interaction ?

30 Analysis of Variance for Multi-Factor Experiments Total Sum of Squares Model 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 y ijk =  +  i +  j + (  ) ij + e ijk Balanced Design Balanced Design

31 Analysis of Variance for Multi-Factor Experiments

32 Analysis of Variance for Multi-Factor Experiments

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 Show

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

37 Analysis of Variance Table Understand the degrees of freedom Understand the degrees of freedom

38 Viscosity Data Conclusions ?

39 Sums of Squares: Connections to Model Parameters

40 Unbalanced Experiments (including r ij = 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 “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