1 Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 9 Review

2 Experimental Design Terminology Covariate Covariate Design (layout) Design (layout) Experimental Region (factor space) Experimental Region (factor space) Factor Factor Experimental units Experimental units Interaction Interaction Levels Levels Repeat Tests Repeat Tests Response Response Test Run Test Run MGH Table 4.1

3 Statistical Experimental Design Principles Systematically change known, controllable influences on the response What does this mean? Measure known, uncontrollable influences on the response What does this mean? Estimate the effects of all sources of variability What are these? Estimate experimental variability What types of experimental variability can occur?

4 Common Design Problems : Erroneous Principles of Efficiency Change factor levels in the most convenient manner, fime-wise or budget-wise Advantages? Disadvantages? Test many levels of inexpensive factors, few levels of expensive ones Advantages? Disadvantages? Run duplicate tests (if any) back-to-back Advantages? Disadvantages?

5 One Factor-at-a-Time Testing What is it? Advantages? Disadvantages

6 Randomization Inexpensive insurance Validates key assumptions (Independence, Randomization distributions) Validates key assumptions (Independence, Randomization distributions) WHY?

7 Complete Factorial Experiments, Completely Randomized Designs All combinations of the factor levels appear in the design at least once All combinations of the factor levels appear in the design at least once Randomize the sequence of test runs, assignment to experimental units Randomize the sequence of test runs, assignment to experimental units

8 Interactions Effects of the levels of one factor on the response depend on the levels of one or more other factors Interaction effects cannot be properly evaluated if the design does not permit their estimation Interaction effects cannot be properly evaluated if the design does not permit their estimation

9 Solving the Normal Equations Solutions are not estimates Estimable functions All solutions provide one unique estimator Estimators are unbiased All solutions to the normal equations produce the same estimates of “estimable functions” of the model means All solutions to the normal equations produce the same estimates of “estimable functions” of the model means

10 Cell Means Models : Estimable Functions All cell means are estimable All linear combinations of cell means are estimable Does not depend on parameter constraints Does not depend on parameter constraints

11 Cell Means Models : Estimable Functions All cell means are estimable Some linear combinations of cell means are uninterpretable Some linear combinations of cell means are essential

12 Parameter Equivalence: Effects Representation & Cell Means Model Parameter constraints Means and Mean Effects

13 Contrasts Contrasts often eliminate nuisance parameters; e.g., 

14 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

15 Analysis of Variance for Multi- Factor Experiments

16 Sums of Squares: Connections to Model Parameters

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 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