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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 9 Review. Experimental Design Terminology. Covariate Design (layout) Experimental Region (factor space) Factor Experimental units Interaction Levels Repeat Tests Response Test Run. MGH Table 4.1.

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Statistical Analysis

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

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

  3. Statistical Experimental Design Principles • Systematically change known, controllableinfluences on the response • What does this mean? • Measure known, uncontrollableinfluences 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 WHY? Inexpensive insurance Validates key assumptions (Independence, Randomization distributions)

  7. Complete Factorial Experiments,Completely Randomized Designs All combinations of the factor levels appear in the design at least once 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

  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

  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

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

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

  14. 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

  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: SSE Factor levels: i = 1, 2, ... , a Sample variances: Pooled variance estimate:

  18. 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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