22-1©2011 Raj JainCSE567MWashington University in St. LouisTwo Factor Two Factor Full Factorial Design Full Factorial Design with Replicationswith ReplicationsRaj Jain Washington University in Saint LouisSaint Louis, MO [email protected] slides are available on-line at:http://www.cse.wustl.edu/~jain/cse567-11/22-2©2011 Raj JainCSE567MWashington University in St. LouisOverviewOverview Model Computation of Effects Estimating Experimental Errors Allocation of Variation ANOVA Table and F-Test Confidence Intervals For Effects22-3©2011 Raj JainCSE567MWashington University in St. LouisModelModel Replications allow separating out the interactions from experimental errors. Model: With r replications Here,22-4©2011 Raj JainCSE567MWashington University in St. LouisModel (Cont)Model (Cont) The effects are computed so that their sum is zero: The interactions are computed so that their row as well as column sums are zero: The errors in each experiment add up to zero:22-5©2011 Raj JainCSE567MWashington University in St. LouisComputation of EffectsComputation of Effects Averaging the observations in each cell: Similarly, Use cell means to compute row and column effects.22-6©2011 Raj JainCSE567MWashington University in St. LouisExample 22.1: Code SizeExample 22.1: Code Size22-7©2011 Raj JainCSE567MWashington University in St. LouisExample 22.1: Log TransformationExample 22.1: Log Transformation22-8©2011 Raj JainCSE567MWashington University in St. LouisExample 22.1: Computation of EffectsExample 22.1: Computation of Effects An average workload on an average processor requires a code size of 103.94(8710 instructions). Processor W requires 100.23 (=1.69) less code than avg processor. Processor X requires 100.02(=1.05) less than an average processor and so on. The ratio of code sizes of an average workload on processor W and X is 100.21(= 1.62).22-9©2011 Raj JainCSE567MWashington University in St. LouisExample 22.1: InteractionsExample 22.1: Interactions Check: The row as well column sums of interactions are zero. Interpretation: Workload I on processor W requires 0.02 less log code size than an average workload on processor W or equivalently 0.02 less log code size than I on an average processor.22-10©2011 Raj JainCSE567MWashington University in St. LouisComputation of ErrorsComputation of Errors Estimated Response: Error in the kth replication: Example 22.2: Cell mean for (1,1) = 3.8427Errors in the observations in this cell are:3.8455-3.8427 = 0.00283.8191-3.8427 = -0.0236, and3.8634-3.8427 = 0.0208Check: Sum of the three errors is zero.22-11©2011 Raj JainCSE567MWashington University in St. LouisAllocation of VariationAllocation of Variation Interactions explain less than 5% of variation may be ignored.22-12©2011 Raj JainCSE567MWashington University in St. LouisAnalysis of VarianceAnalysis of Variance Degrees of freedoms: 22-13©2011 Raj JainCSE567MWashington University in St. LouisANOVA for Two Factors w ReplicationsANOVA for Two Factors w Replications22-14©2011 Raj JainCSE567MWashington University in St. LouisExample 22.4: Code Size StudyExample 22.4: Code Size Study All three effects are statistically significant at a significance level of 0.10.22-15©2011 Raj JainCSE567MWashington University in St. LouisConfidence Intervals For EffectsConfidence Intervals For Effects Use t values at ab(r-1) degrees of freedom for confidence intervals22-16©2011 Raj JainCSE567MWashington University in St. LouisExample 22.5: Code Size StudyExample 22.5: Code Size Study From ANOVA table: se=0.03. The standard deviation of processor effects: The error degrees of freedom:ab(r-1) = 40 use Normal tablesFor 90% confidence, z0.95= 1.64590% confidence interval for the effect of processor W is:1∓ t s1= -0.2304 ∓ 1.645 × 0.0060 = -0.2304 ∓ 0.00987= (-0.2406, -0.2203) The effect is significant.22-17©2011 Raj JainCSE567MWashington University in St. LouisExample 22.5: Conf. Intervals (Cont)Example 22.5: Conf. Intervals (Cont) The intervals are very narrow.22-18©2011 Raj JainCSE567MWashington University in St. LouisExample 22.5: CI for InteractionsExample 22.5: CI for Interactions22-19©2011 Raj JainCSE567MWashington University in St. LouisExample 22.5: Visual TestsExample 22.5: Visual Tests No visible trend. Approximately linear ⇒ normality is valid.22-20©2011 Raj JainCSE567MWashington University in St. LouisSummarySummary Replications allow interactions to be estimated SSE has ab(r-1) degrees of freedom Need to conduct F-tests for MSA/MSE, MSB/MSE, MSAB/MSE22-21©2011 Raj JainCSE567MWashington University in St. LouisExercise 22.1Exercise 22.1Measured CPU times for three processors A1, A2, and A3, on five workloads B1, B2, through B5 are shown in the table. Three replications of each experiment are shown. Analyze the data and answer the following: Are the processors different from each other at 90% level of confidence? What percent of variation is explained by the processor-workload interaction? Which effects in the model are not significant at 90% confidence.22-22©2011 Raj JainCSE567MWashington University in St. LouisHomework 22Homework 22 Submit answer to Exercise 22.1. Show all numerical
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