Unformatted text preview:

18-1©2006 Raj JainCSE567MWashington University in St. Louis22kkr Factorial r Factorial DesignsDesignsRaj Jain Washington University in Saint LouisSaint Louis, MO [email protected] slides are available on-line at:http://www.cse.wustl.edu/~jain/cse567-06/18-2©2006 Raj JainCSE567MWashington University in St. LouisOverviewOverview! Computation of Effects! Estimation of Experimental Errors! Allocation of Variation! Confidence Intervals for Effects! Confidence Intervals for Predicted Responses! Visual Tests for Verifying the assumptions! Multiplicative Models18-3©2006 Raj JainCSE567MWashington University in St. Louis22kkr Factorial Designsr Factorial Designs! r replications of 2kExperiments⇒ 2kr observations.⇒ Allows estimation of experimental errors.! Model:! e = Experimental error18-4©2006 Raj JainCSE567MWashington University in St. LouisComputation of EffectsComputation of EffectsSimply use means of r measurements! Effects: q0= 41, qA= 21.5, qB= 9.5, qAB= 5.18-5©2006 Raj JainCSE567MWashington University in St. LouisEstimation of Experimental ErrorsEstimation of Experimental Errors! Estimated Response:Experimental Error = Estimated - Measured∑i,jeij= 0# Sum of Squared Errors:18-6©2006 Raj JainCSE567MWashington University in St. LouisExperimental Errors: ExampleExperimental Errors: Example! Estimated Response:! Experimental errors:18-7©2006 Raj JainCSE567MWashington University in St. LouisAllocation of VariationAllocation of Variation! Total variation or total sum of squares:18-8©2006 Raj JainCSE567MWashington University in St. LouisDerivationDerivation! Model:Since x's, their products, and all errors add to zeroMean response:18-9©2006 Raj JainCSE567MWashington University in St. LouisDerivation (Cont)Derivation (Cont)Squaring both sides of the model and ignoring cross product terms:18-10©2006 Raj JainCSE567MWashington University in St. LouisDerivation (Cont)Derivation (Cont)Total variation:One way to compute SSE:18-11©2006 Raj JainCSE567MWashington University in St. LouisExample 18.3: MemoryExample 18.3: Memory--Cache StudyCache Study18-12©2006 Raj JainCSE567MWashington University in St. LouisExample 18.3 (Cont)Example 18.3 (Cont)Factor A explains 5547/7032 or 78.88%Factor B explains 15.40%Interaction AB explains 4.27%1.45% is unexplained and is attributed to errors.18-13©2006 Raj JainCSE567MWashington University in St. LouisConfidence Intervals For EffectsConfidence Intervals For Effects! Effects are random variables.! Errors ∼ N(0,σe) ⇒ y ∼ N( , σe)! q0= Linear combination of normal variates⇒ q0is normal with variance σe2/(22r)Variance of errors:! Denominator = 22(r-1) = # of independent terms in SSE⇒ SSE has 22(r-1) degrees of freedom. Estimated variance of q0: sq02=se2/(22r)18-14©2006 Raj JainCSE567MWashington University in St. LouisConfidence Intervals For Effects (Cont)Confidence Intervals For Effects (Cont)! Similarly,! Confidence intervals (CI) for the effects:! CI does not include a zero ⇒ significant18-15©2006 Raj JainCSE567MWashington University in St. LouisExample 18.4Example 18.4! For Memory-cache study: Standard deviation of errors:! Standard deviation of effects:! For 90% Confidence: t[0.95,8]= 1.86 ! Confidence intervals: qi∓ (1.86)(1.03) = qi∓ 1.92q0= (39.08, 42.91)qA=(19.58, 23.41)qB=(7.58, 11.41)qAB= (3.08, 6.91)! No zero crossing ⇒ All effects are significant.18-16©2006 Raj JainCSE567MWashington University in St. LouisConfidence Intervals for ContrastsConfidence Intervals for Contrasts! Contrast M Linear combination with ∑ coefficients = 0! Variance of ∑ h! For 100(1-α)% confidence interval, use t[1-α/2; 22(r-1)].18-17©2006 Raj JainCSE567MWashington University in St. LouisExample 18.5Example 18.5Memory-cache studyu = qA+ qB -2qABCoefficients= 0, 1, 1, and -2 ⇒ Contrastt[0.95;8]=1.8690% Confidence interval for u:18-18©2006 Raj JainCSE567MWashington University in St. LouisConf. Interval For Predicted ResponsesConf. Interval For Predicted Responses! Mean response :! The standard deviation of the mean of m responses:18-19©2006 Raj JainCSE567MWashington University in St. LouisConf. Interval for Predicted Responses (Cont)Conf. Interval for Predicted Responses (Cont)100(1-α)% confidence interval:! A single run (m=1):! Population mean (m=∞18-20©2006 Raj JainCSE567MWashington University in St. LouisExample 18.6: MemoryExample 18.6: Memory--cache Studycache Study! For xA= -1 and xB = -1:! A single confirmation experiment:! Standard deviation of the prediction:! Using t[0.95;8]= 1.86, the 90% confidence interval is:18-21©2006 Raj JainCSE567MWashington University in St. LouisExample 18.6 (Cont)Example 18.6 (Cont)! Mean response for 5 experiments in future:! The 90% confidence interval is:18-22©2006 Raj JainCSE567MWashington University in St. LouisExample 18.6 (Cont)Example 18.6 (Cont)! Mean response for a large number of experiments in future:! The 90% confidence interval is:! Current mean response: Not for future. Use contrasts formula.! 90% confidence interval:18-23©2006 Raj JainCSE567MWashington University in St. LouisAssumptionsAssumptions1. Errors are statistically independent.2. Errors are additive. 3. Errors are normally distributed.4. Errors have a constant standard deviation σe.5. Effects of factors are additive⇒ observations are independent and normally distributed with constant variance.18-24©2006 Raj JainCSE567MWashington University in St. LouisVisual TestsVisual Tests1. Independent Errors:! Scatter plot of residuals versus the predicted response ! Magnitude of residuals < Magnitude of responses/10 ⇒ Ignore trends ! Plot the residuals as a function of the experiment number! Trend up or down ⇒ other factors or side effects 2. Normally distributed errors: Normal quantile-quantile plot of errors 3. Constant Standard Deviation of Errors: Scatter plot of y for various levels of the factor Spread at one level significantly different than that at other⇒ Need transformation18-25©2006 Raj JainCSE567MWashington University in St. LouisExample 18.7: MemoryExample 18.7: Memory--cachecache18-26©2006 Raj JainCSE567MWashington University in St. LouisMultiplicative ModelsMultiplicative Models! Additive model:! Not valid if effects do not add. E.g., execution time of workloads.ith processor speed= viinstructions/second.jth workload Size= wjinstructions! The two effects multiply. Logarithm ⇒ additive model:! Correct Model:Where, y'ij=log(yij)18-27©2006 Raj JainCSE567MWashington


View Full Document

WUSTL CSE 567M - 2kr Factorial Designs

Documents in this Course
Load more
Download 2kr Factorial Designs
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view 2kr Factorial Designs and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view 2kr Factorial Designs 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?