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# Experimental Statistics For Engineers II ST 516

NCS

GPA 3.79

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This 46 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 516 at North Carolina State University taught by Peter Bloomfield in Fall. Since its upload, it has received 10 views. For similar materials see /class/223929/st-516-north-carolina-state-university in Statistics at North Carolina State University.

## Reviews for Experimental Statistics For Engineers II

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Date Created: 10/15/15

Practical Interpretation 0 With a quantitative factor like power in the etch rate exam ple typically use regression modeling 9 50 31 6 or perhaps y5o 1x 2x2e 0 With a qualitative factor like method in the peak dis charge rate example we typically focus on comparisons among means Comparing Means o A contrast is a linear combination of treatment means L a Z Ciluiz With 2 Ci 2 0 quoti1 i1 0 Examples are M1 M2 where c1 1c2 1C3 2 ca 2 O M1 am 3 where c1 1cQ C3 2 c4 ca Testing a Contrast o Many hypotheses about treatment means can be written in terms of a contrast CL HO I Z CrL ILLrL 0 i1 for appropriate contrast constants c1c2 ca 0 We estimate 231 aim by o C is an unbiased estimator with a2 a 2 VC Zc i1 which we estimate by VC MSE a 2 2 Ci n i1 u The test statistic is therefore 0 231 01 to iVro iMTSE 231 c3 0 Compare to with the t distribution with df N a o Equivalently compare F0 2 25 with the F distribution with df 1N a o Confidence interval for C MSE Z 01 i ta2N a i1 n M ultiple Contrasts 0 Sometimes we test several contrasts in the same experiment or equivalently set up CIs for those contrasts error rate be comes an issue 0 Control experiment Wise error rate for all possible contrasts using Scheffe s method replace tel271 with Wa 1Fa7a17Na o Confidence interval becomes CL CL 2 Z Ciy39l i a 1Faa 1N a 2 Cr 1 i1 Pairwise Comparisons o Often the only contrasts we consider are pairwise compar isons M W a To control experiment wise error rate for all pairwise compar isons Tukey s method gives shorter intervals than Scheff s CI is MSE yi I qua 7 7 where qaaf is a percent point of the studentzed range statistic and f N a is the degrees of freedom in MSE Fisher s Least Significant Difference o The basic t statistic for comparing M with W is 7 Qj e a to So we declare W and W to be significantly different if 1 1 J z J 0 That iS 1 1 is the least significant difference or LSD 0 Notes In a balanced design m 2 nj 2 n so the LSD is the same for every pair M and Lj The LSD method has comparison wise error rate or it does not control experimehtwise error rate In R gt TukeyHSDaovEtchRate factorPower etchRateLong Tukey multiple comparisons of means 95 familywise confidence level Fit aovformula EtchRate factorPower data etchRateLong factorPower diff lwr upr p adj 180160 362 3145624 6925438 00294279 200160 742 41145624 10725438 00000455 220160 1558 122745624 18885438 00000000 200180 380 4945624 7105438 00215995 220180 1196 86545624 15265438 00000001 220200 816 48545624 11465438 00000146 10 gt plot TukeyHSD aov EtChRate factor Power etchRateLong 220180 200180 220160 200160 180160 220200 95 familywise confidence level ll I I 1 I l l 1 l I I 1 l l I 1 l l l 1 l l l i I I I 0 50 100 150 Differences in mean levels of factorPower 11 sample Size 0 Using Operating Characteristic curves 3 is the probability of type II error 3 1 PReject HOHO is false I 1 PFO gt Faya17NaHO iS false 0 Charts plot 3 against CD where a 2 D2 2 quot211 71 CLO392 0 So if we know 02 and a set of treatment effects 7172 Ta for which we want the type II error to be we can find the smallest acceptable n 12 Note often the resulting n is too large for the experiment to be feasible The experimenter must accept higher 3 or larger T s Alternative to using OC charts using length of confidence interval for comparing two means confidence interval is l 2 yi II ta27Na X E If we know 02 can choose n to give desired width Still often gives infeasible too large n 13 Practical Interpretation 0 With a quantitative factor like power in the etch rate exam ple typically use regression modeling 9 50 31 6 or perhaps y5o 1x 2x2e 0 With a qualitative factor like method in the peak dis charge rate example we typically focus on comparisons among means Comparing Means o A contrast is a linear combination of treatment means L a Z Ciluiz With 2 Ci 2 0 quoti1 i1 0 Examples are M1 M2 where c1 1c2 1C3 2 ca 2 O M1 am 3 where c1 1cQ C3 2 c4 ca Testing a Contrast o Many hypotheses about treatment means can be written in terms of a contrast CL HO I Z CrL ILLrL 0 i1 for appropriate contrast constants c1c2 ca 0 We estimate 231 aim by o C is an unbiased estimator with a2 a 2 VC Zc i1 which we estimate by VC MSE a 2 2 Ci n i1 u The test statistic is therefore 0 231 01 to iVro iMTSE 231 c3 0 Compare to with the t distribution with df N a o Equivalently compare F0 2 25 with the F distribution with df 1N a o Confidence interval for C MSE Z 01 i ta2N a i1 n M ultiple Contrasts 0 Sometimes we test several contrasts in the same experiment or equivalently set up CIs for those contrasts error rate be comes an issue 0 Control experiment Wise error rate for all possible contrasts using Scheffe s method replace tel271 with Wa 1Fa7a17Na o Confidence interval becomes CL CL 2 Z Ciy39l i a 1Faa 1N a 2 Cr 1 i1 Pairwise Comparisons o Often the only contrasts we consider are pairwise compar isons M W a To control experiment wise error rate for all pairwise compar isons Tukey s method gives shorter intervals than Scheff s CI is MSE yi I qua 7 7 where qaaf is a percent point of the studentzed range statistic and f N a is the degrees of freedom in MSE Fisher s Least Significant Difference o The basic t statistic for comparing M with W is 7 Qj e a to So we declare W and W to be significantly different if 1 1 J z J 0 That iS 1 1 is the least significant difference or LSD 0 Notes In a balanced design m 2 nj 2 n so the LSD is the same for every pair M and Lj The LSD method has comparison wise error rate or it does not control experimehtwise error rate In R gt TukeyHSDaovEtchRate factorPower etchRate Tukey multiple comparisons of means 95 familywise confidence level Fit aovformula EtchRate factorPower data etchRate factorPower diff lwr upr p adj 180160 362 3145624 6925438 00294279 200160 742 41145624 10725438 00000455 220160 1558 122745624 18885438 00000000 200180 380 4945624 7105438 00215995 220180 1196 86545624 15265438 00000001 220200 816 48545624 11465438 00000146 10 gt plot TukeyHSD aov EtChRate factor Power etChRate 95 familywise confidence level o 9 II o 00 I I I OI EI I I I I I 8I NI I OI 8 I I I I I I RI N I I CI 3 TII I l 8 N I o I T I I I I RI NI I o I O I FI I I I O I Q I I I I 0 50 100 150 Differences in mean levels of factorPower 11 sample Size 0 Using Operating Characteristic curves 3 is the probability of type II error 3 1 PReject HOHO is false I 1 PFO gt Faya17NaHO iS false 0 Charts plot 3 against CD where a 2 D2 2 quot211 71 CLO392 0 So if we know 02 and a set of treatment effects 7172 Ta for which we want the type II error to be we can find the smallest acceptable n 12 Note often the resulting n is too large for the experiment to be feasible The experimenter must accept higher 3 or larger T s Alternative to using OC charts using length of confidence interval for comparing two means confidence interval is l 2 yi II ta27Na X E If we know 02 can choose n to give desired width Still often gives infeasible too large n 13 The General 2k P Design 0 Requires p independent generators 0 Defining relation consists of these and their products gen eralized interactions 21 1 in total a Each effect has 21 1 aliases find them by multiplying the given effect by all words in the defining relation Design Criteria a High resolution 2 length of shortest word in full defining relation o Low aberration number of words with that length 0 Example 2372 design defining relation contains at least one 4 letter word 0 Each 4 letter word introduces 4 aliases of a main effect with a 3 factor interaction and 6 aliases of 2 factor interactions with each other 0 Choices I ABCF BCDG ADFG I ABCF ADEG BCDEFG I ABCDF ABDEG CEFG has minimum aberra tion Blocking a Fractional Factorial o Needed as always when the design has more runs than can be carried out under homogeneous conditions 0 Eg for 2 blocks choose an effect to be confounded with blocks 0 All of its aliases are then also confounded choose carefully 0 Table X has recommended choices Resolution 111 Designs 0 Main effects are aliased with 2 factor interactions 0 Designs exist for K N 1 factors in only N runs when N is a multiple of 4 saturated designs 3 1 7 4 15 11 31 26 39 Eg 2111 v 2111 v 2111 v 2111 Sequential Experiments Fold Over a Begin with the principal fraction for a resolution III design 0 If one factor is of special interest follow up with the alternate fraction in which signs for that factor are reversed a Combined experiment a single factor fold over gives main effect for that factor and all its 2 factor interactions free of 2 factor aliases 0 Or if all main effects are of interest follow up with the alternate fraction in which signs for all factors are reversed a Combined experiment a full fold over or reflection gives all main effects free of 2 factor aliases gt a resolution IV design 0 Often the two halves should be treated as blocks with those effects in the complete defining relation that change sign confounded with blocks Plackett Burman Designs 0 Two level fractional factorial designs for k N 1 variables in N runs with N a multiple of 4 saturated designs 0 When N is a power of 2 these are 251 19 designs with k 2 1 1 andek q for q234 o Plackett Burman designs for other N have more complicated aliasing structure Checking Assumptions 0 Normal distributions use probability plot or quantile quantile plot straight line implies normal distribution Normal QQ Plot 1 5 0 El Theoretical Quantiles 15 05 05 0 El 164 166 168 170 172 174 Sample Quantiles 0 Equal variances informally compare standard deviations also compare slopes in q d plots formally use F test for ratio of variances If Variances are Unequal 0 Estimated standard error of Q1 32 is now 32 32 1 2 711 712 so the test statistic is Q1 Q2 2 2 i 52 n1 n2 to 0 Under Ho M1 2 to is approximately t distributed with degrees of freedom o R command and output for equal variances gt ttestcement quotModifiedquot cement quotUnmodifiedquot varequal TRUE Two Sample ttest data cement quotModifiedquot and cement quotUnmodifiedquot t 21869 df 18 pvalue 00422 alternative hypothesis true difference in means is not equal to 0 95 percent confidence interval 054507339 001092661 sample estimates mean of X mean of y 16764 17042 o R command and output for unequal variances gt ttestcement quotModifiedquot cement quotUnmodifiedquot varequal FALSE Welch Two Sample ttest data cement quotModifiedquot and cement quotUnmodifiedquot t 21869 df 17025 pvalue 0043 alternative hypothesis true difference in means is not equal to 0 95 percent confidence interval 054617414 000982586 sample estimates mean of X mean of y 16764 17042 0 Note fewer non integer degrees of freedom slightly higher p value slightly wider confidence interval Paired Comparisons o Tensile bond strength of mortar depends on the condition of the items being bonded eg moisture content 0 Comparison is more precise if the formulations are tested on matched pairs with similar condition 0 Statistical model ymm je j i12 j12n Here gm measurement for formulation 139 in pair j o W mean strength for formulation 139 averaged across con ditions o 57 deviation from overall mean strength for pair j Eiyj N NOO39Z2 Analysis 0 form within pair differences dj 9m 92a M1 M2 1j 6291 0 pair deviation j cancels out so dj N NM1 M2aa 05 standard error of J is wagn 0 Sample variance 8 estimates 0 o The test statistic d to SalW 0 Under HO M1 M2 to is t distributed with n 1 degrees of freedom 10 Comparison with Unpaired Design 0 If each run was made on randomly selected materials with no pairing yijMi ij ija 139 12 j 12m b 0 HOW Vary 7j I 0 03912 o denominator of to is larger making test less sensitive 0 degrees of freedom are 201 1 making test more sensitive 11 c on balance test is usually more sensitive so pairing is good a pairing helps when within pair variation is much less than among pair variation 12

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