Comparing Alternatives

Transkript

1 Coarg Alteratve Sauel Kouev Referece Meaurg Couter Perforace A Practtoer' Gude by Davd J. Lla, Cabrdge Uverty Pre, New Yor, NY, 000, ISBN The uleetal teachg ateral rovded at htt:// by Davd J. Lla

2 Roada Coarg two alteratve Before-ad-after coaro No-correodg eaureet Coarg roorto Coarg ore tha two alteratve Oe-Factor aaly of varace ANOVA 3 Coarg Two Alteratve Dd a chage to the yte have a tattcally gfcat act o erforace? I there a tattcally gfcat dfferece betwee two dfferet yte?. Before-ad-after coaro ared obervato. No-correodg eaureet uared obervato 4

3 Before-ad-After Coaro Before-ad-after eaureet are ot deedet They for correodg ar Meaureet wth the et before ad after eed ot be detcally dtrbuted Procedure Coute ea ad tadard devato of the dfferece d b a b before eaureet a after eaureet d ea value of d d td. devato of d Fd cofdece terval for the ea of dfferece d If 30, a CI for d gve by c, c d z α / d If < 30, a CI for d gve by c, c d t α / ; No tattcally gfcat dfferece betwee yte f terval clude 0 5 E: Before-ad-After Coaro Meaureet Before b After a Dfferece d b a

4 E: Before-ad-After Coaro Mea of dfferece d Std. devato 4.5 d Fro ea of dfferece, aear that chage reduced erforace. However, tadard devato large! 95% Cofdece Iterval for the ea of dfferece: t α / ; t0.975;5 c d t.57 d [ 6, α / ; 5.36,3.36] Iterval clude 0 Wth 95% cofdece, there o tattcally gfcat dfferece betwee the two yte. 7 No-Correodg Meaureet No drect correodece betwee ar of eaureet Uared obervato eaureet of yte eaureet of yte Meaureet wth each et are IID rado varable Procedure. Coute ea. Coute dfferece of ea 3. Coute tadard devato of dfferece of ea 4. Fd cofdece terval for th dfferece 5. No tattcally gfcat dfferece betwee yte f terval clude 0 Th rocedure ow a t-tet 8 4

5 are both ad Aug that ug we ca etate the tadard devato of Therefore, ] [ the R.V. are de. Y If Note :,,,, > Y a a Y σ σ σ σ σ σ σ σ σ σ σ σ Cof. Iterval for Dfferece of Mea 0 Why Add Stadard Devato?

6 Cof. Iterval for Dfferece of Mea ad alo orally dtrbuted,.e. N µ, σ - µ N0, σ If 30 ad c z are aro.orally dtrbuted CLT tadard oral dtrbuto Ele we ca ue thestudet- t dtrbuto c t c t 30, we ca aroate σ wth α / c α / α / ; α / However, df df ; df - z tead df Eale eaureet eaureet

7 Eale cot. df c c,, t α / ; t α / ; df df t 3.4 [6,00] 0.95; Secal Cae If oly a few eaureet avalable.e. < 30 or < 30, but t ow that Error are orally dtrbuted ad σ σ or The ecal cae ale c, c df t α / ; df Tycally roduce tghter cofdece terval Soete ueful after obtag addtoal eaureet to teae out all dfferece 4 7

8 8 5 teret eaured yte the evet of the roorto of #evet yte total teret yte #evet of # evet yte total teret yte #evet of Coarg Proorto The uber of evet of teret follow a boal dtrbuto wth araeter ad If > 0 we ca aroate the boal dtrbuto ug oral dtrbuto: ] [ E σ,, N N 6,,, for the ea of CI Need a Let N N N - E E E E Coarg Proorto

9 9 7 α α α α α α Pr Pr / / / / z z z z / where, z c c α Coarg Proorto 3 8 Ital oeratg yte OS,300,03 terrut 3.5 hour 4,89 terrut occurred OS code 0.099, or % of te eecutg OS Ugraded OS 999,38 84, , or 8.5% of te eecutg OS Stattcally gfcat roveet? % cofdece terval 0.04, Stattcally gfcat dfferece? Eale

10 Roada Coarg two alteratve Before-ad-after coaro No-correodg eaureet Coarg roorto Coarg ore tha two alteratve Oe-Factor aaly of varace ANOVA 9 Coarg More Tha Two Alteratve Wat to fd out f there tattcally gfcat dfferece betwee the alteratve Naïve aroach Coare yte two-by-two ug revou techque 0 0

11 Oe-Factor Aaly of Varace Very geeral techque, alo called Oe-factor ANOVA Oe-factor eeretal deg Oe-way clafcato Searate total varato oberved a et of eaureet to:. Varato wth dvdual yte Due to rado eaureet error. Varato betwee yte Due to real dfferece rado error A to detere f tattcally greater tha? Oe-Factor Aaly of Varace Mae eaureet of alteratve y th eaureet o th alteratve Aue eaureet error for the dfferet alteratve are: Ideedet Gaua orally dtrbuted

12 Meaureet for All Alteratve Alteratve Meaure et Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 3 Colu Mea Alteratve Meaur eet Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 4

13 Devato fro Colu Mea Colu ea are average value of all eaureet wth a gle alteratve Average erforace of oe alteratve y. y Each eaureet ca be rereeted a y e y. e devato of y error eaureet fro colu ea 5 Overall Mea Alteratve Meaure et Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 6 3

14 Devato Fro Overall Mea Average of all eaureet ade of all alteratve y.. y Colu ea ca be rereeted a y. y α α devato of colu ea fro overallea.. effect of alteratve It ca be how that α 0 7 Effect Devato Fro Overall Mea Alteratve Meaure et Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 8 4

15 Effect ad Error Effect dtace fro overall ea Horzotally acro alteratve Error dtace fro colu ea Vertcally wth oe alteratve Error acro alteratve, too Idvdual eaureet are the y y α e.. 9 Su of Square of Dfferece: SSE y e SSE y y. e y. e y y. SSE characterze the varato due to error 30 5

16 Su of Square of Dfferece: SSA y. y α y... SSA α α y. y.. y.. SSA characterze the varato due to the effect 3 Su of Square of Dfferece: SST y t SST y.. α e α e t y y.. y y.. SST characterze the total varato 3 6

17 Su of Square of Dfferece SSA SST y. y.. SSE y y. y y.. Su of quare of dfferece SSA varato due to effect of alteratve error SSE varato due to error eaureet SST total varato Eadg SST ad ug SST SSA SSE α 0 we ca how 33 ANOVA Fudaetal Idea Searate varato eaured value to:. Varato due to effect of alteratve SSA varato acro colu. Varato due to error SSE varato wth a gle colu If dfferece aog alteratve are due to real dfferece, SSA hould be tattcally > SSE 34 7

18 Coarg SSE ad SSA Sle aroach SSA / SST fracto of total varato elaed by dfferece aog alteratve SSE / SST fracto of total varato due to eeretal error But t tattcally gfcat? 35 Stattcally Coarg SSE ad SSA Varace ea quare value SS df dfssa, ce alteratve total varato degree of freedo dfsse, ce alteratve, each wth df dfsst dfssa dfsse a SSA e SSE t SST 36 8

19 Degree of Freedo for SSA Alteratve Meaure et Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 37 Degree of Freedo for SSE Alteratve Meaure et Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 38 9

20 Degree of Freedo for SST Alteratve Meaure et Col ea Effect y y y y y. α y y y y y. α y y y y y. α y y y y y. α 39 Varace fro Su of Square Mea Square Value Ue F-tet to coare rato of varace F F a e [α ; df u, df deo] tabulated crtcal value If F couted > F table we have α * 00% cofdece that the varato due to actual dfferece alteratve, SSA, tattcally greater tha the varato due to error, SSE. 40 0

21 4 ], ; [ Tabulated Couted ] [ Mea quare Deg freedo quare Su of Total Error Alteratve Varato e a e a F F F SSE SSA SST SSE SSA α Oe-Factor ANOVA Suary Effect Colu ea Overall ea 3 Meaureet Alteratve ANOVA Eale

22 ANOVA Eale cot. Varato Su of quare Deg freedo Mea quare Couted F Tabulated F Alteratve SSA a F [0.95;,] 3.89 Error SSE e Total SST ANOVA Eale cot. SSA/SST / % of total varato eaureet due to dfferece aog alteratve SSE/SST / % of total varato eaureet due to oe eaureet Couted F tattc > tabulated F tattc 95% cofdece that dfferece aog alteratve are tattcally gfcat. 44

23 The Method of Cotrat ANOVA tell u that there a tattcally gfcat dfferece aog alteratve But t doe ot tell u where the dfferece Ue ethod of cotrat to coare ubet of alteratve A v B {A, B} v {C} Etc. 45 Cotrat Cotrat lear cobato of effect of alteratve c w α w 0 Cotrat are ued to coare effect of a ubet of the alteratve E.g. Coare effect of yte to effect of yte w w c α α 0 α α α w

24 Cofdece Iterval For Cotrat Need Etate of varace of cotrat Arorate value fro Studet t table Coute cofdece terval a before If terval clude 0 The o tattcally gfcat dfferece et betwee the alteratve cluded the cotrat 47 Cofdece Iterval For Cotrat Aug varato due to error equally dtrbuted aog total eaureet Var[ c] Var[ w Var[ w α ] w α ] Var[ α ] c e w SSE df c e c, c c t α / ; c c w e 48 4

25 Eale 90% cofdece terval for cotrat of [Sy-Sy] α α 0.44 α c 3 [ ] c e 90% : c, c ,0.096 Wth 90% cofdece, the dfferece betwee yte ad yte tattcally gfcat 49 Suary Coarg two alteratve Ue cofdece terval to detere f there are tattcally gfcat dfferece Before-ad-after coaro Fd terval for ea of dfferece No-correodg eaureet Fd terval for dfferece of ea If terval clude zero No tattcally gfcat dfferece Coarg roorto 50 5

26 Suary cot. Coarg ore tha two alteratve Ue oe-factor ANOVA to earate total varato to: Varato wth dvdual yte Due to rado error Varato betwee yte Due to real dfferece rado error I the varato due to real dfferece tattcally greater tha the varato due to error? Ue cotrat to coare effect of ubet of alteratve 5 Further Readg The Art of Couter Syte Perforace Aaly: Techque for Eeretal Deg, Meaureet, Sulato, ad Modelg by R. K. Ja, Wley Arl 99, ISBN: , 99 Perforace Evaluato ad Becharg, edted by Lzy Kura Joh, Leve Eechout, CRC Pre Ic., ISBN: , 005 Probablty ad Stattc for Egeer ad Scett 7th Edto by Roald E. Walole, Rayod H. Myer, Sharo L. Myer, Keyg Ye, Keyg Yee, Pretce Hall, 7 edto Jauary 00, ISBN-0: , ISBN-3:

27 Eerce Ug the before-ad-after coaro techque wth both a 90% ad a 99% cofdece level, detere whether turg a ecfc coler otzato o ae a tattcally gfcat dfferece. Reeat your aaly ug ANOVA tet wth alteratve. Ela your reult. Ue the ANOVA tet to coare the erforace of three dfferet, but roughly coarable, couter yte eaured ter of eecuto te of a arorate bechar rogra. The ANOVA tet how oly whether there a tattcally gfcat dfferece aog the yte, ot how large the dfferece really. Ue arorate cotrat to coare the dfferece betwee all oble ar of the yte. Ela ad terret your reult. 53 7