Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:

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1 Soo Kg Lm 1.0 Nested Fctorl Desg Two-Fctor Nested Desg Alss of Vrce... Exmple Stggered Nested Desg for Equlzg Degree of Freedom Three-Fctor Nested Desg Alss of Vrce... 9 Exmple

2 Soo Kg Lm Fgure 1: Two-fctor ested desg... Fgure : Three-fctor ested desg... Fgure : ANOVA tle for two-fctor ested desg... 5 Fgure : Dt of two-fctor ested desg of exmple Fgure 5: ANOVA results of exmple Fgure 6: A stggered ested desg... 8 Fgure 7: ANOVA tle for three fctor ested desg Fgure 8: Lekge curret test results of the three fctor ested desg, me d sum of k th tester of th fclt, d me d sum of th devce.. 11 Fgure 9: The me for th devce, th fclt, d k th tester Fgure 10: Me d sum of th devce for th fclt Fgure 11: ANOVA tle for three fctor ested expermet showg results of F-test

3 Soo Kg Lm 1.0 Nested Fctorl Desg For stdrd fctorl desgs, where ech level of ever fctor occurs wth ll levels of the other fctors d desg wth more th oe duplcte, ll the tercto effects c e studed. I ested fctor desg, the levels of oe fctor lke fctor B re smlr ut ot detcl for dfferet levels of other fctor lke fctor A. These re lso clled herrchcl desgs. The two-fctor d three-fctor ested desgs re show Fg. 1 d respectvel. Fgure 1: Two-fctor ested desg I ths exmple, mche s the fxed fctor, whle opertor s rdom fctor. Fgure : Three-fctor ested desg I ths exmple, fctor A s cosdered s fxed fctor whle, fctor B d C s cosdered s rdom fctor. 1.1 Two-Fctor Nested Desg The ler model for two fctor ested desg s - -

4 Soo Kg Lm = μ + α + β () + e k() (1) where μ s the grd me, α s the effect of level of fctor A, β () s the effect of th level of fctor B ested wth th level of fctor A, d e k() s the error ested wth th levels of fctor A d th level of fctor B. k s the umer of duplcte. Replcg the prmeters equto (1) ther respectve estmtors elds equto (). () Oe ssumes tht the oservtos re depedet d follows orml dstruto wth me μ d vrce σ Alss of Vrce If the umer of levels of fctor A s, the umer of levels of fctor B ested uder ech level of fctor A s, d the umer of duplcte s, the clculto of the sums of squres d ts degree of freedom re show equto () to (5). The totl sum of squre T s T k 1 1 k k k 1 () d t hs ( - 1) degree of freedom. The sum of squre due to fctor A A s A k1 1 1 k1 () d t hs ( - 1) degree of freedom. The sum of squre due to B ested wth A B/A s B/ A 1 1 k k 1 d t hs ( - 1) degree of freedom. The sum of squre due to error E s clculted equl to (5)

5 Soo Kg Lm E = T - ( A + B/A ) (6) whch hs ( - 1) degree of freedom. The sum of squre due to error E c lso e clculted usg of equto (7), whch s E = 1 1 k 1 k (7) The ANOVA tle of two fctor ested desg showg ther respectve sum of squre, degree of freedom, me squre vlue, d clculted F-vlue s show Fg.. Fctor Sum of Degree of Clculted F-vlue from Me Squre Squre Freedom F-vlue F-Tle A A ( - 1) A/( - 1) MSA/MSE F[( - 1), ( - 1)] B B ( - 1) B( - 1) MSB/MSE F[( - 1), ( - 1)] B/A B/A ( - 1) AB/(( - 1)) MSB/A/MSE F[(( - 1)), ( - 1)] Error E ( - 1) E/(( - 1)) - - Totl T ( - 1) Fgure : ANOVA tle for two-fctor ested desg Exmple 1 A comp s terested to test f there s dfferece mog the percetge of defect produced three equpmet d sx opertors mufcturg floor. The egeer uses ested fctor desg wth sx opertors 1,,,, 5, d 6, who operte the mches two tmes. The percetge of defect produced d the ssocte me vlues re show Fg.. Note tht ths desg the equpmet s fxed, wheres the opertor s the rdom fctor. The reso eg the equpmet s fxed for two dfferet selected groups of opertors. Equpmet () 1 Opertor () % of Defect 5, 8,, 0 5,, 1 1, k k

6 Soo Kg Lm 1 k k x 1 1 k k xx Fgure : Dt of two-fctor ested desg of exmple 1 Oservto s the k th replcte for k = 1 d o equpmet, where = 1,, d, wth opertor, where = 1,,,, 5, d 6. Ths s two-stge ested desg. If there re equl umer of levels of B wth ech level of A d equl umer of duplctes the the desg s lced ested desg. The effects tht c e tested ths desg re the effect due to equpmet, whch s fctor A d the effect of opertor ested wth the equpmet B/A. The error term s ested wth levels of A d B. I ths desg, the tercto etwee A d B cot e tested ecuse ever level of B does ot pper wth ever level of A. Soluto For ths expermet; =, =, d =. The totl sum of squre s equl to = = T k1 1 1 k1 xx = 5.5. Its degree of freedom s ( - 1) = 1-1 = 11. The sum of squre of fctor A s The sum of squre of B/A s 1 6-1) =. 8 A k1 1 1 k1 x xx = Its degree of freedom s ( - 1) = - 1 = B/ A k k 1 x 9 xx = Its degree of freedom s ( - 1) = ( - 6 -

7 Soo Kg Lm The sum of squre due to error s E = = 1.5. Its degree of freedom s ( - 1) = x( - 1) = 6. The ANOVA results of the expermet for = 0.05 re show Fg. 5. Fctor Equpmet Opertor wth equpmet Sum of Squre Degree of Freedom Me Squre Clculted F-vlue F-vlue from F- Tle for = 0.05 F0.05(, 6) = 5.1 F0.05(, 6) =.76 p-vlue > > Error Totl Fgure 5: ANOVA results of exmple 1 The results show tht there s o dfferece mog the umer of defect produced equpmet 1,, d d there s o dfferece mog sx opertor 1,,,, 5, d 6 ested wth ech equpmet. I ested expermet, the fctors tested c e fxed or rdom or comto of oth fctors. The clculto of the sums of squres d the test sttstcs do ot chge rrespectve of whether these fctors re fxed or rdom tpes. However, the terpretto of the results depeds upo the tpes of fctors Stggered Nested Desg for Equlzg Degree of Freedom The ested fctor desg cots more formto o fctors t lower levels the herrch of the desg th t hgher levels ecuse of the lrger degree of freedom. I hgher level studes, the dscrepces degrees of freedom mog sources of vrto c e cosderle. Stggered ested desgs were developed to equlze the degrees of freedom for the me squres t ech level of the herrch. The stggered desgs hve uequl umers of levels for fctors tht re ested wth other fctors. The levels for fctor B ested wth fctor A re vred from oe level of fctor A to other such w tht the degree of freedom for MSA d MS(B/A) re lmost equl. A stggered ested desg s gve Fg. 6. The degree of freedom for the equpmet s. For three opertor/equpmet comto, the degree of freedom opertor/equpmet s + + = 6. I ths desg, the degree of freedom for opertor/equpmet s =

8 Soo Kg Lm 1.1. Three-Fctor Nested Desg Fgure 6: A stggered ested desg The ler model for three fctor ested desg s l = μ + α + β + k() + (αβ) + (α) k() + e ()l (8) where μ s the grd me, α s the effect of level of fctor A, β s the effect of th level of fctor B, k() s the effect of k th level of fctor C ested wth th level of fctor B, (αβ) s the effect of th level of fctor tercts wth th level of fctor B, (α) k() s the effect of th level of fctor A tercts wth the k th level of fctor C ested wth th level of fctor B, d e k() s the error ested wth th levels of fctor A, th level of fctor B, d k th level of fctor C. l s umer of duplcte. Replcg the prmeters equto (.61) ther respectve estmtors elds equto (9). kl k k l Oe ssumes tht the oservtos l re depedet d follows orml dstruto wth me μ d vrce σ. Tkg the summto for = 1 to, = 1 to, k = 1 to c, d l = 1 to, t elds the sum of squre equto (10). (9) 1 1 k1 l1 c c 1 k1 1 1 k1 c c 1 c c k l c k 1 1 k1 l1 l (10) - 8 -

9 Soo Kg Lm If we use the smol A, B, d C fctor, the smplfed form of equto (11) c e wrtte s T = A + B + C/B + AB + AC/B + E (11) Alss of Vrce If the umer of levels of fctor A s, the umer of levels of fctor B ested uder ech level of fctor A s, d the umer of replctos s, the clculto of the sums of squres d the ssocted degrees of freedom re show equto (1) to (.70). The totl sum of squre T s T k c 1 1 k 1 l1 l c 1 1 k 1 l1 l c 1 1 k 1 l1 c l (1) d t hs (c - 1) degree of freedom. The sum of squre due to fctor A A s c (1) A 1 d t hs ( - 1) degree of freedom. The sum of squre due to fctor B B s B 1 c (1) d t hs ( - 1) degree of freedom. The sum of squre due to C ested wth B C/B s c k (15) C/ B 1 k 1 d t hs (c - 1) degree of freedom. The sum of squre due to tercto of fctor A d fctor B s c (16) AB

10 Soo Kg Lm d ts degree of freedom s AB s ( - 1)( - 1). The sum of squre due to fctor A tercts wth fctor C ested fctor B s c k (17) AC/ B 1 1 k 1 It hs ( - 1)(c - 1) degree of freedom. The sum of squre due to error E c e clculted usg l of equto (9), whch s c E = 1 1 k 1 l1 (18) l d ts degree of freedom s c( - 1). The ANOVA tle of three fctor ested desg showg ther respectve sum of squre, degree of freedom, me squre vlue, clculted F-vlue, d α = 0.05 s show Fg. 7. Fctor Sum of Degree of Me Clculted F-vlue from Squre Freedom Squre F-vlue F-Tle A A ( - 1) A/( - 1) MSA/MSE F[( - 1), c( - 1)] B B ( - 1) B/( - 1) MSB/MSE F[( - 1), c( - 1)] C/B/ C/B C/B (c - 1) ((c - 1)) MSC/B/MSE F[((c - 1)), c( - 1)] AB AB ( - 1)( - 1) AB/ F[(( - 1)( - 1)), MAB/MSE ( - 1)( - 1) c( - 1)] AC/B AC/B ( - 1)(c -1) AC/B/ F[(( - 1)( - 1), MAC/B/MSE ( - 1)(c -1) c( - 1)] Error E c( - 1) E/ (c( - 1)) - - Totl T (c - 1) Fgure 7: ANOVA tle for three fctor ested desg Exmple A test egeer hs three correlto devces, where he wts to test them usg four testers rdoml selected ech from two fcltes. The ech correlto devce s test twce for ts lekge curret. Alze the vrce for ll fctors. The results of lekge curret test re show Fg

11 Soo Kg Lm () Fclt () 1 = Tester (k) 1 1 Devce 1 Devce Devce Tester - Devce Totl k Me of k = 1 k 1 l l k 1 l1 l k 1 l1 = 11.9 Me of totl..5 Fgure 8: Lekge curret test results of the three fctor ested desg, me d sum of k th tester of th fclt, d me d sum of th devce Soluto The me for th devce, th fclt d k th l l tester s 1 s show Fg. 9, whle the me d sum of th devce for th fclt s show Fg. 10. B: Fclt () 1 C :Tester (k) 1 1 Devce 1 () A Devce () Devce () Fgure 9: The me for th devce, th fclt, d k th tester B Fclt () 1 1 = C Tester (k) 1 1 A Devce 1 () Devce () Devce () k 1 l1 1 1kl l = k 1 l1 =.7 kl Fgure 10: Me d sum of th devce for th fclt Bsed o equto (1) to (18), the sums of squre d degrees of freedom of the expermet re clculted.

12 Soo Kg Lm The vlue of s, s, c s, d s. The sum of squre due to C ested wth B C/B s C x C/ B c k 1 k 1 (.8.) (.5.) (1.95.) 6 (.8.5) (.7.5) = 6x = (..) / B 1 k 1 (.65.5). k The sum of squre due to fctor A tercts wth fctor C ested fctor B AC/B s AC / B k 1 1 k 1 ( ) ( )... ( = x0.91 = The sum of squre due to tercto of fctor A d fctor B AB s AB x 1 1 ( ) (...9.7) ( ) 8 ( ) ( ) ( ) = 8x0.0 = The sum of squre for fctor A A s A xx 1 16 (.8.7) (.8.7) (.5.7) = The sum of squre for fctor B B s B xx 1 (..7) (.5.7) = The totl sum of squre T s T 1 1 k 1 l1 l 1 1 k 1 l1 xxx =.095 l 8.5) - 1 -

13 Soo Kg Lm The sum of squre due to error E s equl to = ANOVA tle showg ther respectve sum of squre, degree of freedom, me squre vlue, clculted F-vlue, d F-tle α = 0.05 s show Fg. 11. Fctor Sum of Degree of Me Clculted F-vlue from Squre Freedom Squre F-vlue α = 0.05 p-vlue A F0.05(, ) =.0 > B F0.05(1, ) =.6 < C/B F0.05( 6, ) =.51 < 0.05 AB F0.05(, ) =.0 > AC/B F0.05(1, ) =.18 > Error Totl Fgure 11: ANOVA tle for three fctor ested expermet showg results of F-test The results show tht there s o correlto etwee fclt oe d two d there s dfferece for tester ested fclt. For other fctors, the tests show tht the re ot 95.0% cofdece level

14 Soo Kg Lm A Alss of vrce..., 8 F Fxed fctor... H Herrchcl desg... N Nested fctor desg... P p-vlue... 6, 1 R Rdom fctor... S Stggered ested desg... 6 T Three-fctor ested desg... 7 Two-fctor ested desg... V Vrce..., 7-1 -

Section 7.2 Two-way ANOVA with random effect(s)

Section 7.2 Two-way ANOVA with random effect(s) Secto 7. Two-wy ANOVA wth rdom effect(s) 1 1. Model wth Two Rdom ffects The fctors hgher-wy ANOVAs c g e cosdered fxed or rdom depedg o the cotext of the study. or ech fctor: Are the levels of tht fctor