Examples of non-orthogonal designs

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1 Examples of non-orthogonal designs Incomplete block designs > treatments,, blocks of size 5, 5 > The condition of proportional frequencies cannot be satisfied by the treatment and block factors. ¾ g Z has nontrivial projections onto ¼ f œ U Z and f œ U. " # -1-

2 There is treatment information in both inter- and intrablock strata. The best linear unbiased estimators of treatment contrasts in f" and f# are called interand intrablock estimators, respectively. -2-

3 W œ F K, W œ M F " # Normal equations in inter- and intrablock strata: where G G " " " Xα^ œ UX ^ X # # α œ UX #, " w " w X X X X X G œ \ ÐF K Ñ\, U œ \ ÐF KÑC, # w # w X X X X X G œ \ ÐM F Ñ\., U œ \ ÐM FÑC. -3-

4 The intra-block estimator of a treatment contrast under the randomization model is the same as its best linear unbiased estimator under the usual two-way layout model with fixed block effects. G œ? R? R, X # X " F w # " X X F U œ W R? W, WX and WF: vectors of treatment totals and block totals, respectively. F -4-

5 ? F œ5m ¾ G œ? X RR # " U œ W RW X X # X 5 " 5 F w? X : > > diagonal matrix with the 3th diagonal entry equal to < 3, where < 3 is the number of replications of the 3th treatment. -5-

6 Similarly, X " " w " w 5,5 G œ RR <<, " " X 5 F where < œð< ", á, <Ñ > w. U œ RW C.. <, -6-

7 ANOVA for a general randomized block design: Sources of variation SS d.f. MS E(MS) Interblock " " X X 0" rank ÐG " X Ñ X Treatments ^ w Ðα Ñ G " α^ rank Ð G " " w " Ñ â α G α Residual By subtraction, 5ÐC C Ñ 3œ" 3... #, 1 ã 0 " -7-

8 Intrablock # # X X 0# rank ÐG # X Ñ X Treatments ^ w Ðα Ñ G # α^ rank Ð G # " w # Ñ â α G α Rresidual By subtraction, 5 # ÐC C Ñ,( 5 1) 3œ" 4œ" ã 0 # Total, 5 # ÐC C Ñ,5 " 3œ" 4œ" 34 ÞÞ -8-

9 Balanced incomplete block designs An incomplete block design is said to be binary if 834 œ0 or 1, a"ÿ3ÿ>, "Ÿ4Ÿ,, where 834 is the number of times the 3th treatment appears in the 4th block. 3th diagonal entry of RR w œ < 3 ( 34, )-th entry œ- 34, the # of blocks in which the 3th and 4th treatments appear together. -9-

10 A block design is equi-replicate if all treatments appear the same number of times. Let <œ,5î>. A balanced incomplete block design (BIBD) is a binary and equireplicate design in which all pairs of treatments appear together in the same number of blocks -34 œ

11 A BIBD with,œ>œ7 and 5œ3:

12 For a BIBD, G œ ÒÖ<Ð5 1 Ñ - M -N Ó. X # " 5 > > The matrix has row sum Ò<Ð5 "Ñ Ð> 1 Ñ - Ó œ!. " 5 G X # ¾ <Ð5 "Ñ œ Ð> 1Ñ- G œ Ò> -M -N Ó X # " 5 > > -12-

13 # # X X Clearly rankðg Ñ œ > 1. A g-inverse of G is ÐG Ñ œ X # 5 > - M> A solution for ^α # is α^ # 5 œ > X # - U # w " The 3th entry of U X is X X 8 F, where œ" X 3 X w is the th treatment total. is called the th adjusted treatment total., -13-

14 ¾ the intrablock estimator of any treatment contrast - w α is w # 5 w # 5 > w > - X > œ" - α^ œ - U œ - X, with w # var Ð ^ w # 5 w - α Ñ œ 0# - ÐGX Ñ - œ 0# > All normalized contrasts are estimated with the same precision. In particular, each pairwise comparison is estimated with variance # # > - # varðα^ α^ Ñ œ

15 The treatment sum of squares in the intrablock stratum is # w # 5 # w # 5 > w # X > - X X > - 3 3œ" Ðα^ Ñ U œ ÐU Ñ U œ ÐX Ñ. -15-

16 For estimation in the interblock stratum, G œ ÒÐ< -ÑM -N Ó N X " " < 5 > >,5 > 5 This matrix also has rank > 1, with < - M> as a g- inverse. Therefore a solution for ^α " is " 5 < X " - U α^ œ, # -16-

17 The interblock estimator of any treatment contrast - w α is " w " 5 w " - α^ œ < - - UX 5 w " œ < - - Ð5 RW F C.. < Ñ 5 w " œ < - - Ð5 RW F C.. < 1> Ñ œ " w < F >, " < œ" 4œ" - -RW œ - Ð 8 F Ñ, w var Ð ^ w " 5 w - α Ñ œ 0" - ÐGX Ñ - œ 0" <

18 Again all normalized contrasts are estimated with the same precision. The treatment sum of squares in the interblock stratum is " w " 5 " w " X < - X X Ðα^ Ñ U œ ÐU Ñ U. -18-

19 Sources SS d.f. MS E(MS) Interblock 5 " w " " w " Treatments < - ÐU X Ñ U X > " â 0 " > " α G X α Residual By subtraction â 0 " Intrablock Treatments, 5ÐC C Ñ 3œ" 3... Residual By subtraction, 5 #, 1 5 > " > 3 w # w # - ÐX Ñ > " â # > " GX 3œ" # ÐC C Ñ,Ð5 1 Ñ 3œ" 4œ" Total lc KCl,5 1 # â 0 # 0 α α -19-

20 Compare with a hypothetical randomized complete block design in which each treatment appears < times. Under such a design, var( ^ w -α) œ # / < ; or since it would involve a different grouping of the units, let's write this variance as --0 / <. Then w w # w var Ð- α^ Ñ for a randomized complete block design var Ð-w # α^ Ñ for a BIBD > - 0# <5 0. œ w # The ratio > -Î<5, denoted by /, is called the efficiency factor (or more precisely, the efficiency factor in the intrablock stratum) of the BIBD. -20-

21 w var Ð- α^ Ñ for a randomized complete block design var Ð-w " α^ Ñ for a BIBD < - 0 <5 0 œ w # ". The factor Ð< -ÑÎ<5 may be called the efficiency factor in the interblock stratum. > -Î<5 Ð< -ÑÎ<5 œ ÒÐ> 1Ñ- <ÓÎ<5 œ ÒÐ5 1Ñ < <ÓÎ<5œ1. ¾ the efficiency factor in the interblock stratum is 1 / -21-

22 0 / 1. For a BIBD, var Ð- var Ð- α^ α^ w # w " Ñ Ñ 1 / / œ 0 0 # ". The efficiency factor can also be expressed as > - >Ð5 1Ñ <5 Ð> 1 Ñ5. /œ œ -22-

23 Recovery of interblock information The interblock and intrablock estimators are uncorrelated. [In general, estimators in different strata are uncorrelated: 3Á4 Ê w w w cov Ð+WC,,WCÑ œ +WÐ = 0 WÑW, 4 œ 0] œ!

24 Suppose ) ^ and ^ " )# are two uncorrelated unbiased estimators of a certain parameter ). Among unbiased estimators of the form + ^ ^ ")" + #)#, var( + ) ^ + ) ^ ) is minimized by " " # # " + œ 3 Ñ, 3 œ 1, 2. 3 " ^ " ^ # Ñ -24-

25 If 0" and 0# were known, then the best linear combination of the inter- and intrablock estimators of a treatment contrast would be > - w # < - w " 50 - α^ # 50 - α^ " > - < - 50 # 50". Since 0" and 0# are unknown, they need to be estimated. -25-

26 Wu and Hamada, p.76: an experiment to assess the effect of tires made out of different compounds on the lifetimes of the tires. A tire can be divided into at most three sections with each section being made out of a different compound. Wear data: Compound Tire A B C D

27 units [nvalues=12] factor [levels=4; values=3(1...4)] tire factor [labels=!t(a, B, C, D)] compound read compound; frepresentation=labels A B C A B D A C D B C D: blocks tire treatments compound anova adisplay [print=effects] stop -27-

28 ***** Analysis of variance ***** Source of variation d.f. tire stratum compound 3 tire.*units* stratum compound 3 Residual 5 Total

29 ***** Information summary ***** Model term e.f. non-orthogonal terms tire stratum compound tire.*units* stratum compound

30 units [nvalues=12] factor [levels=4; values=3(1...4)] tire factor [labels=!t(a, B, C, D)] compound read compound, resistance; frepresentation=labels A 238 B 238 C 279 A 196 B 213 D 308 A 254 C 334 D 367 B 312 C 421 D 412: blocks tire treatments compound anova resistance adisplay [print=effects] stop -30-

31 Source of variation d.f. s.s. m.s. v.r. tire stratum compound tire.*units* stratum compound Residual Total

32 ***** Tables of effects ***** tire stratum compound effects e.s.e. * rep. 3 compound A B C D ***** tire.*units* stratum ***** compound effects e.s.e rep. 3 compound A B C D

33 > y=scan() 1: : Read 12 items > compound=factor(scan(what="")) 1: A B C A B D A C D B C D 13: Read 12 items > tire=gl(4,3) > out=aov(y~compound+error(tire)) -33-

34 > out[[3]] Terms: compound Residuals Sum of Squares Deg. of Freedom 3 5 Residual standard error: Estimated effects may be unbalanced > coef(out[[3]]) compoundb compoundc compoundd

35 > out[[2]] Terms: compound Sum of Squares Deg. of Freedom 3 Estimated effects may be unbalanced > coef(out[[2]]) compoundb compoundc compoundd

36 Non-orthogonal row-column designs W" œ V K W# œ G K W$ œ M V G K. Normal equations: G G G " " " X X # # # X œ UX $ $ $ X œ UX α^ œ U (1) α^ (2) α^. -36-

37 where " w " w X X X X X G œ \ ÐV K Ñ\, U œ \ ÐV KÑC, # w # w X X X X X G œ \ ÐG K Ñ\, U œ \ ÐG KÑC, G X $ œ \ X w ÐM V G K Ñ \ X, U œ \ ÐM V G KÑC $ w X X -37-

38 Equations (1) and (2) are the same as the equation for calculating the interblock estimators except that F is replaced by V and G, respectively. ¾ The inter-row (respectively, inter-column) estimators are the same as the interblock estimators when the rows (respectively, columns) are considered as blocks. -38-

39 and " " w " w " " X - V V <- X - V V G œ R R <<, U œ R W C <, # " w " w # " X < G G <- X < G G G œ R R <<, U œ R W C <, where RV is the treatment-row incidence matrix, R is the treatment-column incidence matrix, W V is the vector of row totals, and W G is the vector of column totals. The ANOVA in the inter-row and intercolumn strata are the same as that in the inter-block stratum..... G -39-

40 Calculation of the estimates in f $ : Estimators of treatment contrasts in stratum f $ are the same as the best linear unbiased (LS) estimators under the usual additive fixed effects model: C œ. α " # %, 34 >Ð3ß4Ñ where C34 is the observation at the 3th row and 4th column, >Ð3ß 4Ñ is the treatment assigned there, the α, " and #'s are unknown constants, EÐ% 34Ñ œ 0, and the observations are uncorrelated with constant variance. -40-

41 By direct computation, and X $ " w " X V G w " w - V < G <- G œ? R R R R << $ X " " X - V V < G G U œ W R W R W C.. <., -41-

42 Designs with RV œ7n À Each treatment appears 7 times in each row Ê the condition of proportional frequencies is satisfied by the treatment and row factors. It follows that g Z ¼ e Z. ¾ There is no treatment information in. f " We already know that the inter-column estimators are the same as the inter-block estimators with the columns considered as blocks. -42-

43 For the estimators in f $, \ X w ÐV KÑ œ 0 Ê G X $ œ \ X w ÐM V G K Ñ\ X œ \ X w ÐM G Ñ\ $ w w U œ \ ÐM V G KÑC œ \ ÐM GÑC. X X -43- ¾ The estimators in f $ are the same as the intrablock estimators with the columns considered as blocks. If RV œ N and RG is the incidence matrix of a BIBD, then it is called a Youden Square. X X

44 Compound Tire A B C D Section Tire A B C 2 B A D 3 C D A 4 D C B -44-

45 units [nvalues=12] factor [levels=4; values=3(1...4)] tire factor [labels=!t(a, B, C, D)] compound factor [levels=4; values=(1,2,3)4] section read compound, resistance; frepresentation=labels A 238 B 238 C 279 B 213 A 196 D 308 C 334 D 367 A 254 D 412 C 421 B 312: blocks tire*section treatments compound anova resistance adisplay [print=effects] stop -45-

46 ***** Analysis of variance ***** Variate: resistan Source of variation d.f. s.s. m.s. v.r. tire stratum compound section stratum tire.section stratum compound Residual Total

47 ***** Information summary ***** Model term e.f. non-orthogonal terms tire stratum compound tire.section stratum compound tire -47-

48 ***** Tables of effects ***** Variate: resistan ***** tire stratum ***** compound effects e.s.e. * rep. 3 compound A B C D ***** tire.section stratum ***** compound effects e.s.e rep. 3 compound A B C D

From Handout #1, the randomization model for a design with a simple block structure can be written as

From Handout #1, the randomization model for a design with a simple block structure can be written as Hout 4 Strata Null ANOVA From Hout 1 the romization model for a design with a simple block structure can be written as C œ.1 \ X α % (4.1) w where α œðα á α > Ñ E ÐÑœ % 0 Z œcov ÐÑ % with cov( % 3 % 4