BAS DESIGN ANLY EXPER
BAS DESIGN ANLY EXPER STA 6208
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Balanced Incomplete Block Designs STA 6208 Course Notes S pri n g 2009 An incomplete block design is a design containing blocks that have fewer EUs than the number of treatments In a balanced incomplete block design BIBD gt all blocks have the same of EUs lt treatments gt every EU in a block gets a different treatment gt all treatments are equally replicated applied to the same of EUs over the entire design gt every pair of treatments appears in the same of blocks Note This is not balanced in the usual sense Randomized complete block designs require EUsblock treatments What if available EUsblock lt treatments Example Evaluating Eyedrops gt treatments 5 brands of eyedrops gt EUs eyes gt blocks human subjects Example Reinforcing Concrete Pillars gt treatments 6 ways to reinforce a pillar gt EUs pillars gt blocks batches of concrete one batch makes 4 pillars Notation g treatments b blocks k EUsblock lt g r reps of each trt A blocks containing a particular pair of trts Example Block 1 Block 2 Block 3 Block 4 g b k r A Relations 1 N total EUs kb rg 2g1 rk71 True because both equal same block pairs of EUs such that one of them gets treatment 1 Same for 2 Or 3 Theorem Fisher BIBDs always have b 2 g or equivalently r 2 k When equality holds that is b g or equiv r k the BIBD is symmetric lntrablock Analysis Let yij response of EU in blockj receiving treatment 139 Model yyuai jeij oz iH1 treatment effect 6 quot block effect Za 0 6 iidN002 fixed blocks 26 0 1 random blocks 6 iid N0Uf9 and indep of 605 A BIBD is unreduced if each of the possible subsets of k treatments appears in some block Often undesirable because may require too many blocks Appendix C2 in textbook lists a catalog of BlBDs Randomization Beginning with a reference BIBD plan Appendix C2 1 Randomize order of blocks 2 Randomize treatment codes to EUs within each block 3 Randomize treatments to treatment codes Because BlBDs are not balanced in the usual sense there are no Hasse diagrams and the usual rules for constructing effect estimates and ANOVA do not apply Just use a computer to run the analysis SAS Example 142 Dish Detergent Note When specifying the model in computer code you must list the block effect term first if you want sequential Type I 555 to be correct Relative Efficiency of BIBD to RCB The relative efficiency of a BIBD to a RCB design is usually cataloged as kil A EBIBDRCB ff which assumes incorrectly but purposefully that the error variance is the same for both designs Note k lt g and so EBIBD RCB lt 1 Interpretation A RCB design with about r EBIBD RCB blocks would provide the same information precision as this BIBD assuming the same error variance and no adjustment for different dfg Let 55E be the usual residual sum of squares for the model with fixed block effects with its usual df de N717g717b71 N7g7b1 M a Test for Treatment Effects H0oz0alli Haa3 0somei Reject H0 only if F M5 Ralu g71 A 7 MSE MSE gt Fag71N7gib1 ANOVA For all ij such that trt 139 occurs in block j define Vij Yij Thy Fact The least squares estimate of a is A V rEBIBD RCB and the partial sum of squares for treatment adjusted for blocks is 5512133 Ral rEBIBDRCB 232 with dfm g 7 1 Contrast Inference Just as for RCB designs a treatment contrast has the form 2mm where ZW 0 i i and is estimated by 2 ma unbiased A 1 7 a100 CI for Z Wiai Z Wiai i ta2N7gibl MSE Z WiZr EBIBD RCB Can make simultaneous Cls using Bonferroni or Scheff Can also use Tukey if pairwise Derivations of Sche and Tukey Simultaneous Con dence Intervals Note This is optional supplementary material which you may nd useful for understanding the theory behind the Scheffe and Tukey methods for simultaneous contrast con dence intervals under the usual model for a completely randomized design Scheff Simultaneous Con dence Intervals for All Contrasts The supplement entitled Formal Details About Contrasts77 shows how the usual treatment effects model for a completely randomized design CRD with g gt 1 treatment groups of sizes 711 ng all nonzero can be expressed in the standard vector form for a regression model yX e eN0Iazgt and states that the least squares estimators of contrasts are exactly the quantities of the form t y for some t E M O 1 where M is the column space of X and N m I 719 Since the least squares estimators are unbiased the contrasts themselves can be written ct t Ey t X In this notation the Scheffe method produces simultaneous con dence intervals for all ct such that t E M O 11 that is for all contrasts The derivation relies on the Cauchy Schwarz inequality from linear algebra If t and 39v are two column vectors of length N If12 1515 11 Let Pr PiJN where P is the projection onto M the column space of X and J 1N 19V Then Pr is a projection in fact the projection that de nes the treatment sum of squares SSTrt You may easily verify that Pr projects onto M O 11 and thus for any contrast estimator t y t PT t It follows that with probability 1 every t E M O 1 satis es 2 t y 002 ty XI 56V t Pr 62 S W Pr GYPT 6 t t E Pr 6 applying Cauchy Schwarz with 39v Pr 6 that is Pt yict2 e Pr et t for all t evmijt 1 1 1 regardless of the values of M1 Hay and 02 gt 0 Now it follows from theory of the multivariate normal distribution that e PT e 2 7 N X971 since the projection Pr has rank 9 7 1 Also since I 7 PX 0 it is routine to verify that SSE yI7 Py eI7 Pe and thus since Pr and I7 P are orthogonal to each other it follows from theory of the multivariate normal that e PT e and SSE are independent Therefore E Pr 69 1 E Pr 60297 1 MSE SSE02N7g N F9 1gtN 9 and thus e Pre 71 1704 P g Fag71N7g PiePTE g 9 1Fag1N7gMSE 2 regardless of the values of M1 Hay and 02 gt 0 Thus using 1 and 2 1704 Pltt y7ct2 ePrett for all teM l E Pre S 9 1Fag71N79 MSEgt lt Pt y7ct2 g7lFag1NgMSEt t for all t EM l and so Plty7ctl 4g71Fag1Ng MSEt t for all teM l 2 1704 In fact it can be shown that this probability equals 1 7 04 It follows that if the interval t y i g7 1Fag1Ng MSEt t is associated with the contrast ct for every t E M O 11 then these intervals are 1 7 a100 simultaneous con dence intervals for all contrasts Using the explicit expressions for y and t provided in the supplement Formal Details About Contrasts77 it is easy to show that 9 9 t y 2w and t t i1 i1 and thus these intervals coincide with those presented in the textbook when applied to the analysis of a CRD Tukey Simultaneous Con dence Intervals for All Pairwise Comparisons Assume the usual treatment means or effects model for a completely randomized design CRD with g gt 1 treatment groups of the same size n yijuiqj EijNiidN002 i1g j17l Let N git be the total number of experimental units as usual Suppose Z1 Z9 and W are independent random variables with Z1 Zg each having a standard normal distribution and W having a chi square distribution with 1 degrees of freedom Then the distribution of maxi Z5 7 mini Z5 W 1 is called the Studentized range distribution with parameters 9 and 1 which shall be denoted by qg 1 Its upper 04 quantile percent point is well de ned for all 04 and shall be denoted qag 1 From this de nition it follows that maxyl M 7 mm M W max yhiw minlyl Ml l V0 71 VUZn N qg N79 MSEn SSEaZN 1 g under any values of M1 ug and 02 Notice that with probability 1 Mm7mgt7ltm7im mme7wgt7lgni7im fmauiwdi with equality for some pair ofi and j Therefore for any 1 Hay and 02 max 77 7min 77 H S qagNg MMSEn Pmm7mwwm7w gamN7vaambummm P yi yju MiHM S 97 MSE1n1n for alliandj and thus a 1 7 a100 con dence interval for M 7 W simultaneous over all i and j is 1 1 My N 7 9 1 1 7 i MS 7 7 yi 24 E n n In the general case with treatment groups of possibly different sizes n1 ng and N m 1 my total experimental units we often use the Tukey Kramer form 7 7 My N 7 9 1 1 4 4 7 i M S 7 7 f 11 d yquot 24 E m W or a i an 3 While these are no longer necessarily exact 1 7 a100 simultaneous con dence intervals it has been proven that they are conservative Formal Details About Contrasts Note This is optional supplementary material7 which you may nd useful for developing a more formal technical understanding of contrasts Vector Representation The usual treatment effects model for a completely randomized design with g gt 1 treatment groups of sizes 711712 719 all nonzero can be expressed in the standard vector form for a regression model7 y X e 6 N0 102 by letting yii ylm 2421 11 11 0 0 6 3 1m 0 1 2 0 1 y X B C 92712 I I g 1 0 0 1 9 2991 9W9 Where7 as usual7 y denotes the response of the jth experimental unit Within the ith treatment group The matrix X does not have full column rank but that Will not matter For this model relative to the intercept only reduced model7 a contrast is any linear combina tion 9 9 Zwmi such that Z w O 1 i1 ii lts least squares estimator 9 2 will 11 can be written as a linear function of y wllnlnl g 7 w21n2n2 Z wig t y7 where t i1 39 wglng ng Note 0 LettingNnlnz 7197 9 m g g m 22m 2w 0 i1 7391 i1 i1 39 ObViOUSIY t E M where M is the column space of X Together these conditions imply that t e M m 11 that is t is in the subspace of M that is orthogonal to the span of 1N Moreover you may verify the following Fact If t is any vector of length N then ty is the least squares estimator of a contrast if and only if t E M 1 2 in which case the contrast it estimates is nontrivial if and only if t 7 0 Properties of Contrasts You may verify the following facts 1 If t y is the least squares estimator of a nontrivial contrast then the sum of squares for the contrast it estimates is 71 7 tylz 55w 7 y ttt ty 7 t Notice that tt t 1t is the projection onto the span of t Therefore 55w is a quadratic form of y that has one degree of freedom and the distribution of SSwa2 is chi square possibly noncentral with one degree of freedom 3 If each of y and t zy is the least squares estimator of some contrast then the two contrasts they estimate are orthogonal if and only if tltz O that is if and only if t1 and 752 are orthogonal as vectors Since the subspace M 1 has dimension gel it follows from 2 that at most 97 1 contrasts may be in a collection of nontrivial contrasts that are all pairwise orthogonal Using representation 2 makes it easy to derive formulas and facts concerning contrast estima tors For example y y Vt y t Vyt t Iazt t ta2 aZZnuin21i1m aZwan i1 i1 Also if tag and t zy estimate orthogonal contrasts then Covt 1yt 2y t1Vyt2 110th 75175202 0 Since t ly and t zy are jointly multivariate normal this further implies that they are independent Let t ly t gily be estimators of nontrivial pairwise orthogonal contrasts and let T t1 t Because of orthogonality t ltl 0 0 t giltynl Also T has full rank 9 7 1 and the column space of T is all of M 1 It follows that TTT 1T is the projection onto 2 ljiv Recall that if P is the projection onto M the column space of and if J 1N1V then P7 JN is a projection You may further verify that P 7 JN projects onto M 1 Hence from the uniqueness of a projection onto a given subspace it follows that TT T 1T P7 JN Therefore since P 7 JN is the de ning matrix for the treatment ie regression sum of squares SSTrL y P7JNy y TT T 1T y 1t 1t1 0 7 tly 7 M y tgill s s 0 1tg t94 tg y tly2 ital2 tltl tgnltg71 558 ssgil where SSS 513971 are the sums of squares for the g 7 1 orthogonal contrasts This shows why a full collection of nontrivial orthogonal contrast sums of squares decomposes the treatment sum of squares as stated in lecture Orthogonal Polynomial Contrasts When the treatment groups are de ned by distinct values of a single observed quantitative variable call it 2 it is possible to reparameterize to an orthogonal polynomial model in 2 for which the non intercept coe icients are orthogonal contrasts Suppose that the distinct 2 values that de ne groups 1 2 g are respectively 21 22 29 Then the usual non orthogonal full degree polynomial model for the response versus 2 can be written in vector form as 1 0 11 211 251 2f 111 0 1 221 221 29 1 yZ0e where Z T m 27 2 m and 0 02 2 Q7 3 1ng 291 g 291 g 2g 1ng 0971 The fact that X and Z have the same column space shows that this polynomial model is simply a reparameterization of the original treatment effects model That is for every possible 8 there exists a 0 such that X 8 Z 0 and vice versa Using this relationship it can be veri ed that 01 02 4991 are contrasts How ever they are generally not orthogonal You may be familiar with the thin Q R decomposition from matrix analysis There exists an N X 9 matrix Q having orthonormal columns and a g x g upper triangular matrix R such that ZQR Since Z is non singular R is invertible It follows that Q and Z have the same column space and Q can therefore be used to de ne yet another reparameterization of the treatment effects model yQca c C 091 This is the orthogonal polynomial model The parameters in c are related to those in 6 through X l5 Q07 from which it follows that c Q X You may verify from this expression that cl 2 4391 are indeed contrasts and also easily obtain their coefficients w in representation 1 assuming you have access to software that can perform the Q R decomposition Their least squares estimates are the corresponding elements of E Q ero y Q y Therefore columns 2 3 g of Q are the t vectors associated with the contrasts 0102 cgnl respectively and hence 01 02 4391 are orthogonal contrasts since the columns of Q are orthog onal We therefore refer to c1 c2 4391 as orthogonal polynomial contrasts The contrast cl is called the linear orthogonal polynomial contrast 62 is called the quadratic orthogonal polynomial contrast 03 is called the cubic orthogonal polynomial contrast and so forth This terminology is motivated by the relationship QcZ0QRO from which it follows that c R0 and 0 R lc Now R is an upper triangular matrix and it can be shown that R 1 is also upper triangular Therefore for each k O g 7 1 Ckck1cgnl0 0k0k1 0g10 For example if we set 02 091 equal to zero but allow co and cl to vary freely this is equivalent to setting 02 4994 equal to zero that is to reducing the model to a simple linear regression of the response y on 2 In this reduced model7 01 0 if and only if 01 07 that is7 if and only if the slope of the simple linear regression is zero So 01 is associated With testing for a relationship in the simple linear regression model Similarly setting 03 091 equal to zero is equivalent to reducing the model to a quadratic regression of y on 2 Under this model 02 0 if and only if 02 07 that is7 if and only if the quadratic term is not needed So 62 is associated with testing whether the quadratic term is needed in a quadratic model In fact7 it can be shown that the contrast sums of squares for 0102 7391 equal the respective sequential sums of squares for 917 027 7091 in that order Therefore7 orthogonal polynomial contrast sums of squares may be used to select the degree of the polynomial Via sequential statistical tests7 as is common practice in polynomial regression H F 9 7 U H E0 00 4 U 03 Rules for Finding Test Denominators The denominator for testing a term U is the mean square for the leading eligible random term below U in the Hasse diagram An eligible random term V below U is leading if there is no eligible random term that is above V and below U If there are two or more leading eligible random terms then we must use an approximate test Satterthwaite In the unrestricted model all random terms below U are eligible In the restricted model all random terms below U are eligible except those that contain a xed factor not contained in U Rules for Finding Expected Mean Squares The representative element for a random term is its variance component The representative element for a red term is a function Q equal to the sum of the squared effects for the term divided by the degrees of freedom The contribution of a term is the number of data values N divided by the number of effects for that term the superscript for the term in the Hasse diagram times the representative element for that term The expected mean square for a term U is the sum of the contributions for U and all eligible random terms below U In the unrestricted model all random terms below U are eligible In the restricted model all random terms below U are eligible except those that contain a xed factor not contained in U Introduction to Two Series Factorials STA 6208 Course Notes S pri n g 2009 A 2quot factorial treatment structure has k fully crossed fixed treatment factors at 2 levels each Can be used in various designs eg gt balanced CRD need n2k EUs gt RCB design need 2quot EUs per block Notation gt Codes for levels 0 low 1 high gt Labels for factors A B C gt Labels for treatments lowercase letter strings indicating factors at high eg ac only A and C at high Exception 1 all factors at lowquot Motivation Consider an experiment with gt many fixed crossed quantitative or binary factors but probably few of importance gt a limited of EUs available either overall or per block Can afford only 2 levels for each factor quotlowquot and quothighquot Example Chemical Manufacture gt Many process parameters temperatures pressures agitation rates amounts of catalysts reaction time gt Experiments are expensive production overhead costs loss of production time gt can afford few EUs gt Process takes time to run gt few EUs per day Example 23 Factorial 2232 treatment A Bi C 000 1 7 7 7 100 a i i 010 b 7 7 110 ab i 001 C 7 7 101 ac 7 011 bc 7 111 abc These treatments are listed in standard order Factorial Contrasts In a 2k factorial every term main effect or interaction has 1 df A one contrast can represent it Example 23 Factorial Table w Contrast Coefs factorial contrast coefs 1 or 71 treatment A B C AB AC BC ABC 1 7 7 7 7 a 7 7 7 7 b 7 7 7 7 ab 7 7 7 7 c 7 7 7 7 ac 7 7 7 7 bc 7 7 7 abc 5 Note The usual effect estimates are multiples of these factorial contrast estimates Examples Suppose low is level 1 and high is level 2 for each factor gt The usual main effect estimates of A A 1 042 7041 0 gt The usual BC interaction effect estimates k 2 3 A A A A 1 3722 6721 B Yu 6711 eBC And so forth Notation 7mabe average response of EUs receiving trt trt label eg 1717 1737 17m ew estimate of the factorial contrast for term term by substituting 75 for trt means Example 23 Factorial Factorial contrast estimates for A and BC A 70 73 17b Tab Tc T ac 7m 73m BC 71 173 Yb Tab Tc Vac ch 73m Terminology Our textbook adopts the following convention I 27 hem is called the effect of term or the term effectquot Eg ZBC is the BC effectquot 1 F hem is the total effect of term For a main effect term this estimates the averaged mean difference in response between its factor39s high and low levels 1 eg the total effect of A is zkil A 2622 6227621 Sums of Squares Form these as usual for the design eg for a balanced CRD A 1 2 n ssA EWWWO Wi data 5535 2 EC similarly For a RCB design replace n with r blocks Tests As usual eg test for AB interaction using MSAB withCRD 1 and 2kn71 df MSE or RCB 1 and 2k71r71 df FAB SAS Example 106 Pacemaker Substrates Remark If there is only one replication ie n 1 or r 1 then there is no MSE Must use special analysis methods see textbook Sec 1042 Variances and Cls For a balanced CRD 2 7 7 039 VZterm V i trtgt Z Vytrt 2k m at so the variance of the corresponding effect is 1 1 1 k 02 039 V Zterm VZterm 39 2 7 n2k For a RCB design replace n with r Form Cls for EZtem2k as usual eg for a CRD n gt 1 1 term i ta22kn71 MSE n 2k 9 Derivation of the Satterthvvaite Approximation Note This is optional supplementary material which you may nd useful for understanding the Satterthwaite approximation Recall that the chi square distribution with 1 degrees of freedom has mean 1 and variance 21 In ordinary statistical use the degrees of freedom parameter of the chi square is an integer but the chi square family can be naturally extended to non integer degrees of freedom through the more general gamma family of distributions of which the chi square is a sub family This extension preserves the formulas for mean and variance Suppose that M51 MSK are independent and MSk 2 ukizwka k1K 0k for some constants 11 lK positive and usually integers and 0f a Then EMSk ag and VMSk 2 aguk as follows from using the moments ofthe chi square distribution stated above Note that M51 MSK are de ned to behave like mean squares from a balanced random effects model in the sense that they are independent and satisfy lkMSk 2 7 N k 1 K HMS XW 7 7 Now let M 91M51 9KMSK for some constants gl gK Then 4 U4 EM 910f 9K0f and WM 2ltg 31g ylgt 1 K using independence to get the variance We wish to nd an effective degrees of freedom 1 such that 1 M W so that M behaves like a mean square and therefore may be used as the numerator or denomi nator of an approximate F statistic for example We will allow 1 to assume non integer values in the hope of achieving a better approximation The Satterthwaite approximation attempts to choose 1 such that the variance of matches that of the X12 distribution that is vlt VM VZWM 21 has approximately a X12 distribution Note that the mean of already matches that of the X12 distribution by de nition Solving the preceding equation for 1 gives EltMgt2 910 91 1 7 2 7 VM 2 Ti1 2 0k 91 171 9K In practice 0 a are unknown usually linear combinations of variance components so the Satterthwaite approximation proposes to replace them with the unbiased estimates M51 MSK yielding the usual formula for the Satterthwaite degrees of freedom M2 M32 9amp4 VK M32 g 71 11 Remarks 0 The Satterthwaite method is not the only way to approximate 1 Some software packages provide other methods eg options to the SAS PROC MIXED MODEL statement include the KenwaTd Roger method 0 The Satterthwaite approximation tends to be reasonable if 91 gK are positive but not necessarily if some of them are negative 0 lfgl gK are positive it can be shown that the estimated 1 is always at least minz1 1K Therefore if 11 UK are positive integers 1 is at least 1 This is not necessarily true if some of 91 gK are negative