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 Ñœcov( % 3 % 4 Ñif there is an allowable permutation 1 such that 1Ð3 Ñ œ 3 1Ð4 Ñ œ 4. Nelder (1965) showed that Z can be expressed in the following form Z œ 0 W (4.2) 3œ! 3 3 where W! á W are symmetric idempotent matrices such that W3W4 œ 0 for all 3 Á 4 3œ! W3 œ M W! œ K œ R N. The spaces f3 œ eðw3ñ which only depend on the block structure do not depend on the entries of Z are called strata dim Ðf 3 Ñ is called the degree of freedom (d.f.) associated with f3. Note that 00 â 0 are the eigenvalues of Z f! á f are the associated eigenspaces each W 3 is the orthogonal projection matrix onto f. Furthermore V œ Š f. R 3 3œ! 3 Example 1. For a completely romized design for some Z œ Ð+ ÑM N + ; see Example 1 of Hout 1. Then Z œ Ð+ ÑÐM RNÑ RÒ+ ÐR ÑÓN œ Ð+ ÑÐM KÑ Ò+ ÐR ÑÓK. Therefore œ W œk W œm K f œz f œz.!! Example 2. Block designs ÐÎ5ÑÞ In this case Z has three different entries corresponding to the three equivalence classes given in (1.2) of Hout 1 can be written as + M ÐVF MÑ -ÐN VFÑ where VF is as in Hout 2 the three (0 1)-matrices M VF M N VF describe the three kinds of relations between any pair of unit labels induced by º. For example the Ð3ß 4Ñth entry of VF M is 1 if only if 3 Á 4 unit labels 3 4 belong to the same block; the Ð3ß 4Ñth entry of N V F is 1 if only if unit labels 3 4 belong to different blocks. Then Z œ+ M ÐVF MÑ -ÐN VFÑ œ+ M Ð5F MÑ -ÐRK 5F Ñ (By (2.2) Hout 2) 1-1-
œð+ ÑÐM FÑ Ò+ Ð5 Ñ -5ÓÐF KÑ Ò+ Ð5 Ñ -5 -RÓK œ 0 W 0 W 0 W!! where W œ K W œ F K W œ M F f œ Z f œ U Z f œ U.!! Example 3. Similarly for < - Z œ 0! W 0 W 0 W 0$ W$ where W! œ K W œ V K W œ G K W$ œ ÐM V G KÑ f! œ Z f œ e Z f œ V Z f œ ( e V). $ The observation vector following decomposition Then we have C can be projected onto different strata resulting in the C KC œ W C 3œ 3. lc KCl œ lw3 Cl. (4.3) Suppose α œâ œα> (in which case lc KCl does not contain the treatment effects therefore measures the variability among the experimental units). Then for 3œ1 â w 3œ E( lwc 3 l ) œ E( CWC 3 ) w œ E( C) W E( C) trðw Z Ñ 3 3 œ tròw Ð 0 W ÑÓ 3 4 4 4œ! œ trð0 3 W 3 Ñ œ 0 dim Ðf Ñ. 3 3 The second equality holds since when α œâ œα> E ÐCÑ Z is orthogonal to f3 for all 3œ1 â s. Then E Ð.37Ð f 3 Ñ lwc 3 l Ñ œ 03. (4.4) This leads to the following null ANOVA table: -2-
Sources of variation Sums of Squares degrees of freedom Mean square E(MS) f lwcl dim Ðf Ñ.37Ðf Ñ lwcl 0 ã ã ã ã ã f lwcl dim Ðf Ñ.37Ðf Ñ lwcl 0 Total lc KCl R 1 Example 1 continued. In a completely romized design there is only one stratum other than R Z: f œ Z. When α œ â œ α> E ÐR lc KC l Ñ œ E Ò R 3œ ÐC3 CÑ. Ó œ 0. In this case measures the overall variability among the units. 0 Example 2 continued. In the block structure Î5 there are two strata other than Z: f œ U Z f œ U with dimensions 1 Ð5 Ñ respectively. The sums of squares lwc l œ lðf KC Ñ l lwc l œ lðm FC Ñ l can be computed by using the formulas given in Hout 2. For convenience index the 4th unit in the 3th block by double subscripts Ð3ß 4Ñ. Then 3. 3. 3œ 3œ lwcl œ 5ÐC C Ñ œ 5ÐC Ñ 5ÐC Ñ 5 lwcl œ ÐC C Ñ. 34 3. 3œ 4œ Therefore when α œâ œα > 0 3. 3œ œ E Ò 5ÐC C Ñ Ó 5 0 Ð5 Ñ 34 3 3œ 4œ œ E Ò ÐC C. Ñ Ó. Thus 0 is the between-block variance 0 is the within-block variance. In a successful blocking 0 is less than 0. The two strata f f are called interblock intrablock strata respectively. We have the following null ANOVA table for a block design: -3-
Sources of variation Sums of Squares d.f. Mean square E(MS) 3. 3. 0 3œ 3œ interblock 5ÐC C Ñ 5ÐC C Ñ 5 5 34 3. Ð5 Ñ 34 3. 0 3œ 4œ 3œ 4œ intrablock ÐC C Ñ Ð5 Ñ ÐC C Ñ Total 5 ÐC C Ñ 5 3œ 4œ 34 Example 3 continued. In the block structure < - there are three strata other than Z: f œ e Z f œ V Z f$ œ ( e V) with dimensions < 1-1 Ð< ÑÐ- Ñ respectively. For convenience index the unit in the 3th row 4th column by double subscripts Ð3ß 4Ñ. Then < < 3. 3. 3œ 3œ lwcl œ -ÐC C Ñ œ -ÐC Ñ <-ÐC Ñ - -. 4. 4 4œ 4œ lwcl œ <ÐC C Ñ œ <ÐC Ñ <-ÐC Ñ < - lwcl œ ÐC C C C Ñ. $ 34 3.. 4 3œ 4œ Therefore when α œâ œα > < 0 < 3. 3œ œ E Ò -ÐC C Ñ Ó - 0 -. 4 4œ œ E Ò <ÐC C Ñ Ó < - 0 $ Ð< ÑÐ- Ñ 34 3.. 4 3œ 4œ œe Ò ÐC C C C Ñ Ó. -4-
Thus 0 is the between-row variance 0 is the between-column variance 0$ is the betweenunit variance with row-to-row column-to-column variations eliminated. We have the following null ANOVA table for a row-column design: Sources of variation Sums of Squares d.f. Mean square E(MS) < < 3. < 3. 0 3œ 3œ rows -ÐC C Ñ < 1 -ÐC C Ñ - -. 4 -. 4 0 4œ 4œ columns <ÐC C Ñ - 1 <ÐC C Ñ < - < - 34 3.. 4 Ð< -ÑÐ--Ñ 34 3.. 4 0$ 3œ 4œ 3œ 4œ units ÐC C C C Ñ Ð< ÑÐ- Ñ ÐC C C C Ñ < - Total ÐC C Ñ <- 1 3œ 4œ 34 Nelder (1965) gave simple rules for determining the degrees of freedom projections onto the strata. The first step is to determine how the total degrees of freedom are split up. For the block structure 8Î8 we have the following d.f. identity: 88 œ1 / 8/ where / 3 œ8 3 1. The three terms on the right-h side specify the degrees of freedom of the three strata. The d.f. identity for the block structure 8 8 is 8 8 œ 1 / / / /. We define the nesting crossing functions as a Ð8 8 Ñ œ 1 / 8 / (4.5) V Ð8 8 Ñ œ 1 / / / / (4.6) respectively. The arguments in the a V functions may be substituted by other a V functions. The d.f. identities for other more complex block structures can be obtained from the block structure formulas by exping the corresponding a V functions. For example the d.f. identity for 8Î8Î8 $ can be obtained by exping aðað8 8Ñ 8Ñ $. Generally terms cannot be destroyed by algebraic manipulation (e.g. 1 / cannot be replaced by 8) except that like terms may be subtracted to become zero which is deleted -5-
that any unity appearing in a product is suppressed (1 BœB). Another important rule is that the 8 term that appears in the right-h side of (4.5) must be the algebraic sum of all the terms in the expansion of 8. It is useful to note that the crossing function in (4.6) can also be expressed as VÐ8 8ÑœaÐ1 8ÑaÐ1 8Ñ. (4.7) Example 4. To determine the d.f identity for 8Î8Î8 we exp $ aðað8 8 Ñ 8 Ñœ1 Ða Ð8 8 Ñ 1Ñ 8 8 / $ $ œ1 / 8 / 8 8 /. $ So there are four strata with degrees of freedom 1 8 1 8 Ð8 1 Ñ 8 8 Ð8$ 1 Ñ. Example 5. The d.f identity of 8 Ð8 Î8 Ñ $ follows from the expansion of VÐ8 að8 8 ÑÑ œ að1 8 ÑaÐ1 að8 8 ÑÑ (by (4.7)) $ $ œ Ð1 / ÑÒ1 að8 8 Ñ 1Ó $ œð1 / ÑÒ1 / 8/ Ó $ œ 1 / / / / 8 / / 8 /. (4.8) $ $ From the d.f. identity we can write down a yield identity which gives projections to all the strata. For convenience we index each unit by multi-subscripts as before dot notation is used for averaging. The following is the rule given by Nelder: exp each term in the d.f. identity as a function of the n 's; then to each term in the expansion corresponds a 3 mean of the y's with the same sign averaged over the subscripts for which the corresponding n 's are absent. 3 For example from (4.8) we obtain 8 8 8$ œ 1 Ð8 1Ñ Ð8 1Ñ Ð8 8 8 8 1Ñ Ð8 8$ 8 Ñ Ð8 8 8$ 8 8 8 8$ 8 Ñ. This gives the following yield identity: C345 œc. ÐC3 C. Ñ ÐC. 4. C. Ñ ÐC34. C3 C. 4. C. Ñ ÐC. 45 C. 4. Ñ ÐC345 C34. C. 45 C. 4. Ñ. Therefore for the block structure 8 Ð8 Î8$ Ñ the strata other than Z have degrees of freedom equal to 8 1 8 1 8 8 8 8 1 8 8$ 8 8 8 8$ 8 8 8 8$ 8. The corresponding sums of squares in the null ANOVA are 8 8 8 8 88ÐC C Ñ 88ÐC C Ñ 8ÐC C C C Ñ 3œ $ 3. 4œ $. 4.. 3œ 4œ $ 34. 3. 4.. -6-
8 8$ 8 8 8$ 8 ÐC C Ñ ÐC C C C Ñ. 4œ 5œ. 45. 4. 3œ 4œ 5œ 345 34.. 45. 4. -7-