Ebook Design and analysis of experiments (Volume 2: Advanced experimental design): Part 2

We can no longer talk about the main effect of a factor or the interaction between two or more factors but shall talk instead about main effect components or comparisonsbelonging to a ce

10.1 INTRODUCTION

In discussing the 2nfactorial design in Chapter 7 we saw that main effects andinteractions can be defined simply as linear combinations of the true responses,more specifically as the average response of one set of 2n−1 treatment combi-nations minus the average response of the complementary set of 2n−1treatmentcombinations And even more specifically, the main effect of a certain factor isthe average response with that factor at the 1 level minus the average responsewith that factor at the 0 level Turning now to the situation where each factorhas three levels, which we shall refer to as 0 level, 1 level, and 2 level, such asimple definition of main effects and interactions no longer exists We can no

longer talk about the main effect of a factor or the interaction between two or

more factors but shall talk instead about main effect components or comparisonsbelonging to a certain factor and about interaction components We shall see howall this can be developed as a generalization of the formal approach describedfor the 2nexperiment in Section 7.4

10.2.1 The 3 2 Case

To introduce the concepts we shall consider first the simplest case, namely that

of two factors, A and B say, each having three levels, denoted by 0, 1, 2 A

treatment combination of this 32 factorial is then represented by x
= (x1, x2)

where x i = 0, 1, 2(i = 1, 2), with x1 referring to factor A and x2to factor B.

Design and Analysis of Experiments Volume 2: Advanced Experimental Design

By Klaus Hinkelmann and Oscar Kempthorne

ISBN 0-471-55177-5 Copyright  2005 John Wiley & Sons, Inc.

359

More formally, we can define these three sets by the three equations:

set I: x1= 0set II: x1= 1set III: x1= 2

(10.1)

Comparisons among the mean true responses for these three sets are then said

to belong to main effect A Since there are three sets, there are two linearly

independent comparisons among these three sets (i.e., their mean responses),

and these comparisons represent the 2 d.f for main effect A For example, the

comparisons could be (set I − set II) and (set I − III), or (set I − set II) and(set I+ set II − 2 set III)

Similarly, we can divide the nine treatment combinations into three sets

corre-sponding to the levels of factor B or, equivalently, correcorre-sponding to the equations:

As in the 2n case, the interaction between factors A and B will be defined in

terms of comparisons of sets (of treatment combinations), which are determined

by equations involving both x1 and x2 One such partitioning is given by

set I: x1+ x2= 0 mod 3: {(0, 0), (1, 2), (2, 1)}

set II: x1+ x2= 1 mod 3: {(1, 0), (0, 1), (2, 2)}

set III: x1+ x2= 2 mod 3: {(2, 0), (0, 2), (1, 1)}

(10.3)

Comparisons among these three sets account for 2 of the 4 d.f for the A × B

interaction The remaining 2 d.f are accounted for by comparisons among thesets based on the following partition:

set I: x1+ 2x2= 0 mod 3: {(0, 0), (1, 1), (2, 2)}

set II: x1+ 2x2= 1 mod 3: {(1, 0), (0, 2), (2, 1)}

set III: x1+ 2x2= 2 mod 3: {(2, 0), (0, 1), (1, 2)}

(10.4)

for each i, where τ ij is the true response for the treatment combination (x1=

i, x2= j) With the model (10.5), a contrast among sets (10.1), that is,

c1(τ00+ τ11+ τ22) + c2(τ10+ τ02+ τ21) + c3(τ20+ τ01+ τ12)

(c1+ c2+ c3 = 0)

The reader will notice that the last two comparisons have no particular meaning

or interpretation for any choice of the c i ’s, except that they each belong to the

2-factor interaction A × B, and that each represents 2 d.f of that interaction This is

in contrast to the parameterization given in Section I.11.8.1 in terms of orthogonal

To sum up our discussion so far, the effects and interactions for a 3 experimentare given in pairs of degrees of freedom by comparisons among three sets oftreatment combinations as follows:

It is convenient to denote the pairs of degrees of freedom corresponding to

x1+ x2= 0, 1, 2 by the symbol AB and the pair corresponding to x1+ 2x2=

0, 1, 2 by AB2

It is easy to see that the groups given by the symbols AB2 and A2B arethe same It is, therefore, convenient, in order to obtain a complete and uniqueenumeration of the pairs of degrees of freedom, to adopt the rule that an order

of the letters is to be chosen in advance and that the power of the first letter in asymbol must be unity This latter is obtained by taking the square of the symbolwith the rule that the cube of any letter is to be replaced by unity, that is, if the

initial letter of the symbol occurs raised to the power 2, for example, A2B, wethen obtain

A2B ≡ (A2B)2 ≡ A4B2≡ AB2This procedure follows from the fact that the partitioning produced by

symbols, each representing 2 d.f For example, for the 33 experiment there will

be 13 symbols as given in Table 10.1 together with their defining equations of

the form α1x1+ α2x2+ α3x3= 0, 1, 2 mod 3.

For the general case of the 3n experiment, denoting the factors by

A1, A2, , A n, the (3 n − 1)/2 symbols can be written as A α1

1 , A α2

2 , , A α n

n with α i = 0, 1, 2 (i = 1, 2, , n) and the convention that (1) any letter Ai with

α i = 0 is dropped from the expression, (2) the first nonzero α is equal to one (this can always be achieved by multiplying each α i by 2), and (3) any α i = 1

is not written explicitly in the expression (This is illustrated in Table 10.1 by

α1x1+ α2x2+ · · · + αn x n = δ1 mod 3 (10.8)and

β1x1+ β2x2+ · · · + βn x n = δ2 mod 3 (10.9)

are satisfied by exactly 3n−2treatment combinations x
= (x1, x2, , x n )

This implies that the set of treatment combinations determined by α
x = δ1has exactly 3n−2 treatment combinations in common with each of the

three sets determined by the equations β
x = 0, 1, 2 mod 3, respectively.

It is in this sense that the two partitions α and β are orthogonal to

each other

3 If a treatment combination x
= (x1, x2, , x n ) satisfies both Eqs

(10.8) and (10.9) for a particular choice of δ1, δ2, then x also satisfies

the equation

(α1+ β1)x1+ (α2+ β2)x2+ · · · + (αn + βn )x n = δ1+ δ2 mod 3

(10.10)Equation (10.10) is, of course, one of the three equations associated

with the partition α
+ β
= (α1+ β1, α2+ β2, , α n + βn ), in whicheach component is reduced mod 3, and hence with the interaction

In addition to satisfying (10.10), the treatment combination x, which

sat-isfies (10.8) and (10.9), also satsat-isfies the equation

n This interaction is therefore another

GI of E α and E β To summarize then, any two interactions E α and E β

have two GIs E α +β and E α +2β , where α + β and α + 2β are formed

mod 3 and are subject to the rules stated earlier We illustrate this by thefollowing example

and hence the GIs of AB and ABC2are ABC and C Another way of obtaining

this result is through formal multiplication and reduction mod 3, that is,

of 3n treatment combinations into three sets of 3n− 1 treatment combinationseach The symbol, with a subscript that is the right-hand side of the equationdetermining the particular one of the three sets in which the treatment combina-tions lie, will denote the mean response of that set as a deviation from the overall

satisfying α
x = i mod 3



− M (10.12)

We shall also use the notation E α α
x for given α and x to denote one of the

quantities E0α , E1α , E α2 depending on whether α
x = 0, 1, 2 mod 3, respectively.

We note that a comparison belonging to E α is, of course, given by

c0E0α + c1E1α + c2E2α (c0+ c1+ c2= 0) (10.13)Also, it follows from (10.12) that

E α0 + E α

1 + E α

so that any comparison of the form (10.13) could be expressed in terms of only

two E i α Such a procedure was, in fact, adopted for the 2n factorial, but as we

where summation is over all α
= (α1, α2, , α n ) = (0, 0, , 0), subject to the rule that the first nonzero α i equals one, and α
x is reduced mod 3 Theproof of (10.15) follows that of (7.42) and will be given for the general case inSection 11.5

We illustrate (10.15) with the following example

Example 10.2 Consider the 33 factorial with factors A, B, C and denote the true response of the treatment combination (i, j, k) by a i b j c k Then (10.15)can be written as

a i b j c k = M + Ai + Bj + ABi +j + AB2

i +2j + Ck + ACi +k + AC2

i +2k + BCj +k + BC2

j +2k + ABCi +j+k + ABC2

i +j+2k + AB2C i +2j+k + AB2C i2+2j+2k For i = 1, j = 1, k = 2, for example, this becomes

We emphasize again that the parameterization (10.15), which because

of (10.14) is a non-full-rank parameterization, becomes important in tion with systems of confounding (Section 10.5) and fractional factorials(Section 13.4)

Suppose that each treatment combination is replicated r times in an appropriate

error control design, such as a CRD or a RCBD Comparisons of treatments arethen achieved by simply comparing the observed treatment means, and tests formain effects and interactions are done in an appropriate ANOVA

ij k

y ij k − y i··· − y ·jk + y···· 2

For purposes of illustration we consider a 33 experiment in an RCBD with r

blocks With the usual model

y ij k = µ + βi + τj k + eij k

or

y = µI + Xβ β + Xτ τ + e

where (j k) denotes the level combinations for the three factors A, B, C, and

with the factorial structure of the treatments, we obtain the usual ANOVA given

in Table 10.2

An alternative way of computing the various components of the treatment sum

of squares is based upon the definition of the 2-d.f components of any interaction

and the corresponding symbols defined in Section 10.3 Let E α denote any such

interaction component, such as AB or AB2C, and let E0α, E1α, E α2 be the mean

AB2C 2 9r

[
AB2C0]2+ [
AB2C1]2+ [
AB2C2]2

AB2C2 2 9r

 [
AB2C2]2+ [
AB2C2]2+ [
AB2C2]2



observed responses (as a deviation from the overall mean) of the three sets

defining E α The sum of squares associated with E α, accounting for 2 d.f., isthen given by

SS(E α ) = r3 n−1

E α02+E α12

+E α22

(10.16a)with

+E α22

(10.16b)Specifically, for the 33experiment the sum of squares due to 3-factor interaction,for example, can be broken down as given in Table 10.3 The usefulness of thisprocedure will become apparent when we consider systems of confounding inSection 10.5 The SS given in (10.16a) is simply the SS (among sets) for the

sets defined by α
x = 0, 1, 2 Since the various partitions are orthogonal to each

other, so are their associated SSs It is not difficult to show that the sum of the

four SSs in Table 10.3 is the same as SS(A × B × C) in Table 10.2.

Generally, it is also useful to list the quantities E0α ,  E1α ,  E α2 for main effects andinteractions as they can be used to estimate the yield of any treatment combination

or comparisons among treatment combinations (see Section 10.3)

x1+ x2= 0, 1, 2 mod 3 and that AC2 is similarly represented by the equations

x1+ 2x3= 0, 1, 2 mod 3

It is obvious then that considering jointly

x1+ x2= k mod 3

for all possible combinations of k, , = 0, 1, 2, we partition the 27 treatment

combinations into 9 sets of 3 treatment combinations each, these sets being theblocks for the desired system of confounding Now, any treatment combination

which satisfies (10.17) for given (k, ), also satisfies

2x1+ x2+ 2x3= k +  mod 3

or, equivalently,

x1+ 2x2+ x3= 2(k + ) mod 3 and since 2(k + ) ≡ 0, 1, 2 mod 3 it follows that there are three sets of three

blocks which satisfy the equation

x1+ 2x2+ x3= 0, 1, 2 mod 3

respectively Comparisons among these sets, however, define the interaction

com-ponent AB2C Hence AB2Cis also confounded with blocks, and we recognize

immediately that AB2C is a GI of AB and AC2; that is,

(AB) × (AC2) = A2BC2= A4B2C4= AB2C

Similarly, any treatment combination that satisfies (10.17) also satisfies theequation

(x1+ x2) + 2(x1+ 2x3) = k + 2 mod 3

x2+ x3= k + 2 mod 3

Hence the other GI

(AB)(AC2)2= A3BC4= BC

is also confounded with blocks These four interactions, AB, AC2, AB2C, and

BC, then account for the 8 d.f for comparisons among blocks

The composition of the blocks for the above system of confounding can be

obtained from Eqs (10.17) with (k, ) assuming all possible values Alternatively,

we can construct first the intrablock subgroup (IBSG) from

x1+ x2= 0 mod 3

x1+ 2x3= 0 mod 3

and then, using the x representation for treatment combinations, add

(componen-twise and mod 3) a treatment combination, not already contained in the IBSG, toeach element in the IBSG This process is continued as described in Section 8.3,until all blocks have been constructed in this manner as given in Table 10.4

A similar design can be obtained by using SAS PROC FACTEX and is given inTable 10.5

We shall comment briefly here on some aspects of the SAS output and how

it relates to our discussion in this chapter:

1 We note that rather than using 0, 1, 2 for the factor levels, SAS uses−1,

0, 1, respectively, as commonly used in response surface and regressionmethodology

2 The single degree of freedom associated with the main effects and

interac-tions are listed formally akin to the linear-quadratic effects representationfor quantitative factors (see I.11.8.1), for example,

A −→ A linear

2∗ A −→ A quadratic

We should point out, however, that these representations are not identical

[see also (4) below]

3 The confounding rules are essentially the same as those we have explained

earlier In this example the block compositions are obtained by satisfyingthe equations

2∗ A + 2 ∗ B + 2 ∗ C = δ1

B + 2 ∗ C = δ2

for some δ1, δ2( = 0, 1, 2 mod 3), where A, B, and C are the levels of those

factors In our notation this is equivalent to satisfying the equations

2x1+ 2x2+ 2x3= γ1

x2+ 2x3= γ2or

x1+ x2+ x3= γ∗

1

x2+ 2x3= γ2

Hence, in this example we confound ABC and BC2and, hence, AB2and

AC2 with blocks We only need to remember that −1 ≡ 2 mod 3 and

−2 ≡ 1 mod 3

4 The aliasing structure gives a list of the main effects and 2-factor

interac-tions that are either estimable or confounded with blocks, the latter being

identified by [B] More precisely, we should really say that the aliasing

structure represents a list of the number of degrees of freedom ated with estimable and confounded effects, respectively For example,

associ-the output identifies A + B and 2 ∗ A + 2 ∗ B as estimable This does not mean, however, that A linear ×B linear, or A quadratic ×B quadratic are

estimable since there is no relationship between these components and the

2-d.f component AB.

2 If two interactions E and E are confounded with blocks, then their GIs

E α +β and E α +2β are also confounded with blocks.

3 To find a system of confounding, one needs to specify only q = n − p independent main effects and/or interactions E α1, E α2, , E α q since the



q

4

+ · · · + 2q−1

q q

with blocks

4 The composition of the blocks is obtained by means of the IBSG, which

is composed of the treatment combinations satisfying the equations

α j x1+ αj x2+ · · · + αj n x n= 0

(j = 1, 2, , q = n − p) as determined by the independent confounded interactions E α1, E α2, , E α q in (3) The remaining blocks are thenobtained as described in Section 10.5.1

As we have mentioned earlier, the number of treatment combinations is quitelarge even for a moderate number of factors This would call in most cases forincomplete blocks and hence for a system of confounding But even this maylead to certain difficulties since at this time we are only considering blocks thesize of which is a power of 3, so that the choice is quite limited (For otherblock sizes we refer to Section 11.14.4.) To complicate matters, according toFisher’s (1942, 1945) theorem (see Section 11.7) confounding of main effectsand/or 2-factor interactions can be avoided only if the block size is larger than

twice the number of factors, that is, k > 2n For purposes of reference we list

in Table 10.6 possible types of confounding involving up to five factors andvarious block sizes Further systems can be obtained from this list by permutingthe letters

, AB2C2, BC2D2

27 Any main effect or interaction

5 9 BE∗, ABC∗, AB2CE, ACE2, CDE∗

BCDE2, BC2D2, ABC2DE, ABD2E2

AB2C2DE2, AB2D2, AC2D, AD2E

27 ABC∗, AB2DE∗, AC2D2E2, BC2DE

aEffects with an asterisk ( ∗) are the independent effects.

Generally not all components of a particular interaction are confounded withblocks and, hence, limited intrablock information on that interaction is still avail-able (see also Sections 10.7.2 and 10.7.3) Even so, in most practical cases it will

be useful to resort to partial confounding These can be obtained easily from thesystems provided in Table 10.6 In the following we shall comment briefly onsome such systems

in blocks and the structure of the analysis of variance with q repetitions of the

basic pattern are given in Table 10.7

10.6.2 Three Factors

1 Blocks of Size 3 Consulting Table 10.6 suggests that a suitable system ofconfounding consists of a basic pattern of four replicates of the followingtypes:

Type I confound AB, AC, BC2, AB2C2

Type II confound AB, AC2, BC, AB2C

Type III confound AB2, AC, BC, ABC2

Type IV confound AB2, AC2, BC2, ABC

This will yield full information on main effects, 12 relative information on2-factor interactions, and 34 relative information on 3-factor interactions

2 Blocks of Size 9 The most useful design consists of one or more repetitions

of a basic pattern of four replicates confounding ABC, ABC2, AB2C,

and AB2C2, respectively This will result in full information on all maineffects and 2-factor interactions and 34 relative information on the 3-factorinteraction components The arrangement of the blocks is given in Table

10.8, each column (level of C) combined with the levels of A and B giving

a block and each set of three columns a replicate

10.6.3 Treatment Comparisons

At this point we comment briefly on evaluating the amount of information onany treatment comparison provided by the confounded design relative to theunconfounded design To do so we make use of the fact that the yield of atreatment combination can be represented in terms of main effect and interac-tion components (see Section 10.3) For purposes of illustration suppose that we

are interested in the comparison (a0b0c0− a0b0c1) , using q repetitions of the basic pattern given in Table 10.8 Notice that a0b0c0 and a0b0c1 never occurtogether in the same block, so that a simple comparison among their mean yields

+BC20− BC22

+ABC0−ABC1



+ABC20−ABC22

+AB
2C0−AB
2C1

Each quantity in parentheses is statistically independent of the others (because

of the orthogonality of the partitions), with a variance depending on the tem of confounding used In the present case any difference among main effect

sys-components (like C0− C1) and among 2-factor interaction components (like

AC0− AC1) is estimated with variance

is estimated with variance

62 =31.

10.6.4 Four Factors

1 Blocks of Size 3 A suitable system consists of a basic pattern of fourreplicates, using the four systems of confounding given in Table 10.6.This design will result in 34 information on main effects and all 2-factorinteraction components

2 Blocks of Size 9 The most useful type of confounding is obviously thelast one given in Table 10.6 since it confounds only 3-factor interactioncomponents, one from each type of 3-factor interaction Altogether thereexist eight such systems of confounding

3 Blocks of Size 27 In general, the experimenter will wish to avoid blocks ofsize as large as 27, though, in some fields of experimentation and with sometypes of experimental material, the effect on error variance of reducingblock size from 27 to 9 may be so small as not to offset any loss inrelative information that results from confounding If blocks of size 27are being used, any of the eight 4-factor interaction components may beconfounded

10.6.5 Five Factors

1 Blocks of Size 9 It is not possible to avoid confounding a main effect or factor interactions Under these circumstances, the system of confoundinggiven in Table 10.6 and permutation of that set would be most useful

2-2 Blocks of Size 27 One can show that it is not possible to find a designconfounding only 4- and 5-factor interactions The most useful system ofconfounding is then one of the form given in Table 10.6

10.6.6 Double Confounding

Occasionally it may be desirable to impose a double restriction on the pattern of

a 3nexperiment This leads to systems of double confounding Suitable systemscan be found by consulting Table 10.6 The actual arrangement of the treatmentcombinations can be obtained by first constructing the IBSGs for confoundingwith “rows” and “columns,” respectively, and then adding the elements of the firstrow and first column termwise mod 3 As an example consider the 33experiment

x1+ x2= 0 mod 3

x1+ x3= 0 mod 3The final arrangement (apart from randomization of rows and columns) then is

In the course of our discussion of 3n factorial experiments we have alreadycommented on some aspects of the analysis for particular situations We shallnow make some remarks about the general 3n factorial experiment in blocks

of size 3p using some system of complete or partial confounding Since thedevelopment parallels that of Section 9.7, we shall not repeat all the details, butonly those that are specific to the 3ncase

The basic model underlying the analysis is as before:

y = µI + Xρ ρ + Xβ∗β∗+ Xτ τ + e where ρ represents the replicate effects, β∗ the block within replicate effects,

and τ the treatment effects, or in its reparameterized form

y = µI + Xρ ρ + Xβ∗β∗+ Xτ∗τ ∗+ e where τ∗ represents the interaction components E α

i for all admissible α
=

(α1, α2, , α n ) and i = 0, 1, 2 A basic pattern of partial confounding consists

of s types of replicates, each replicate consisting of 3 n −p blocks, and the basicpattern is repeated q times.

and E γ ∈ E2 is confounded in c(γ  ) replicates and not confounded in u(γ  )

replicates of the basic pattern We denote by N= 3n sq the total number ofobservations

The following statements concerning the analysis are then obvious extensions

of those made in Section 9.7

1

2+Eδ m

1

2+E γ 

the corresponding interaction is confounded SS(E∗

ij) is obtained only from the

j th replicate in the ith repetition, and 

ij and 

ij denote summation over

The B|T-ANOVA in its basic form is given in Table 10.10a with a partitioning

of the block sum of squares in Table 10.10b The only sum of squares that needs

to be explained is that associated withE γ  vs E γ 

More specifically this sum

of squares is associated with the comparisons

where V γ  is the variance–covariance matrix of

which is a sum of squares with 2n2 d.f and which depends only on block effectsand error

by using, in the T|B-ANOVA, the F tests as approximations to the randomization

tests (see, e.g., I.6.6 and I.9.2):

total interaction among s factors has 2 s d.f and hence has 2s− 1 components

of the form E α For example, for s = 3 and factors A, B, C, the components are ABC, ABC2, AB2C , AB2C2 To test then the hypothesis that there is no

A × B × C interaction we use the F test:

with 8 and ν R d.f In this case an F test of the form (10.25) or (10.26) may not

tell us very much about the 3-factor interaction, except that when one or the other

of those tests is significant then we can conclude that A × B × C interaction is

possibly present

squares associated with the components inE2 and/orE3 Great care has to

be exercised in interpreting such a test

3 All components belong to E1 In this case no test exists in the context ofthe T|B-ANOVA (but see Section 10.7.4)

h (h = 0, 1, 2) obtained from those replicates

in which these interactions are confounded We then have

For the practical use of (10.30) and (10.31) in connection with, for example,

the combined estimate of a(x) − a(z) for two treatment combinations x
and z
,

using (10.15), we usually need to estimate w and w
(or ρ) As always

σ e2= 1



w = MS(I|I, Xρ , X β∗, Xτ∗) (10.32)which is obtained from the T|B-ANOVA of Table 10.10

For the estimation of w
by the Yates procedure we use the B

|T-ANOVA of Table 10.9 The two components of SS(X β∗|I, Xρ , X τ∗) areSS(Remainder) as defined in Section 10.7.4 above and SS E γ  vs Eγ 

There are, of course, other methods of estimating the weights as described

in Section 1.11 In any case, Satterthwaite’s procedure (Satterthwaite, 1946; seealso I.9.7.7) must be used to obtain the degrees of freedom associated with theestimator given in (10.37)

We consider the 32 factorial in blocks of size 3, confounding the 2-factor

inter-action component AB with blocks The data as well as the intrablock analysis

(using SAS PROC GLM) and combined intra- and interblock analysis (usingSAS PROC MIXED) are given in Table 10.11 We shall comment briefly onthese analyses

10.8.1 Intrablock Analysis

We findσ2= MS(I|I, X ρ , X β∗, X τ∗) = MS(E) = 8333 MS(E) (with 6 d.f.)

is used to test hypotheses about A, B, and A × B by forming F ratios with the respective type III mean squares in the numerator Concerning the A × B

interaction, we know, of course, that it has only 2 d.f., which are associated with

the interaction component AB2

10.8.2 Combined Analysis

In this example AB is confounded in both replicates, that is, AB belongs toE1,

whereas A, B, and AB2 belong to E3 This means that only interblock

infor-mation is available for AB and only intrablock inforinfor-mation is available for A,

B , and AB2 Thus, the combined analysis for A and B yields the same results

as the intrablock analysis We can verify this easily by comparing the F ratios and P values using the type III MS in PROC GLM and PROC MIXED, respec- tively In both cases we find for A: F = 126.87, P = 0001, and for B: F = 2.07,

P = 2076.

To test the hypothesis that there is no A × B interaction we obtain

options nodate pageno=1;

proc print data=three;

title1 'TABLE 10.11';

title2 '3**2 FACTORIAL IN BLOCKS OF SIZE 3';

title3 '(WITH AB COMPLETELY CONFOUNDED)';

run;

proc glm data=three;

class rep block A B;

model y=rep A|B block(rep);

title3 'INTRA-BLOCK ANALYSIS';

run;

proc mixed data=three;

class rep block A B;

model y=rep A|B/ddfm=satterth;

Class Levels Values

Dependent Variable y

Class Level Information

Type 3 Tests of Fixed Effects

$(10.41)

Under H0(10.41) reduces to σ2+3

2σ β2 We therefore need to obtain an estimator

for this quantity that will then provide the denominator for the F ratio to test

H0 Using (10.37) and (10.32) it is easy to see that

using results from the ANOVA table Finally,

F = MS(A × B)

σ2+3

2σ β2 =116.06

7.95 = 14.60

This value is comparable to the corresponding value 13.03 obtained with PROC

MIXED, which uses REML to estimate the variance components σ2 and σ β2

The degrees of freedom for the denominator of the F ratio are obtained by

= 2.22

which is comparable to Den DF= 2 in the PROC MIXED output

11.1 INTRODUCTION

In the preceding chapters we have discussed at great length the design and sis of factorial experiments with two and three qualitative levels In both cases thedevelopment is based upon orthogonal partitions of the complete set of treatmentcombinations and comparisons among the resulting subsets These partitions arebased on solving certain equations or sets of equations using only elements fromthe (mathematical) field of residue classes mod 2 and mod 3, respectively andelementary facts about ordinary arithmetic mod 2 and mod 3, respectively Thequestion then is whether this can be extended

analy-Consider, say, arithmetic mod 4 Addition with an additive identity and tiplication with a multiplicative identity are easily defined, for example,

mul-2+ 3 = 1and

2× 3 = 2However, this is not enough We want to set up families of hyperplanes defined

by, for example,

x1+ 2x2= 0, 1, 2, 3

and in order to achieve orthogonal partitions of the 4n treatment combinations,

we need the result that the equation in the unknown x

ax = b (a
= 0)

Design and Analysis of Experiments Volume 2: Advanced Experimental Design

By Klaus Hinkelmann and Oscar Kempthorne

ISBN 0-471-55177-5 Copyright  2005 John Wiley & Sons, Inc.

393

2x= 1

is not satisfied by any x So we see that the simple arithmetic mod k does not extend immediately to any k > 3 It does so, however, if k is a prime number,

psay

The addition and multiplication properties mod p of the set S = {0, 1, 2, ,

p− 1} are obvious It only remains to show that

GF(p m )(see Appendix A)

In this chapter we shall discuss the p n factorial as a generalization of the

2n and 3n factorials and indicate how this, in turn, can be generalized to the

s n = (p m ) n factorial Much of the development in this chapter is due to Bose(1947b), Bose and Kishen (1940), Fisher (1942, 1945), Kempthorne (1947, 1952),Rao (1946a, 1947b), and Yates (1937b)

combinations each through the equations

α1x1+ α2x2+ · · · + αn x n = δ (11.1)

where δ takes on all values in GF(p) That this is true can be seen as follows:

Suppose for fixed α and δ we have αi
= 0 for some i We can then choose all

Because of uniqueness of division in GF(p), x i is then determined uniquely

Since each x j (j
= i) can take on p different values x∗

j , we have p n−1solutions

to (11.1) Comparisons among the p sets of treatment combinations generated

by (11.1) define the p − 1 d.f associated with the effect or interaction E α =

A α1

1 A α2

2 · · · A α n

n

Let S(x ; α, δ) denote the set of treatment combinations x= (x1, x2, , x n )

satisfying (11.1) Then a contrast belonging to E α can be defined formally as

(p − 1) partitions account for the p n− 1 d.f among treatments In order to have

a unique enumeration, we restrict the first nonzero α i in a partition α to be equal

to 1 The partitions then define the main effects, 2-factor interactions, , n-factor interaction, designated in general by E α = A α1

1 A α2

2 · · · A α n

n with the convention

that a letter Ai with αi = 0 is dropped from the expression

As a consequence of partitioning the totality of p n − 1 d.f into (p n − 1)/

(p − 1) sets of p − 1 d.f each, a k-factor interaction, for example, A1×

A2× · · · × Ak , consists of (p − 1) k−1 components denoted by, for example,

A1A α2

2 · · · A α k

k , where α2, α3, , α k take on all nonzero values in GF(p) As

an example, in a 5n design the 2-factor interaction A × B consists of AB, AB2,

AB3, AB4

We also note that for two distinct partitions α= (α1, α2, , α n ) and β=

(β1, β2, , β n )the equations

α1x1+ α2x2+ · · · + αn x n = δ1 (11.2)

These equations have a unique solution in x i and x j , and since each x k can take

on p different values x∗

k , we have p n−2 different solutions to (11.2) and (11.3).

This is true for all δ2= 0, 1, , p − 1 and δ1fixed, which implies that the p n−1

treatment combinations satisfying (11.2) can be divided into p distinct sets of

p n−2treatment combinations each satisfying one of the p equations (11.3) with

δ2= 0, 1, , p − 1 Any contrast belonging to E α is therefore orthogonal to

any contrast belonging to E β It is in this sense then that the partitions α and β,

and hence the interactions E α and E β, are orthogonal

Any treatment combination x that satisfies the equations (11.2) and (11.3) also

satisfies the equation

(α1+ β1)x1+ (α2+ β2)x2+ · · · + (αn + βn )x n = δ1+ δ2 (11.4)

As δ1and δ2 take on all values in GF(p), δ1+ δ2 takes on each value in GF(p)

exactly p times Combining all x that satisfy (11.4) for a particular value of

δ1+ δ2, we have partitioned the treatment combinations according to

(α1+ β1)x1+ (α2+ β2)x2+ · · · + (αn + βn )x n = δ

with δ ∈ GF(p) This is, of course, the partition α + β, and the corresponding

interaction E α +β is, in conformity with previous usage, a GI of E α and E β

(α1+ λβ1)x1+ (α2+ λβ2)x2+ · · · + (αn + λβn )x n = δ (11.5)

with δ ∈ GF(p) and each λ ∈ GF(p), λ
= 0.

Formally the GIs are obtained by multiplying the corresponding letters raised

to certain powers into each other, reducing the powers mod p and modifying

the powers (if necessary) such that the first letter included appears with power

unity by multiplying every power by the same appropriate value in GF(p) For

example, in a 53 factorial the GIs of AB and AC2 are

(AB) × (AC2) = A2BC2= (A2BC2)3 = AB3C

(AB) × (AC2)2 = A3BC = (A3BC)2 = AB2C2

(AB) × (AC2)3 = A4BC = (A4BC)4 = AB4C4

(AB) × (AC2)4 = BC3

More generally we have the following theorem

Theorem 11.1 The total number of GIs among q interactions E α1, E α2, ,



Proof The q interactions are defined by the q sets of equations

α j x1+ αj x2+ · · · + αj n x n = δj with δ j ∈ GF(p), j = 1, 2, , q, or for short

α

Any treatment combination x that satisfies (11.7) for a given set δ1, δ2, , δ q

also satisfies any of the equations

components then take on all possible values in GF(p) Hence the possible number

With a p n factorial and blocks of equal size the only block sizes are k = p  (≤

n) If  < n, we need to confound certain interactions with blocks More precisely, for a p n factorial in blocks of size p  we have p n − blocks and hence we mustconfound (p n − − 1)/(p − 1) interactions with blocks To find such a system of

confounding, we first state the following theorem

Theorem 11.2 If in a p n factorial with equal block sizes p  ( ≤ p n−2) two

interactions E α and E β are confounded with blocks, then so are their GIs E α +λβ,

λ ∈ GF(p), λ
= 0.

Proof Consider the equations associated with E α and E β:

α1x1+ α2x2+ · · · + αn x n = δ1

β1x1+ β2x2+ · · · + βn x n = δ2 (11.9)

For any pair (δ1, δ2) the equations (11.9) are satisfied by a set of p n−2treatment

combinations x= (x1, x2, , x n ) denoted by S(x ; α, δ1; β, δ2) say Each set

S (x; α, δ1; β, δ2) makes up one or several blocks A contrast belonging to E α isgiven by