Much of the effort is concentrated on the construction of incomplete block designs for various types of treatment structures, including "regular" treatments, control and test treatments, and factorial treatments. Further, we discuss various kinds of row-column designs as examples of the use of additional blocking factors.
INTRODUCTION AND EXAMPLES
A number of incomplete block models are available for this type of situation, for example, Kempthorne (1953) and Zoellner and Kempthorne (1954). The origin of incomplete block designs goes back to Yates (1936a) who introduced the concept of balanced incomplete block designs and their analysis using within- and between-block information (Yates, 1940a).
GENERAL REMARKS ON THE ANALYSIS OF INCOMPLETE BLOCK DESIGNS
Other incomplete block designs were also proposed by Yates (1936b, 1937a, 1940b), who called these designs quasi-factorial or lattice designs. Further contributions in the early history of incomplete block designs were made by Bose and Fisher (1940) on the structure and construction of balanced incomplete block designs.
THE INTRABLOCK ANALYSIS .1 Notation and Model
- Normal and Reduced Normal Equations The normal equations (NE) for µ, τ i , and β j are then
- The C Matrix and Estimable Functions
- Solving the Reduced Normal Equations
- Estimable Functions of Treatment Effects
- Analyses of Variance
We will now perform a within-block analysis (Section 1.3), a between-block analysis (Section 1.7), and a combined analysis (Section 1.8) for the general incomplete block design. From the Gauss–Mark theorem (see Section I.4.16.2) we know that for any linearly estimable function of treatment effects, saycτ,.
INCOMPLETE DESIGNS WITH VARIABLE BLOCK SIZE
Before we do, we mention that its validity depends on one very important assumption, and that is the constancy of the variance σe2 for all blocks. In general, the size of σe2 will depend on the size of the blocks: the larger the blocks, the larger σe2 will be, as it is partly a measure of the variability of the experimental units within blocks (see I.9.2.4) .
DISCONNECTED INCOMPLETE BLOCK DESIGNS
In Example 1, we may want to reduce all blocks to a constant size, thereby reducing the number of experimental units available. Therefore, it is quite reasonable to assume that σe2 is constant for blocks of different sizes if the number of experimental units varies little.
RANDOMIZATION ANALYSIS
Derived Linear Model
When deriving the properties for the random variables βu and ωuv, we have of course used the well-known distribution properties for the design random variables αju andδj uv, which e.g.
Randomization Analysis of ANOVA Tables
Thus, for the mean squares (MS) from tables 1.1 and 1.2, we have the expected values according to randomization theory, as given in tables 1.5 and 1.6, respectively. Considering (1.36) and (1.37), there is no exact test for the equality of block effects.
INTERBLOCK INFORMATION IN AN INCOMPLETE BLOCK DESIGN
Introduction and Rationale
Interblock Normal Equations Consider the model equation (1.1)
The quantities w and wj of (1.42) and (1.43) are referred to as the within- and between-block weights, respectively, as well as the reciprocal of the within-block variance, . We note here that typically (see Kempthorne, 1952) cross-block analysis proceeds not in terms of "observations" z [as given in (1.45)] but rather in terms of block totals Xβy.
Nonavailability of Interblock Information
COMBINED INTRA- AND INTERBLOCK ANALYSIS .1 Combining Intra- and Interblock Information
Linear Model
The expression (1.65) looks deceptively simple, but the reader should note that the elements of A−1 depend on σb2 and σe2. Finally, we note that equations (1.60) show a striking similarity to the NE block (1.7), except that the system (1.60) is of full order and the elements of its coefficient matrix depend on the unknown parametersσb2 and σe2.
Some Special Cases
RELATIONSHIPS AMONG INTRABLOCK, INTERBLOCK, AND COMBINED ESTIMATION
General Case For the full model
The second expression in (1.72) is actually the quadratic form to be minimized to obtain the RNE for (see Section I.4.7.1), and this is equivalent to minimizing. Q1≡(y−Xττ)(I−P)(y−Xττ) (1.75) So we summarize: To obtain the intrablock estimator, we minimize Q1; to obtain the interblock estimator we minimize Q2; and to obtain the combined estimator, we minimize Q1+Q2.
Case of Proper, Equireplicate Designs
If other roots are of equal torque, then the intrablock estimators for the corresponding treatment parameters do not exist. Similarly, if other roots are equal to zero, then the interblock estimators for the corresponding treatment parameters do not exist.
ESTIMATION OF WEIGHTS FOR THE COMBINED ANALYSIS The estimator for the treatment effects as given by (1.61) depends on the weights
Yates Procedure
One way to estimate andwj is to first estimateσe2andσb2 and then use these estimators to estimate w and wj.
Properties of Combined Estimators
The problem therefore remains to find methods to construct estimators for ρ such that the combined treatment contrast estimator is uniformly better than the corresponding intrablock estimator, in that it has smaller variances for all values of ρ. The only general advice we give at this time in conjunction with the use of the Yates estimator is the somewhat imprecise advice to use the intrablock estimator instead of the combined estimator if the number of treatments is "small". The reason for this is that in such a situation the degrees of freedom for MS(I|I,Xβ,Xτ) and MS(Xβ|I,Xτ) are probably also small, which would imply that σe2 and σβ2, and hence ρ , cannot be estimated very accurately.
MAXIMUM-LIKELIHOOD TYPE ESTIMATION
- Maximum-Likelihood Estimation
- Restricted Maximum-Likelihood Estimation
- Average Variance for Treatment Comparisons for an IBD Let us now consider
- Definition of Efficiency Factor
- Upper Bound for the Efficiency Factor
The basic idea is to obtain estimators for the variance components that are free of the fixed effects in the sense that the likelihood does not contain the fixed effect. For this purpose, we will use a quantity referred to as the efficiency factor of the IBD.
OPTIMAL DESIGNS
Information Function
The problem of finding an optimal design d in D can then be reduced to finding a design that maximizes φ(C∗d) mbid in D (see Cheng, 1996). As shown above, an alternative and historically original approach to finding an optimal design is to consider the minimization of some convex and nonincreasing functions of the dispersion matrices, as shown by the relationship between (1.112) and (1.113).
Optimality Criteria
The statistical significance of these criteria is that minimizing (1.114) minimizes the generalized variance Pτ, (1.115) minimizes the average variance of the set Pτ, and (1.116) minimizes the maximum variance of the individual normalized contrast.
Optimal Symmetric Designs
The moral of the situation is multiple: (1) Researchers must make a choice about problems and often work on unrealistic problems as the most workable approach to real problems and should not be criticized for doing so; (2) almost every optimality problem is artificial and limited to some extent because design value criteria must be introduced, and in almost every research situation it is difficult to value-map the possible designs to the real line; and (3) a solution to a mathematically formulated problem may have limited value, so promoting one design that is only optimal with respect to a particular value criterion, C1, and declaring another design of poor value because it is not optimal may be unfair because that design might be better with respect to another value criterion, says C2say. On the other hand, if a balanced design is an optimal design, but we cannot use that design due to practical constraints and have to use a near-balanced design instead, then we have a way to evaluate the efficiency of the design that we are about to evaluate . .
COMPUTATIONAL PROCEDURES
- Intrablock Analysis Using SAS PROC GLM
- Intrablock Analysis Using the Absorb Option in SAS PROC GLM A computational method as described in Section 1.3 using the RNE can be imple-
- Combined Intra- and Interblock Analysis Using the Yates Procedure
- Combined Intra- and Interblock Analysis Using SAS PROC MIXED
- Comparison of Estimation Procedures
- Testing of Hypotheses To test the hypothesis
The ANOVA table provides the same information as in Table 1.8, except that it does not provide solutions for the intersection and blocks. Their estimated variances and the estimated variances for the treatment contrasts are obtained from A−1×9.09 (see Tables 1.13 and 1.14).
INTRODUCTION
DEFINITION OF THE BIB DESIGN
PROPERTIES OF BIB DESIGNS
Fisher (1940) first proved that (2.6) was a necessary condition for the existence of a balanced incomplete block design. This is only one of several necessary conditions (Shrikhande, 1950) for the existence of a symmetrical BIB design.
ANALYSIS OF BIB DESIGNS .1 Intrablock Analysis
Combined Analysis
From the definition of the BIS design and the form of NN given in (2.7) it follows that the coefficient matrix,Asay, in (1.60) now has the form. We note that σe2 here refers to the error variance in a CRD, while σe2 in case 1 refers to the within-block variance.
ESTIMATION OF ρ
The largest possible value for r∗ is the one corresponding to the combined estimator when ρ is known. Shah (1970) shows that for his estimator (given in Theorem 2.1) the difference between rc∗ and r∗(ρ) is not noticeable for moderate values of ρ.
SIGNIFICANCE TESTS
Since ρ is usually not known and must therefore be estimated from the data, a certain loss of information occurs; that is, r∗(ρ) < rc∗, where r∗(ρ) is the effective number of replications when an estimatorρ is used instead of ρ in the combined analysis. In addition, Table 2.1A provides the SAS solutions to the normal equations, which are then used to compute the set of five orthonormal contrasts based on .
SOME SPECIAL ARRANGEMENTS
Replication Groups Across Blocks
Since the replication groups are orthogonal to blocks and to treatments, the procedure for estimating linear contrasts of treatment effects is the same as before. As can be seen from example 2.3 and model (2.23), the replication groups play the same role as the columns in a Latin square design.
Grouped Blocks
Their proof actually provides a general procedure for constructing a Youden square from a given symmetric BIB design. Proof Since for an α-distinguishable design BIB there are q−1 linearly independent relations between the columns of N, we have.
RESISTANT AND SUSCEPTIBLE BIB DESIGNS .1 Variance-Balanced Designs
Definition of Resistant Designs
An interesting question then is: If one or more treatments are deleted from a BIB design, the resulting design is still variance balanced. Such designs are called resistive BIB designs by Hedayat and John (1974) who also gave a characterization of such designs and showed their existence.
Characterization of Resistant Designs
Existence, construction and properties of resistive BIB designs have been discussed by Hedayat and John (1974) and John (1976), and we will not go into them except for the following statement. They show that the symmetry of the design is both necessary and sufficient for the local resistance of degree k with respect to L consisting of k treatments occurring in exactly one block of the design if we demand D∗ to be a BIB- design (rather than just a variance-balanced design).
Robustness and Connectedness
INTRODUCTION
DIFFERENCE METHODS
Cyclic Development of Difference Sets
If we take the set A as the starting block B0 and develop it cyclically, it generates a symmetric design BIB with parametrist, b=t, r =k, λ. The resulting BIB design with 11 treatments in 55 blocks of size 3 and 15 replications per treatment is as follows:
Method of Symmetrically Repeated Differences
If conditions 1-3 are satisfied, the differences are said to repeat symmetrically in G, where each difference occurs λ times. The cyclic development method is again used to generate BIB models as described in the following theorem.
Formulation in Terms of Galois Field Theory
We shall not pursue this further here, but refer the reader to Raghavarao (1971) for a comprehensive list of such results and further references.
OTHER METHODS
Irreducible BIB Designs
Bb} represents a symmetric design BIB with parameters t, b=t, k, r =k, λ, then design D2 can be considered, which is obtained as follows: Select any block from D, say B1, and delete from each of remaining blocksB2,B3,. To see that this is true, we show that in a symmetric BIB design every two blocks have λ treatments in common.
Orthogonal Series
Because of the repeated application of step 3 above, this method is referred to as the successive diagonalization method (Khare and Federer, 1981).
LISTING OF EXISTING BIB DESIGNS
INTRODUCTION
PRELIMINARIES
- Association Scheme
- Association Matrices
- Solving the RNE
- Parameters of the Second Kind
Recall that the RNEs for a proper equireplicate incomplete block design are of the form [see (1.7)]. Since the right-hand side of (4.11) is a linear combination of Bu's, the left-hand side must also be one.
DEFINITION AND PROPERTIES OF PBIB DESIGNS
Definition of PBIB Designs
If any two treatments are kth associates, then the number of treatments is common to the uth associates of the first and the fifth associates of the second ispkuv, independent of the pair of kth associates. The parameters of the second kind are generally displayed in matrix form as Pk =(pkuv)(k m), and in our notation they are (m+1)×(m+1) matrices (it deviates from the original notation by Evil and .
Relationships Among Parameters of a PBIB Design
Equations (4.23) and (4.24) apply to any PBIB design, whereas the remaining puvk are determined by the particular association scheme for a given PBIB design, as will be shown later. For this reason, we refer to a PBIB design as defined above as an m-associated class PBIB design, which we will henceforth denote by PBIB(m) design.
ASSOCIATION SCHEMES AND LINEAR ASSOCIATIVE ALGEBRAS
Linear Associative Algebra of Association Matrices
Linear Associative Algebra of P Matrices
Since they combine in exactly the same way as B matrices under addition and multiplication, they provide a regular representation in terms of (m+1)×(m+1) matrices of the algebra of B matrices, which are ×t matrices with > m+1. Thus, the set of matrices of the form (4.30) also constitutes a linear associative algebra with unit element P0=Im+1.
Applications of the Algebras
This corresponds to (4.12) and shows that the P-matrices multiply in the same way as the B-matrices. We will also make use of them in Section 4.7 in connection with the combined analysis of PBIB designs.
ANALYSIS OF PBIB DESIGNS .1 Intrablock Analysis
Combined Analysis
As developed in Section 1.8.3, the set of normal equations for the combined analysis is of the form. To obtain τ we use the same type of argument as in section 4.2.1, that is, we use the special form of Ace expressed in terms of association matrices; that is,.
CLASSIFICATION OF PBIB DESIGNS
- Group-Divisible (GD) PBIB(2) Designs
- Triangular PBIB(2) Designs
- Latin Square Type PBIB(2) Designs
- Cyclic PBIB(2) Designs
- Rectangular PBIB(3) Designs
- Generalized Group-Divisible (GGD) PBIB(3) Designs
- Generalized Triangular PBIB(3) Designs
- Cubic PBIB(3) Designs
- Extended Group-Divisible (EGD) PBIB Designs
- Cyclic PBIB Designs
- Some Remarks
For the L2 association scheme, two treatments are said to be associated first if δ=1, and second associated if δ=2. The association scheme for this design is then as follows: Two treatments are said to be jth associations if they differ exactly in components.
ESTIMATION OF ρ FOR PBIB(2) DESIGNS
Shah Estimator
We chose our list of association schemes because they lead to a fairly large class of existing and practical PBIB designs [particularly the PBIB(2) designs and the cyclic PBIB designs are well documented (see Chapter 5)] and/or are particularly useful in the construction of confounding systems for symmetric and asymmetric factorial designs (see Chapters 11 and 12). We emphasize that an association scheme is not a PBIB design and does not automatically lead to a PBIB design.
Application to PBIB(2) Designs
In the previous chapter, we discussed different types of association schemes for PBIB designs with two or more associated classes. Instead, we will discuss only a few methods in detail, focusing mainly on three types of PBIB designs, namely PBIB(2) designs, in particular group shareable PBIB(2) designs, cyclic PBIB designs and EGD designs. PBIB designs. .
GROUP-DIVISIBLE PBIB(2) DESIGNS
- Duals of BIB Designs
- Method of Differences
- Finite Geometries
- Orthogonal Arrays
Their relationship to the GD-PBIB(2) design is as follows: Replace any integer x occurring in the ith row of the array by the treatment (i−1)p+x. This is of course the association scheme of a GD-PBIB(2) design with the parameters as mentioned above.
CONSTRUCTION OF OTHER PBIB(2) DESIGNS
Triangular PBIB(2) Designs
For other specialized methods of constructing GD-PBIB(2) models, the reader is referred to Raghavarao (1971). An essentially complete list of existing plans for GD-PBIB(2) designs is given by Clatworthy (1973) together with references and methods relating to their construction.
Latin Square PBIB(2) Designs
Other methods given by Shrikhande and Chang, Liu and Liu (1965) are based on the existence of certain BIB designs and consider, for example, doubling the BIB design as omitting certain blocks from the BIB design. It is of course clear from the construction method that this PBIB design is a dissolvable design, where the blocks from each language form a complete.
CYCLIC PBIB DESIGNS
Construction of Cyclic Designs
This means that if the initial block contains processing, it must also contain processing i+b, i+2b,. The association scheme for the second design is actually the association scheme for a cyclic PBIB(2) design as discussed in Section 4.6.4, except that we need to relabel the treatments as 0,1,.
Analysis of Cyclic Designs
The second row is then obtained by shifting each element of the first row one position to the right (in a circular fashion) and so on so that C is determined entirely by its first row. Due to the construction of designs and the resulting association scheme, we have for theλ1i in N N.
KRONECKER PRODUCT DESIGNS
Definition of Kronecker Product Designs
Properties of Kronecker Product Designs
Proof We will prove the statement by establishing that the concordance matrix and the association matrices satisfy certain conditions as established by Bose and Mesner (1959) and as used in Chapter 4. Furthermore, the association scheme defined by the association matrices (5.21) is indeed. the same as stated in the theorem, which is immediately clear from the way in which the treatments of N are summarized in (5.17).
Usefulness of Kronecker Product Designs
The method of constructing PBIB models as Kronecker product models can obviously be extended to Kronecker products of incidence matrices for three or more existing PBIB models, for example. This not only makes the PBIB models of Kronecker products more attractive from a practical point of view, but also provides methods for constructing PBIB models with different connection schemes.
EXTENDED GROUP-DIVISIBLE PBIB DESIGNS
EGD-PBIB Designs as Kronecker Product Designs
Method of Balanced Arrays
Proof This proof actually gives the construction of the EGD-PBIB design, assuming that the BAiexist. Continue to form the SIP of the blocks of the EGD/(2q−1)-PBIB design with the columns from Baq+1 to q=ν−1.
Direct Method
