Long page
descriptions

Chapter 6   Incomplete designs

6.1   Missing treatments

6.1.1   Parameters that cannot be estimated

If treatments are missing when there are two or more controlled factors, some interaction parameters cannot be estimated.

6.1.2   Inference with missing treatments

Missing treatments often result in explained sums of squares that depend on the order of adding terms to the model so there may be several possible anova tables. The marginal sum of squares for each term can be used to help find the best model.

6.1.3   Missing treatments in block designs

If treatments are missing in some blocks of a randomised block design, the interactions between blocks and factors cannot be fully estimated. However if it is assumed that there is no block-treatment interaction, the effects of the controlled factors can usually still be estimated.

6.2   Latin square designs

6.2.1   Pairwise orthogonality

Two factors are orthogonal if all combinations of their factor levels are used in the same number of experimental units. If all pairs of factors are orthogonal, their main effects can be reordered in an analysis of variance table without affecting their sums of squares.

6.2.2   Latin squares

A Latin squares design is a pairwise orthogonal design for 3 factors, 2 factors and blocks or 1 factor and two blocking variables. The three variables (factors or blocks) must all have the same number of levels.

6.2.3   Treatment structure in Latin squares

If the levels of a factor in a Latin square experiment have internal structure, the explained sum of squares can be split into components to test hypotheses about this structure.

6.2.4   Randomisation for Latin squares

Randomisation consists of randomly permuting the levels of the factors and order of the blocks in a patterned design.

6.2.5   Designs based on Latin squares

Latin squares can be used as the basis of a few designs for situations where the numbers of blocks and factor levels are not equal.

6.2.6   Graeco-Latin squares

Pairwise orthogonal designs for four blocking variables or factors can be generated from two superimposed Latin squares.

6.3   2**n designs with one replicate

6.3.1   Two-level factors and one replicate

A preliminary experiment may be conducted to assess which of many factors are most important. To limit the number of experimental units, each factor is often allowed only 2 levels and a single replicate is used.

6.3.2   Factors with two levels

Factors with two levels may be treated as numerical with values +1 and -1. The main effects of two factors are orthogonal if the sum of the products of these values is zero.

6.3.3   Interactions

The interaction between two factors can be modelled by a numerical variable whose values are the product of the coded factor values. The variables for main effects and interactions in a complete factorial design are orthogonal.

6.3.4   Hypothesis tests

If there are no replicates, the significance of factor effects and interactions can only be tested if it can be assumed that there are no high-order interactions. Hypothesis tests are however less important in screening experiments.

6.4   Fractional factorial designs

6.4.1   Orthogonal main effects

Designs for k factors in half or quarter of the experimental units required for a complete design should be such that the main effects are orthogonal -- each combination of levels for any two of the factors is used the same number of times.

6.4.2   Confounded main effects and interactions

In fractional factorial designs, main effects are confounded with interactions between other factors.

6.4.3   Design principles

A complete factorial design can be augmented with an extra factor whose levels are defined by an interaction between the initial factors. Each original main effect or interaction is confounded with a term involving the added factor.

6.4.4   Another fractional factorial design

Another example is given in which a complete design is augmented with two additional factors.

6.4.5   Factors with more than two levels

Similar designs can be constructed for factors with more than two levels, but they are beyond the scope of this e-book.

6.5   Fractional designs in blocks

6.5.1   Factorial experiments in two blocks

The treatments in a complete factorial design for factors with 2-levels can be split into two blocks by confounding the blocks with a high-order interaction between the factors. All factor main effects and other interactions can still be estimated if it is assumed that the blocks do not interact with the factors.

6.5.2   Partial confounding

By repeating the design on the previous page with blocks that are defined by a different interaction between the factors, all main effects and interactions can be estimated. However the interactions used to define the blocks are estimated with lower accuracy than the main effects or interactions.

6.5.3   Factorial experiments in four blocks

The blocks can be defined by picking two high-order interactions to be aliased with the blocks. However the interaction between these 2-level blocking factors is also confounded with factors so care must be taken that the main effects of all factors can still be estimated.

6.6   Incomplete block designs

6.6.1   Balance in block designs

If each pair of treatments is used together in the same number of blocks, the design is called balanced and all pairs of treatments can be compared with the same accuracy.

6.6.2   Balanced incomplete block designs

This page gives an example of analysis of variance for a balanced incomplete block experiment.

6.6.3   Adjusted treatment means

The raw treatment means do no describe well the differences between the treatments in incomplete block designs. Adjusted treatment means should be used instead.

6.6.4   Balanced lattice designs

Lattice designs are used in experiments for factors with many levels and relatively small block sizes. In a balanced lattice design, each pair of treatments is used together in one block.

6.6.5   Simple lattice designs

Subsets of a full balanced lattice design still allow all treatments to be compared.

6.6.6   Randomisation

Groups of treatments should be allocated to blocks randomly from the rows of a standard design. These treatments should then be randomly allocated to experimental units within each block.