Latin square factorial design

Although this direct method is more adequate for the given example, because the number of the values that are not available are smaller than the sum of rows and columns, the constant method has also been demonstrated for the case of comparison. It should be noted that both methods are generally used in two-way classification such as designs of completely randomized blocks, Latin squares, factorial experiments, etc. Once the values that are not available are estimated, the averages of individual blocks and factor levels are calculated and calculations by analysis of variance done. The degree of freedom is thereby counted only with respect to the number of experimental values. Results of analysis of variance for this example are... 

In this chapter we explore factorial-based experimental designs in more detail. We will show how these designs can be used in their full factorial form how factorial designs can be taken apart into blocks to minimize the effect of (or, if desired, to estimate the effect of) an additional factor and how only a portion of the full factorial design (a fractional replicate) can be used to screen many potentially useful factors in a very small number of experiments. Finally, we will illustrate the use of a Latin square design, a special type of fractionalized design. 

In some applications, Latin square designs can be thought of as fractional three-level factorial designs that allow the estimation of one main factor effect while... 

As an example of the use of a Latin square as a fractional factorial design, suppose we want to find out the effect of increasing concentrations of a chemical added to retain gloss in an industrial paint formulation. The model is... 

Latin square design as a fractional factorial design. 

Latin square design — three factors at four levels. This is a 1/4 replicate of a 43 = 64 factorial... 

The fractional factorial designs, including the Latin squares, are generally used for screening possible experimental variables in order to find which are the most important for further study. Their use is subject to some fairly severe assumptions which should be known and taken into consideration when interpreting the data ... 

Latin square design three factors at three levels. Run the indicated experiment at the levels of A, B, and C shown in the box This is a l/3 replicate of a 38= 27 factorial design. 

Variables ai (lj and x are actual effects of i rows, j columns and the k factor level. One can notice that the k index is bracketed to indicate that in the design of Latin squares there are no m results, as is the case with a three-factorial design with one design-point replication. Design of Latin squares actually has m2 observations or data. 

Sample size and treatment choice are key design questions for general multifactor experiments. Authors have proposed the use of standard factorial experiments in completely randomized designs, block designs, or Latin squares (see, for example, Chapter 6 and Churchill, 2003). However, the unusual distribution of gene expression data makes one question the relevance of standard orthogonal factorial experiments in this context. 

Another form of fractional repUcation often applicable is the latin square. Suppose that three factors are to be considered at four levels. We may be interested, for example, in a comparison of results obtained in four laboratories on four samples by four methods of analysis. A complete factorial design would require 4 , or 64,... 

The third type of experimental design is the factorial design, in which there are two or more clearly understood treatments, such as exposure level to test chemical, animal age, or temperature. The classical approach to this situation (and to that described under the latin square) is to hold all but one of the treatments constant and at any one time to vary just that one factor. Instead, in the factorial design all levels of a given factor are combined with all levels of every other factor in the experiment. When a change in one factor produces a different change in the response variable at one level of a factor than at other levels of this factor, there is an interaction between these two factors which can then be analyzed as an interaction effect. 

The approach in material development is to systematically perform statistically designed experiments. Techniques include fractional factorial designs, Grecko-Latin squares, and self-directed optimization. Data collected is statistically evaluated to determine primary and combined effects of the... 

A special case of a two-level factorial design is the Latin square design, which was introduced very early on to eliminate more than one blocking variable. A Latin square design for two factors is given in Table 4.7 along with the representation as a fractional factorial design. 

This type of design allows the separation of an additional factor from an equal number of blocks and treatments. If there are more than three blocks and treatments, then a number of Latin square designs are possible. It can be noted that Latin square designs are equivalent to specific fractional factorial designs (e.g., the 4x4 Latin square design is equivalent to a 4 fractional factorial design). 

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