Experimental design Latin-square designs

Latin square designs, 1,52 simplex designs, 56,57,58 Experimental designs for specific problems, 61,62,63... 

Experimental designs are square in forms (mxm), and the researched factor is tested once in each step. Table 2.58 shows an example of 4x4 Latin square design. 

Statistical experimental design is characterized by the three basic principles Replication, Randomization and Blocking (block division, planned grouping). Latin square design is especially useful to separate nonrandom variations from random effects which interfere with the former. An example may be the identification of (slightly) different samples, e.g. sorts of wine, by various testers and at several days. To separate the day-to-day and/or tester-to-tester (laboratory-to-laboratory) variations from that of the wine sorts, an m x m Latin square design may be used. In case of m = 3 all three wine samples (a, b, c) are tested be three testers at three days, e.g. in the way represented in Table 5.8 ... 

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. 

For example, the book by Gomez and Gomez describes many possible designs such as the Latin square and the lattice designs. The former can handle simultaneously two known sources of variation among experimental units. Chapters deal with Sampling in experimental plots, and the Presentation of research results. ... 

Table 15 APAP Content, RH and Pressure Combinations Selected Using the Latin Squares Experimental Design for Preparing Surrogate Tablets...

Figure 7 The first four loading vectors of the NIR PLS model (MSC and mean centering) generated on the NIR data collected on the 10% APAP surrogate tablets prepared according to the Latin square experimental design.

The PLS model generated on samples prepared according to the Latin squares experimental design was used to predict the key compact attributes from the real-time spectral data collected for roller compacted samples (Fig. 8). Good agreement was observed between the NIR-predicted values and the values measured off-line using the reference methods (Table 16). 

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 ... 

Graeco-Latin square an experimental design which permits study of the effects of 4 factors at n levels in n2 runs (n > 4). 

Fractional replicates of experimental designs in which all factors are at the same number of levels can be partially replicated in fractions whose denominators are multiples of the number of levels. These designs are the so-called Latin square designs. 

Another useful experimental design for minimizing the effects of two types of inhomogeneity is the Youden square design. Latin squares must have the same number of levels for both of the blocking factors and the treatment factor Youden squares must have the same number of levels for the treatment factor and one of the blocking factors, but the number of levels for the other blocking factor can be... 

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... 

Design of Latin squares is frequently applied when the effect of one factor on several conditionally the same devices is researched for a long time. In that case, rows of designs correspond to successive time studies, and columns to experimental devices. 

The condition for application of Latin squares are interactions that are negligible with respect to experimental error. As in researching complex system there exist interactions, Latin square designs are not widely applied. 

Studies can be conducted as group comparisons if sufficient numbers of animals are available. Alternatively, the use of a latin square cross-over experimental design allows for studies with fewer animals (eg. N = 4). 

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. 

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