Latin square design, randomization

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 ... 

What is the relationship - Youden square designs Latin square designs balanced incomplete block designs randomized complete block designs ... 

The design of experiment written in this form is a reconstructed Latin square design where one of the diagonals has been left out. Generally speaking, Youdens square is a symmetrically balanced incomplete random block where each factor level appears once and only once in each block position. 

The analysis for the Greco-Latin square design is similar to that of a Latin square design. However, one noticeable difference is that two treatment sum of squares have to be computed (factors C and D) by listing two sets of means outside the design table. As an additional note, Greco-Latin squares are most effective if replicated and are subject to the same randomization rules as for the Latin squares. 

The initial order of presentation of the samples may have an effect on the partitioning given by the subjects. In order to minimize this bias, samples should be presented to each subject in a randomized order or, preferably, using a design balanced for order effect (as a Latin Square design). 

Samples are presented in monadic seqnence, coded with three-digit random numbers, following a balanced rotation order (Williams Latin sqnare design) to avoid presentation order and carry-over bias. Hence, best practice requires the use of experimental designs to minimize sample presentation order bias and within-participant randomization of CATA terms. Typically, both designs will be based on Wiliams Latin square designs bnt be different in order to reflect the actual number of samples and terms used. Designs shoirld be developed to take into accoimt the number of consumers in the study. 

This block design, in which each treatment appears once in each row and once in each column, is known as a Latin square. It allows the separation of the variation into the between-treatment, between-block, between-time-of-day and random experimental error components. More complex designs are possible which remove the constraint of equal numbers of blocks and treatments. If there are more than three blocks and treatments a number of Latin square designs are obviously possible (one can be chosen at random). Experimental designs of the types discussed so far are said to be cross-classified designs, as they provide for measurements for every possible combination of the factors. But in other cases (for example when samples are sent to different laboratories, and are analysed by two or more different experimenters in each laboratory) the designs are said to be nested or hierarchical, because the experimenters do not make measurements in laboratories other than their own. Mixtures between nested and cross-classified designs are also possible. 

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... 

Youdens square is always a Latin square where one or more columns (or rows or diagonals) have been left out however, the opposite is not true a Latin square where one or more columns (or rows or diagonals) have been left out is not always a Youdens square, for by leaving out columns from a Latin square the balance in design is lost. It is, however, possible to construct designs of Youdens squares from all symmetrical balanced random blocks [26]. Youdens squares have the same number of rows and levels of a researched factor but quite a different number of columns. 

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. 

Usually, patients are randomized to a particular treatment order, and all patients are eventually exposed to the same variety of treatments. Large numbers of treatment periods, assigned using a Latin square, have been reported however, the logistics and patient retention in such studies are usually difficult, and these ideal designs are likely to be successful only when treatment periods are short ideal designs are commonest for normal volunteer studies (e.g. Amin et al., 1995). 

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