Latin square

Consider an experiment where durability or wear-out of four types of car tires have to be researched. Sixteen tires are at our disposal, four of each type. The research will be done on four cars. The factor in this case is the type of car tire. There are, however, two additional factors that affect the durability of tires  

Let us mark car type as I, II, III and IV and the position of tires on each car as FR, FL, RL and RR. The latest two factors are singled out as an inequality of experimental conditions into blocks by rows and columns. Car tire types are placed by random choice on cars but so that one type of tire is put on one type of car only once in the same position. 

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. 

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. 

It is clear from the table that A, B, C and D are levels of the researched factor. An important condition for applying design of Latin squares is that in each column and each row one factor level may appear once and only once. 

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In this plan, the effects of both automobile and wheel position are controlled by blocking. It should, however, be kept in mind that for the Latin square design, as for other blocking plans, it is generally assumed that the blocking variables do not interact with the primary variable to be evaluated. 

To round off this section we note a few unusual applications of Polya s Theorem an application to telecommunications network [CatK75], and one to the enumeration of Latin squares [JucA76]. In pure mathematics there is an application in number theory [ChaC82], and one to the study of quadratic forms [CraT80], being the enumeration of isomorphism types of Witt rings of fields. Finally, we note a perhaps unexpected, but quite natural, application in music theory to the enumeration of chords and tone rows for an n-note scale [ReiD85]. In the latter paper it is shown that for the usual chromatic scale of 12 semitones there are 80 essentially different 6-note chords, and 9,985,920 different tone rows. 

JucA76 Jucys, A. - A.A. The number of distinct Latin squares as a group theoretic constant. J. Comb. Theory 20 (1976) 265-272. 

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 ... 

Meals were consumed by the subjects following a Latin Square Design. Statistical analysis was performed by a General Linear Models Procedure (20) using split-plot in time analysis with the following non-orthogonal contrasts ... 

There is the possibility of carry-over effects. This is more crucial in Latin square and other cross-over designs. Knowledge of pharmacokinetics and metabolism of a compound under study generally helps in avoiding this problem. 

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... 

How can we distribute the three paints across the single panel to achieve these goals Using three vertical stripes would confuse the additive factor with the rock factor. Using three horizontal stripes would confuse the additive factor with the sand factor. Fortunately, if we divide the panel into nine smaller panels, a Latin square can be used to block against the influence of these two troublesome factors. 

Latin square design as a fractional factorial design. 

How is the following Youden square design related to the Latin square design of Problem 15.15 ... 

If there are three types of blocking factors, Graeco-Latin square designs can be used to minimize their effects. The following is a 4 x 4 Graeco-Latin square. What do a, p, y, and o represent ... 

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. ... 

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