The Latin Square

The hypothetical experiment discussed in the previous section, though now rigorous in design in that it cannot lead to false conclusions, is in some cases capable of further refinement 

Examining Table 1.5, it will be seen that the four sub-batches of each batch include all four treatments. At the same time, with each of the four treatments all four reactors occur once and only once and all four batches occur once and only once. Possible errors due to differences between Batches and between Reactors are thus eliminated both from the averages, thus allowing unbiassed estimates to be obtained, and also from the error, thus making the latter a minimum. 

The computation of the results of a Latin Square is discussed in Chapter XIL(g). 

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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. 

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

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 ... 

Table 5.58 Data for the Latin squares method for a process with three factors.

The correct use of the Latin squares method imposes a completely random order of execution of the experiments. As far as the experiment required in the box table is randomly chosen and as a single value of the process response is introduced into the box, we guarantee the random spreading of the effect produced by the factor which presents a systematic influence. 

Now every reader knows that to check a hypothesis in which we compare two variances, we have to use the Fischer test. Here the computed value of a Fischer random variable is compared with its theoretical value particularized by the concrete degrees of freedom (Vj, V2) and the confidence level 1 — a. Table 5.59 presents the synthesis of the analysis of variances for this case of the Latin squares method. 

Table 5.60 Factorial plan for the Latin squares method -case of chlorination of an organic liquid.

I 5 Statistical Models in Chemical Engineering Table 5.62 Analysis ofthe variances for the Latin squares method, example 5.6.3. 

It is important to note that the effect of the factor that changes the letter in the Latin squares table is negligible. Then, for the investigated chlorination reaction both the concentration of the catalyst (between 0.1 and 0.3% g/g) and its process of addition do not have any effect on the concentration of the by-products. Nevertheless, this conclusion cannot be definitive because we can find from Table 5.62 that we have a high residual variance. In this case, we can suggest that the interaction effects are certainly included in the residual variance. 

The real residual variance frequently named reproducibility variance can be determined by repeating all the experiments but this can turn out to be quite expensive. The Latin squares method offers the advantage of accepting the repetition of a small number of experiments with the condition to use a totally random procedure for the selection of the experiments. With the data from Table 5.61 and... 

Now, it is dear that the residual variance from Table 5.62 contains one or more interaction effects. Moreover, for this application or, more precisely, for the data given for the particularization of the Latin squares method, a partial response has... 

In this Chapter a number of miscellaneous aspects of the use of the analysis of variance will be discussed. The next five sections contain matter which might be borne in mind in planning and analysing the results of factorial experiments, Section (g) is an account of a useful experimental device (the Latin Square), and the subsequent sections describe various applications of the analysis of variance. 

A statistical device that was originally introduced by Fisher to overcome difficulties arising out of differences in soil fertility in agricultural experiments is the Latin Square. 

One of the assumptions underlying the use of a Latin Square is that there should be no interaction between the factors. Since the existence of interaction in industrial chemical systems appears to be common, the Latin Square may not be of wide applicability in this field. 

Consider the Latin Square in Table 12.10. It is possible to construct another Latin Square, using Greek Letters (Table 12.11) such that when they are superimposed (Table 12.12) to each Latin letter corresponds all Greek letters and vice-versa. Such a square is called Graeco-Latin. ... 

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. 

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. 

Note the balance of the Latin square. All four drugs are given at each period to each person. 

At this point one may suspect that different amount of EXERCISE, say light (L), medium (M), and heavy (H), may have different degrees of effect on WTLOSS and should be treated as another block factor. Under these circumstances, the experiment is often carried out according to the Latin square design, as shown in Table 5. Note that each DIET is assigned only once to each ETHNIC and only once to each EXERCISE. It enables the evaluation of three factors with only nine observations. However, it requires that no interaction exist between the factors. 

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. 

To improve the efficiency of the experiment even further one could go to an incomplete block design. An incomplete block design does not run all treatments in all blocks. The most extreme caise is the Latin square design. In a Latin... 

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