Sequential designs

Sequential design. When a particular procedure is always executed in sequential order, for example, the start-up of a distillation column, a similar sequential arrangement of the controls will help to ensiure that parts of the sequence are not omitted. 

Junctions of arrows are called events or nodes. These are points in time and consume no time. They are numbered to provide a convenient numeric sequential designation for all activities. 

TODD s, WHITEHEAD A, STALLARD N, WHITEHEAD, J (2001), Interim analyses and sequential designs in phase III studies , Br J Clin Pharmacol, 51, 394-9. 

As a third example let us consider the growth kinetics in a chemostat used by Kalogerakis (1984) to evaluate sequential design procedures for model discrimination in dynamic systems. We consider the following four kinetic models for biomass growth and substrate utilization in the continuous baker s yeast fermentation. 

The ability of the sequential design to discriminate among the rival models should be examined as a function of the standard error in the measurements (oe). For this reason, artificial data were generated by integrating the governing ODEs for Model 1 with "true" parameter values kt=0.31, k2=0.18, k3=0.55 and k4=0.03 and by adding noise to the noise free data. The error terms are taken from independent normal distributions with zero mean and constant standard deviation (oE). 

The simplest such type of design is the sequential design, simplest if for no other reason than that the type of design it replaces is one of the simplest designs itself. This design is the simple test for comparison of means, using the Z-test or the t-test as the test statistic we have discussed these in our previous column series and book Statistics in Spectroscopy (now in its second edition [1]). 

The first of these assumptions is the use of the Normal distribution. When we perform an experiment using a sequential design, we are implicitly using the experimentally determined value of s, the sample standard deviation, against which to compare the difference between the data and the hypothesis. As we have discussed previously, the use of the experimental value of s for the standard deviation, rather than the population value of a, means that we must use the f-distribution as the basis of our comparisons, rather than the Normal distribution. This, of course, causes a change in the critical value we must consider, especially at small values of n (which is where we want to be working, after all). 

J.J DiStefano III, Optimized blood sampling protocols and sequential design of kinetic experiments, Am. J. Physiol. 240 (1981) R259-R265. 

Both nonsequential and sequential types of designs will be considered here for several reasons. First, sequential designs consist of elements which are themselves nonsequential. Second, use of sequential designs requires a clear-cut objective which must be defined in quantitative terms. Third, sequential designs, while efficient in terms of calculations done, are subject to certain dangers because of their limited exploration of the area of possible interest. 

Likewise, a good deal of the material stays in the tank longer than is desirable and so reduces the effective tank volume. This is not an effective separator. By comparison, the sequential design indicated earlier is far more effective. 

Fig. 3-2. Factorial experimental designs and sequential designs ys - next step in modified simplex algorithm y9 - next step in weighted simplex algorithm...

In environmental analysis we shall extremely seldom expect questions to be answered by methods of sequential design. It could, however, be possible that one is, for example, asked to locate the maximum contamination within a certain area, so that the decontamination process can start there. In the following discussion we will, therefore, provide only a brief summary of some optimization methods which could be useful. 

Mathematical Models. Secondary variable interactions quantify the synergies which are common in food chemistry. These interactions cannot be computed from pooled primary variable/sequential design studies and interpolations from such pooled data would lack the information given by the secondary interaction terms. Prob > t is an estimate of the relative importance of each model term. Terms with the lowest Prob > t could well be the driving force of the reaction processes accounting for the quantity of the volatiles found. From Table IV, about 25% of the model terms present at >0.05 Prob > t are seen to be interaction terms. 

Clinical trials, with almost no exception, are longitudinal (Chow and Liu, 2004). This means that data are accumulated sequentially over time. From the perspective outlined so far in the book, the statistical analysis takes place once the number of subjects stated in the study protocol have been enrolled, randomized, and completed their participation in the trial. This approach can be called the Fixed design or fixed sample design approach. Another design of interest in clinical trials is the group sequential design, in which interim analysis plays a crucial role. 

By its nature, therefore, the group sequential design involves the possibility of multiple testing. In this example it is possible that five analyses could be conducted on data collected in this clinical trial. As discussed in Section 7.10, there is an inherent problem with multiple testing. As more tests are performed, it becomes increasingly likely that a Type I error will occur, i.e., that a result will erroneously be declared as statistically significant. As also noted at that point, however, the problem can be addressed completely satisfactorily by taking appropriate statistical care. 

To avoid these kinds of problems, some new experimental designs have been proposed [26,27], Called optimal composite sequential designs (OCSD), these designs are an extension of optimal sequential designs (OSD) in that they are optimal for more than one type of model. The structure of a typical OCSD is shown in Figure 8.24. 

Search for sequential designs that are optimal for model M2. In this case we apply the same procedure as described for OSD. 

Search for sequential designs that are optimal for both models Mx and M2. In this case we switch alternatively between the two models. Point number NA + NB+l applies to model Mx, and the next point in the sequence with number NA + NB +2 applies to M2. 

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