Experimental design single-factor

The last of the major varieties of experimental design are the nested designs, where the levels of one factor are nested within (or are subsamples of) another factor. That is, each subfactor is evaluated only within the limits of its single larger factor. 

For the moment, we will investigate the experimental design in which each experiment is carried out at a different level of the single factor. Later, in Section 5.6, we will consider the case in which both experiments are performed at the same level. 

Unfortunately, two experiments at two different levels of a single factor cannot provide an estimate of the purely experimental uncertainty. The difference in the two observed responses might be due to experimental uncertainty, or it might be caused by a sloping response surface, or it might be caused by both. For this particular experimental design, the effects are confused (or confounded) and there is no way to separate the relative importance of these two sources of variation. 

It is evident that Equation 11.71 can be used to plot the variance (or uncertainty) of predicting a single new value of response if Xq is made to vary across the domain of factor space. Such a plot of standard deviation of predicting a single new value of response as a function of pH is shown in Figure 11.7 for the experimental design of Equation 11.15, the data of Equation 11.16, and the second-order model of Equation 11.39. 

In previous chapters, many of the fundamental concepts of experimental design have been presented for single-factor systems. Several of these concepts are now expanded and new ones are introduced to begin the treatment of multifactor systems. 

The art of experimental design is made richer by a knowledge of how the placement of experiments in factor space affects the quality of information in the fitted model. The basic concepts underlying this interaction between experimental design and information quality were introduced in Chapters 7 and 8. Several examples showed the effect of the location of one experiment (in an otherwise fixed design) on the variance and co-variance of parameter estimates in simple single-factor models. 

The simplest experimental design is one in which the effect of a single factor is investigated in two experiments. For example, the effect of method of agitation on the rate of solution of a solid in a liquid is being investigated. The experimental design is a one-factor factorial at two levels, three runs at each level, shown at the left. 

In a factorial experiment, a fixed number of levels are selected for each of a number of variables. For a full factorial, experiments that consist of all possible combinations that can be formed from the different factors and their levels are then performed. This approach allows the investigator to study several factors and examine their interactions simultaneously. The object is to obtain a broad picture of the effects of the selected experimental variables and detect major trends that can determine more promising directions for further experimentation. Advantages of a factorial design over single-factor experiments are (1) more than one factor can be varied at a time to allow the examination of interaction effects and (2) the use of all experimental runs in evaluating an effect increases the efficiency of the experiment and provides more complete information. 

A variable interaction or synergy occurs when the effect on the response caused by one variable can be changed by varying the level of a second variable. RSM provides an estimate of the effect of a single variable at selected fixed conditions of the other variables. If the variables do act additively, the factorial (experimental design) does the job with more precision. If the variables do not act additively, the factorial, unlike the one-factor-at-a-time design, can detect and estimate interactions that measure the nonadditivity (5). 

For a two-level factorial design, only two excipients can be selected for each factor. However, for the filler-binder, a combination of brittle and plastic materials is preferred for optimum compaction properties. Therefore, different combinations such as lactose with MCC or mannitol with starch can count as a single factor. Experimental responses can be powder blend flowability, compactibility, blend uniformity, uniformity of dose unit, dissolution, disintegration, and stability under stressed storage conditions. The major advantage of using a DOE to screen prototype formulations is that it allows evaluation of all potential factors simultaneously, systematically, and efficiently. It helps the scientist understand the effect of each formulation factor on each response, as well as potential interaction between factors. It also helps the scientist identify the critical factors based on statistical analysis. DOE results can define a prototype formulation that will meet the predefined requirements for product performance stability and manufacturing. 

A Factor is an aspect of an experiment that we can alter to see if this changes the endpoint we are measuring. The various different possibilities for each factor are then referred to as levels . While f-tests are used with the simplest experimental designs - a single experimental factor that has just two levels - for more complex designs, analyses of variance (ANOVAs) are called for. 

In summary, RSM is a useful technique for finding the optimum conditions for one or more responses over up to about five factors. The types of experimental designs often used for RSM are the CCD and Box-Behnken designs. The response surface can be well-described by a second-order polynomial model, and thus can be used to readily find the optimum conditions for a single response or to perform a tradeoff analysis among two or more responses. 

Although a statistician should be consulted on all the factors above, in practice the most common reason why a statistician is consulted at the beginning of a trial is to establish the sample size. Fortunately, to answer this single question, all the factors above must be considered, so that one way or another the statistician usually winds up in a collaboration on experimental design. Because the discussion often starts with sample size, we will also start there. 

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