Factorial trial design

Table 5.1 A simple factorial trial design involving two drugs.

Factorial trial designs are used to evaluate, simultaneously, two or more treatments through the use of varying combinations of the treatments. An example of a simple trial to evaluate drugs A and B is illustrated in Table 5.1. These trials are particularly relevant where multiple drug therapies are anticipated. 

Cross-over trial design Factorial design Hybrid design Large simple clinical trials... 

As there now exists a large body of laboratory studies on each of the variable systems, for example the effect of the lime/silica ratio in the slag on the desulphurization of liquid iron, the most appropriate phase compositions can be foreseen to some extent from these laboratory studies when attempting to optimize the complex industrial process. The factorial trials are not therefore a shot in the dark , but should be designed to take into account the laboratory information. Any qualitative difference between the results of a factorial trial, and the expectations predicted from physico-chemical analysis might suggest the presence of a variable which is important, but which was not included in the trials. 

Factorial design represents yet another trial design of interest. This may be used to evaluate the effect of two or more interventions upon participants in a single trial (Table 2.7) and, hence, can be economically attractive. 

The full factorial experimental design, however, quickly expands into an unmanageable number of trials. For n variables and p variations on each variable, t, N is calculated by Equation 15.1. 

The fractional factorial is designed to reduce the number of qualification trials to a more reasonable number, say, 10, while holding the number of randomly assigned processing variables to a reasonable number as well, say, 9. The technique was developed as a nonparametric test for process evaluation by Box and Hunter [18] and reviewed by Hendrix [19]. Ten is a reasonable number of trials in terms of resource and time commitments and should be considered an upper limit in a practical testing program. This particular design as presented in Table 7 does not include interaction effects. 

It has also been the case that clinical trials have not been particularly representative as regards the patients studied. For example, it has been claimed that women have often been under-represented for various reasons, not least because fear of damaging the unborn has led sponsors to exclude many women of child-bearing age from their trials. A particular trial in which this certainly was the case was the Physician s Health Study (Hennekens and Eberlein, 1985) in which 22 071 male US physicians were the subjects in a factorial study designed to look at the possible prophylactic effects of aspirin... 

In the first edition of this book, I wrote of response surface designs as applied in clinical research that the field is still in its infancy . In the second edition I am tempted to report, flippantly, that the infant in question does not seem to have grown much. However, an interesting paper that reports an actual clinical trial is that by Tangen and Koch (2001). They consider an example of a 3 x 4 factorial trial in hypertension in which on average 43 patients were randomized to each of 12 possible combinations of three levels of hydrochlorothiazide (placebo, low, high) and four levels of an ACE inhibitor. (For a remarkably similar trial reported in the medical literature, see the 3x4 factorial trial of felodipine and ramipril in 507 patients reported by Scholze et al. (1999).)... 

Other experimental designs have been developed and involve more than two levels, and also involve the use of replicates used to evaluate the statistical error. Fractional factorial experimental designs use part of the full factorial experimental design and cut down the number of trials needed. 

It was decided that four parameters or factors would cover the range of service conditions. Upper dwell time and temperatures were tested at three different values, lower dwell time at two different values and with two different environments. The number of combinations of conditions is therefore 36. A quarter fractional nine trial design necessarily has some confounding and loss of information compared to a full factorial design, but retains the analysis of main effects and some synergies or interactions between factors. 

Labelling die variables as T, S and M, widi reference symbols, a, b, c respectively in trials involving a low (1) and high (h) levels of each variable, each trial having die result jc, die factorial procedure would produce die design code as follows ... 

In the Multicenter Acute Stroke Trial Italy (MAST-I) study, 622 patients were randomized in a 2 X 2 factorial design to receive either a 1-hour infusion of 1.5 lU streptokinase or 300 mg aspirin or both, or neither. Streptokinase (alone or with aspirin) was associated with a greater number of fatahties at 10 days (OR 2.7,95% Cl 1.7. 3). In MAST-I, neither aspirin monotherapy nor combination therapy reduced the primary outcome of combined 6-month fatahty and severe disability. 

This is a novel feature of factorial design when compared with the classical laboratory procedure which excludes indications of the interaction of the variable. The method of analysis of the data, due to Yates, which is commonly used to evaluate these effects, requires that the trials are conducted in the sequence shown above, and proceeds as follows. 

DOE is a methodical statistics approach to studying the qualitative effects of process variables. Variables of interest are given a number of values based on the expected relationships [8]. For example, if the relationship is expected to be linear over a range, two variations can be used to approximate the effect of the variable. For effects that are expected to be quadratic, three variations may be needed. These variations are then matrixed to create a set of trials that differentiate and quantify the effect of each variable. If the number of variables is small, then the experiments can be designed as a full factorial. An example of a full two factorial design of an experiment for three variables is shown in Table 15.1. 

A full three factorial matrix on the 11 variables in the cure cycle shown in Figure 15.1 would mean 177,147 individual trials. A full two factorial design would still mean 2048 trials. Such a design, however, assumes that all interactions, even between all 11 variables, will be important. DOE provides an ordered means of combining variables to reduce the total number of trials. The assumption made is that high-order interactions (i.e., interactions of three or more variables) are rare and/or insignificant. There are several methods for combining variables by DOE. A detailed discussion of these methods is the subject of another book [9]. 

Factorial design of experiments, combined with statistical methods of data analysis, offers wider and more differentiated information on the system, while conclusions are of greater usability. The results of all the eight runs in the analyzed example serve for determining the factor effects, with seven trials being independent possibilities of testing the effects and one serving for their comparison with the chosen fixed values. Three out of seven independently determined factor effects serve for... 

Table 2.102 Fractional factorial designs Design Trials Effects... 

Based on previous testing of the research subject, the design of the full factorial experiment 23 with one replication to determine experimental error has been chosen. To eliminate the influence of systematic error in doing the experiment, the sequence of doing design point-trials, in accord with theory of design of experiments, has been completely random. The outcomes are given in Table 2.107. 

To obtain the mathematical model of the process, 1/4-replica of a full factorial experiment of type 2s has been realized. Design points-trials have been done in a completely random order. The Table 2.129 shows conditions and outcomes of doing a 26 2 fractional factorial experiment. 

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