Factorial design treatments

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,... 

Historically, factorial designs were introduced by Sir R. A. Fisher to counter the then prevalent idea that if one were to discover the effect of a factor, all other factors must be held constant and only the factor of interest could be varied. Fisher showed that all factors of interest could be varied simultaneously, and the individual factor effects and their interactions could be estimated by proper mathematical treatment. The Yates algorithm and its variations are often used to obtain these estimates, but the use of least squares fitting of linear models gives essentially identical results. 

Classical mathematical treatment of a full 2 factorial design... 

The saturated fractional factorial designs are satisfactory for exactly 3, or 7, or 15, or 31, or 63, or 127 factors, but if the number of factors is different from these, so-called dummy factors can be added to bring the number of factors up to the next largest saturated fractional factorial design. A dummy factor doesn t really exist, but the experimental design and data treatment are allowed to think it exists. At the end of the data treatment, dummy factors should have very small factor effects that express the noise in the data. If the dummy factors have big effects, it usually indicates that the assumption of first-order behavior without interactions or curvature was wrong that is, there is significant lack of fit. 

Factorial designs, in which n chemicals are tested at x dose levels (x treatment groups) have been suggested by the US-EPA (US-EPA 1986) as a statistical approach for risk assessment of chemical mixmres. A 2 factorial design has been used to describe interactions between the carcinogenic activity of five polycyclic aromatic hydrocarbons at two dose levels (Nesnow 1994) and a 5 design to identify nonadditive effects of three chemicals on developmental toxicity at five dose levels (Narotsky et al. 1995). 

When two or more different treatments are used concomitantly, it may be of interest to study in the same experiment both the effects due to each treatment (main effects) and the possible interaction between them (interaction effect). Interaction here means that the effect of one treatment is modified (e.g., decreased or enhanced) by the coadministration of the other treatment. In these cases, a factorial design can be considered. This design has been used, for instance to study the effects of clomidine and methylphenidate administered separately or in combination to children with ADHD and tic disorders (Tourette s Syndrome Study Group 2002). The simplest form is the 2x2... 

Alias in a partial factorial design, certain treatments cannot be distinguished. These ate known as aliases ... 

The most common types of factorial designs are those that have all factors at two levels, let us call them high and low . The number of different treatment combinations or experiments is 2 and if k replicates are performed at each treatment combination, the total number of runs is N = k x To study two-level factorial designs, the levels can be coded — 1 (low level) and + 1 (high level). 

The "design matrix" of a factorial design is a list detailing the total number of treatments, combinations or experiments. Columns represent each of the factors being studied, denoted by capital letters, and each row corresponds to an experiment. Hence a 2 factorial design would have 4 runs with the following design matrix ... 

A "matrix design" is a suitable way of obtaining all the treatment combinations implicated in a 2 factorial design, but it is not a handy system to notate... 

A fraction al factorial design was selected for the study of the five experimental factors retained. The responses measured were the total peaks area and the number of peaks, obtained with an ENICA 10 integrator (Delsi Instruments, Paris, France). The computerized treatment of the results (15) allowed us to quantify the main effect and interaction effects of the five factors under study and subsequently to determine the optimum conditions of analysis (Talou, T. Chromatographia, in press.). 

Steven Gilmour is Professor of Statistics in the School of Mathematical Sciences at Queen Mary, University of London. His interests are in the design and analysis of experiments with complex treatment structures, including supersaturated designs, fractional factorial designs, response surface methodology, nonlinear models, and random treatment effects. 

Enamines can be reduced to the corresponding saturated amines by treatment with formic acid. A very simple experimental procedure can be used in which formic acid is added to the neat enamine at such a rate that foaming due to evolution of carbon dioxide can be kept under control. The reduction of the morpholine enamine from camphor was studied in a two-level factorial design in order to determine whether or not an excess of formic acid should be used and at which temperature the reaction should be run. 

An example of a complete factorial design of 2 = 16 treatments, together with... 

In the factorial design, two or more treatments are evaluated simultaneously through the use of varying combinations of the treatments. The simplest example is the 2x2 factorial design in which subjects are randomly allocated to one of four possible combinations of two treatments A and B. These are A alone, B alone, A + B, neither A nor B (placebo). The main uses of the factorial design are to ... 

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