One-factor-at-a-time

A one-factor-at-a-time optimization is consistent with a commonly held belief that to determine the influence of one factor it is necessary to hold constant all other factors. This is an effective, although not necessarily an efficient, experimental design when the factors are independent. Two factors are considered independent when changing the level of one factor does not influence the effect of changing the other factor s level. Table 14.1 provides an example of two independent factors. When factor B is held at level Bi, changing factor A from level Ai to level A2 increases the response from 40 to 80 thus, the change in response, AR, is... 

Example of a false optimum for a one-factor-at-a-time searching algorithm. 

Tor each of the following equations, determine the optimum response, using the one-factor-at-a-time searching algorithm. Begin the search at (0, 0) with factor A, and use a step size of 1 for both factors. The boundary conditions for each response surface are 0 < A < 10 and 0 < B < 10. Continue the search through as many cycles as necessary until the optimum response is found. Compare your optimum response for each equation with the true optimum. 

The proper location of data is also important in parameter-estimation situations. For the nitric oxide reduction reaction (K11), for example, the relative sizes of the three-dimensional confidence regions calculated after each observation are shown in Fig. 27. The size of the confidence region after 12 points taken according to a one factor at a time variation of hydrogen and nitric oxide partial pressures is seen to be equivalent the size of the region... 

When rerunning an assay while troubleshooting, change only one factor at a time. 

Secondly, with the OVAT approach the importance of interactions is not taken into account. An interaction between two factors is present when the effect of one factor depends on the level of another factor. Since only one factor at a time is varied, the presence or absence of interactions cannot be verified. However, this is not dramatic, since in robustness testing the interaction effects are considered negligible. The evaluation of such interactions is more important in method optimization. 

Empirical Modeling. The effect of process variables on the rate of depKJsition and properties of electrolessly depKJsited metals is usually studied by one-factor-at-a-time experiments (one-factor experiments are discussed further later in the book). In these experiments the effect of a single variable (factor), such as Xj, in the multivariable process with the response y, y = fixi, %2, X3,. .., x ), is studied by varying the value (level) of this variable while holding the values of the other independent variable fixed, y Any prediction (extrapolation) of the effect of a single variable on... 

In a full factorial design all combinations between the different factors and the different levels are made. Suppose one has three factors (A,B,C) which will be tested at two levels (- and +). The possible combinations of these factor levels are shown in Table 3.5. Eight combinations can be made. In general, the total number of experiments in a two-level full factorial design is equal to 2 with /being the number of factors. The advantage of the full factorial design compared to the one-factor-at-a-time procedure is that not only the effect of the factors A, B and C (main effects) on the response can be calculated but also the interaction effects of the factors. The interaction effects that can be considered here are three two-factor interactions (AB,... 

Fig. 11. The response surface of a two-factor system. The lines represent equi-response lines. Optimization by varying one factor at a time. From P. J. Golden and S. N. Deming, Laboratory Microcomputer 3, 44 (1984). Reproduced by permission of Science Technology Letters, England...

For determining the robustness of a method a number of parameters, such as extraction time, mobile-phase pH, mobile-phase composition, injection volume, source of column lots and/or suppliers, temperature, detection wavelength, and the flow rate, are varied within a realistic range and tlie quantitative influence of the variables is determined. If the influence of a parameter is within a previously specified tolerance, this parameter is said to be witliin the robustness range of the method. These method parameters may be evaluated one factor at a time or simultaneously as part of a factorial experiment. 

Common practice consists in investigating the influence of one experimental variable (hereafter we will refer to it as a factor while keeping other factors at a fixed value. Then, another factor is selected and modified to perform the next set of experiments, and so forth. This one-factor-at-a-time strategy has been shown to be inefficient and expensive it lacks the ability to detect the joint influence of two or more factors (z.e. it cannot address interactions) and often needs many experiments. An increase in efficiency can be achieved by studying several factors simultaneously and systematically by means of an appropriate type of experimental design. In such a way, the experiments will be able to detect the influence of each factor and also the influence of two or more factors because every observation gives information about all factors. 

V. Czitrom, One-factor-at-a-time versus designed experiments. Am. Stat., 53(2), 1999, 126-131. 

Consider a GC capillary column where the length, diameter and thickness of the film of the stationary phase could all be modified (one factor at a time and without adjustment of the apparatus physical characteristics, such as temperature and pressure, yet maintaining a flow such that the linear velocity of the gas remains the same). 

Experiments may be designed to investigate one factor at a time so that all other independent variable-factors are held constant. This is the so-called classical experimental design. A classical experiment means researching mutual relationships between variables of a system, under "specially adapted conditions... 

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). 

When two or more factors are shown to be apparently non-significant, do not remove them all simultaneously. Remove one factor at a time until all the remaining factors are statistically significant. 

---