Factorial designs effects from

Factorial designs are especially well suited for blocking. When a factorial design is broken up into blocks by fractionalization, the block effect must be assigned to, or confounded with, one of the effects normally obtained from the model. In practice, the interaction of least concern (usually the highest-order interaction) is sacrificed. 

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. 

AC and BC) and one three-factor interaction (ABC). From the 2 full factorial design shown in Table 3.5 these seven effects can be calculated. An eighth statistic that can be obtained from this design is the mean result. From a 2 full factorial design therefore 2 statistics can be calculated. The... 

If systematic errors due to drift are expected then one can perform the design in a well defined randomized way so that the calculated main effects are not biased by the drift [36]. These designs are called anti-drift designs and they are described for full and fractional factorial designs. However, the interactions effects calculated from these designs are still biased by the drift. 

Using the effects of multiple-factor interactions from full or fractional factorial designs. Multiple-factor interactions (e.g. three- and four-factor interactions) are considered to have a negligible effect. It is then considered that these higher-order interaction effects measure differences arising from experimental error [31]. 

The error term can be approximated in different ways. A first possibility is that, analogous to the above, it is estimated from the multiple-factor interactions (two-, three-factor interactions, etc.) for (fractional) factorial designs [29]. In the example of Table 3.19 the sums of squares of the interactions AB, AC, BC and ABC are summed giving a MS error with 4 degrees of freedom. From this iSmain effects. The ANOVA table and equation (20) give of course the same results. 

The number of experiments required to perform a full factorial design increases dramatically with the number of factors. Consider a two level design for 7 factors the full design requires 128 experiments. From which 128 statistics can be measured to estimate the effects shown in Table 5.7. 

NUMBER OF EFFECTS CALCULATED FROM A FULL FACTORIAL DESIGN FOR SEVEN FACTORS... 

From a minimum number of experiments, the Hadamard matrix gives the possibility of estimating the mean effects of four parameters. Among them, the particle size range had the most important effect in the release of diclofenac sodium. By interpreting data, a factorial design including only two parameters was applied from which an optimum formulation was found. 

Here is an example from my research group s work on biosensors. The sensor has an electrode modified with a peptide that binds a target metal ion, which is then electrochemically reduced, the current being proportional to the concentration of metal. The electrode is calibrated with solutions of known concentration. Two experimental designs were used. The first was a two-level factorial design to find the effects of temperature, pH, added salt, and accumulation time. Spreadsheet 3.1 shows the design. 

Three-level fractional factorial designs are also very useful, and charting the effects can be very helpful especially where there are more than three factors. The Plackett-Burman designs are often used to confirm (or otherwise ) the robustness of a method from the set value. Figure 17 shows some results from a ruggedness study for an HPLC method for salbutamol where the resolution factor, between it and its main degradation product is critical. 

From preliminary assays, the experimental error was estimated as 2.50%, expressed as percentage recovery. Note that the complete factorial design is a 2 , requiring 128 runs, whereas the Plackett-Burman design needs only 8 runs to estimate the effects. The responses to the 8 runs corresponding to the design matrix in Table 2.6 were as follows ... 

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. 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

TABLE 4 Influence of DOC0, Ti02 Concentration, and Temperature on the Photocatalyzed Oxidative Degradation of Waste Water Pollutants in a Pilot Reactor Coefficients (Main Effects and Interactions) Calculated from the Experimental Results of a 23 Factorial Design (Table 3)... 

This model gives estimates of an offset term (/ 0), a first-order effect (/ ,) of the first factor jcj, a first-order effect (/32) of the second factor x2, and a second-order interaction effect (/312) between the two factors. When the model of Equation 11.15 is fit to data from a 22 factorial design, the number of factor combinations (/ = 4) is equal to the number of parameters (p — 4), and the number of degrees of freedom... 

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