Full factorial experiment

An important purpose of a designed experiment is to obtain information about interactions among the primary variables. This is accompbshed by varying factors simultaneously rather than one at a time. Thus in Figure 2, each of the two preparations would be mn at both low and high temperatures using, for example, a full factorial experiment. 

Three-Level, Full-Factorial Experiment Design, Interaction Model Between Pad A and Pad B... 

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. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

Adhesion on HLORIN -type fibers, has been studied as a function of five process factors. The names of factors, with their variation levels, are shown in Table 2.32. Matrix 23 of full factorial experiment has been used in constructing random balance matrix. The design matrix by the method of random balance with experimental results is shown in Table 2.33. Note that each design point was repeated 20 to 50 times due to high non reproducibility of the system. 

Assume that twelve factors, Xx to X12, should be screened. The random balance matrix will consist of two independent semi-replicas of a 26 full factorial experiment, with rows or design points that are randomly distributed. The 32 design points thus synthesized will start with the values taken from a normal population with the mean 100 and the standard deviation 60=2.0. The effects of factors have been introduced in the way that the following values were added to the best values of selected factors in the upper level (+) value -15 added to factor X7 value -12 added to factor X4 value +10 added to factor X10 and Xn value +8 added to factor X value +6 added to factor X5 and Xg value +4 added to factor X2 value -4 added to factor X9... 

Full Factorial Experiments and Fractional Factorial Experiments... 

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. 

The experiment has been done through the matrix of full factorial experiment 24, as shown in Table 2.109. Each trial has been done only once, with no replications. The sequence of doing trials has been completely random. 

A full factorial experiment has been done in a pilot-plant. The research included refinement of a product by steam distillation. Five factors have been analyzed, each one at two levels A concentration, B flow, C volume of solution, D mixing speed and E solvent and water ratio. Acidity of the product in each of 32 trials has been analyzed as the response. Outcomes in coded forms are shown in Table 2.112. 

Data from Table 2.112 have been analyzed by the Yates technique and outcomes are given in Table 2.113. The interesting thing in relation to the former example is that the mechanical method, which does not require knowledge of Eq. (2.67) has been demonstrated. Column (1) is obtained by adding up the response data pairs to the column and then by subtracting the data. For example, 19=9+10, 14=8+6,...,11= 5+6, 1=10-9, -2=6-8,...,1=6-5. As shown, differences are taken from the same data pairs but in this way the second data minus the first, the fourth minus the third and so on to the column end. Column (2) is obtained from the first column in the same way. Column (3) from (2), (4) from (3) and (5) from (4). This calculation is evidently repeated k times for a full factorial experiment of 2k. Column (5) gives... 

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. 

An increase in the number of replicated trials causes a decrease in reproducibility variance or experimental error as well as in the associated variances of regression coefficients. Design points-trials can be replicated in all points of the experiment or in some of them. An upgrade of the design of experiment may be realized by a shift from fractional to full factorial experiment, a switch to bigger replica (from 1/6 to 1 /2 replica), a switch to second-order design (when the optimum region is dose by), etc. 

A process having properties dependent on four factors has been tested. A full factorial experiment and optimization by the method of steepest ascent have brought about the experiment in factor space where only two factors are significant and where an inadequate linear model has been obtained. To analyze the given factor space in detail, a central composite rotatable design has been set up, as shown in Table 2.152. 

As said before, linear models are used to reach (move towards) optimum, so that the significance of regression coefficients is an assumption for successful application of the steepest ascent method. Linear models, therefore, include as many factors as possible, and full factorial experiments are even replicated with increased factor variation intervals. 

In Example 2.60, a full factorial experiment 23 (N=8) for studying the influence of three factors (k=3) on response has been analyzed. All trials have been replicated once (rt=2). After processing the outcomes, these regression coefficient values were... 

Equation (2.173) is used for rotatable designs of second and third order when trials are replicated only in null point. In the case of a full factorial experiment or regular fractional replicas, we use ... 

Full factorial experiment and regular fractional replicas Equal replication (2.169) and (2.128)... 

A case of application of fractional replica 27-3 of a full factorial experiment on studying adhesion of thermoplastic polymer and fiber has been analyzed earlier in Example 2.33. Tensile strength of adhesion has been measured as the system response. The experiment included seven factors, with the nature of fiber being a qualitative-categorical factor. The regression coefficient values and method of steepest ascent are shown Table 2.188. 

These three solutions are possible 1) end of research 2) switch to second-order design and 3) upgrading half-replica to a full factorial experiment. 

It is more convenient to upgrade the half-replica to a full factorial experiment and then apply the method of steepest ascent. The design matrix of FUFE 24 with outcomes of experiments is shown in Table 2.192. 

A 1 /4-replica of the full factorial experiment 26"2 has been selected for the basic experiment, with these defining contrasts ... 

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