All-possible combinations of factors

There are several ways we can expand a design such as this we can increase the number of factors, the number of levels of each factor, or we can do both, of course. There are other differences than can be superimposed over the basic idea of the simple, all-possible combinations of factors, such as to consider the effect of whether we can control the levels of the factors (if we can then do things that are not possible to do if we cannot control the levels of the factors), whether the levels correspond to physical characteristics that can be evaluated and the values described have real physical meaning (temperature, for example, has real physical meaning, while catalyst type does not, even though different catalysts in an experiment may all have different degrees of effectiveness, and reproducibly so). 

A three-level full factorial design contains all possible combinations of / factors at their 3 levels (-1,0, + 1). In total, f experiments will be required to examine the /factors (9 experiments for 2 factors 27 for 3 factors, etc.). 

The two-level factorial design method is very convenient in screening for the important variables affecting a measured quantity, and assessing their relative importance [10]. In this method a number of variables or factors are chosen, and a range of values selected for each. Only the extreme limits of the ranges are used (denoted as + and - values). The experiments or simulations are run for all possible combinations of factor values (2 experiments for n factors) and the values of quantities of interest, the responses, calculated. 

Factorial experiment an experiment designed to examine the effect of two or more factors, each applied at least at two levels of operation. The full factorial investigates all possible combinations of these factors at the indicated levels ... 

The Ufson-Roig matrix theory of the helix-coil transition In polyglycine is extended to situations where side-chain interactions (hydrophobic bonds) are present both In the helix and in the random coil. It is shown that the conditional probabilities of the occurrence of any number and size of hydrophobic pockets In the random coil can be adequately described by a 2x2 matrix. This is combined with the Ufson-Roig 3x3 matrix to produce a 4 x 4 matrix which represents all possible combinations of any amount and size sequence of a-helix with random coil containing all possible types of hydrophobic pockets In molecules of any given chain length. The total set of rules is 11) a state h preceded and followed by states h contributes a factor wo to the partition function 12) a state h preceded and followed by states c contributes a factor v to the partition function (3) a state h preceded or followed by one state c contributes a factor v to the partition function 14) a state c contributes a factor u to the partition function IS) a state d preceded by a state other than d contributes a factor s to the partition function 16) a state d preceded by a state d contributes a factor r to the partition function. 

Very often it is sufficient to start with plans where only the lowest and the highest levels of each factor are considered. In this case we have two levels and n factors, which gives rise to a complete design of type 2". If all possible combinations of the two factors in two levels each are to be performed, a total of k = 2" experiments results. 

Our experiment will then investigate all possible combinations of these two factors, i.e. 5 x 2 = 10 combinations. When we use all combinations of two (or more) factors, it is called a full factorial experiment . We used five replicates of each combination and the results are shown in Table 13.7. 

All possible combinations of the two factors have been studied (full factorial... 

Extending this to eight experiments provides estimates of all interaction terms. When represented by a cube, these experiments are placed on the eight corners, and are consist of a full factorial design. All possible combinations of +1 and — 1 for the three factors are observed. 

A full factorial design is the experimental set-up that contains all possible combinations of variables and levels. The number of experiments (A) in a two-level full factorial design is 2 with / the number of factors considered. The design is also called a 2 design. 

However, equation (49) shows that weight factor must then be assigned to each experimental point (see equation 21) and the various expressions modified accordingly details can be obtained from standard works (e.g. Davies, 1961a). A further complication arises if Eij is calculated for all possible combinations of pairs of temperatures. For a reaction studied at n different temperatures, %( —1)/2 different... 

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