Two-factor interactions

Can the relationship be approximated by an equation involving linear terms for the quantitative independent variables and two-factor interaction terms only or is a more complex model, involving quadratic and perhaps even multifactor interaction terms, necessary As indicated, a more sophisticated statistical model may be required to describe relationships adequately over a relatively large experimental range than over a limited range. A linear relationship may thus be appropriate over a narrow range, but not over a wide one. The more complex the assumed model, the more mns are usually required to estimate model terms. 

The two principal factors governing SCC are tensile stresses and exposure to a specific corrodent. These two factors interact synergistically to produce cracking. Only one factor needs to be removed or sufficiently diminished to prevent cracking. 

The factorial approach to the design of experiments allows all the tests involving several factors to be combined in the calculation of the main effects and their interactions. For a 23 design, there are 3 main effects, 3 two-factor interactions, and 1 three-factor interaction. Yates algorithm can be used to determine the main effects and their interactions (17). The data can also be represented as a multiple linear regression model... 

The two-factor interaction effects and the dummy factor effects in FF and PB designs, respectively, are often considered negligible in robustness testing. Since the estimates for those effects are then caused by method variability and thus by experimental error, they can be used in the statistical analysis of the effects. Requirement is that enough two-factor interaction or dummy factor effects (>3) can be estimated to allow a proper error estimate (see Section VII.B.2.(b)). 

Another way to estimate (SE)e is using effects that are a priori considered negligible, such as two-factor interaction effects and dummy factor effects " in EE and PB designs, respectively (Equation (8)). Such effects are considered solely due to the experimental error of the method. ... 

In reference 69, results were analyzed by drawing response surfaces. However, the data set only allows obtaining flat or twisted surfaces because the factors were only examined at two levels. Curvature cannot be modeled. An alternative is to calculate main and interaction effects with Equation (3), and to interpret the estimated effects statistically, for instance, with error estimates from negligible effects (Equation (8)) or from the algorithm of Dong (Equations (9), (12), and (13)). Eor the error estimation from negligible effects, not only two-factor interactions but also three- and four-factor interactions could be used to calculate (SE)e. 

In references 71 and 72, SST limits are defined based on experience, and the examined responses should fall within these limits. The two papers do not provide much information concerning the robustness test performed. Therefore, it is not evident to comment on the analysis applied, or to suggest alternatives. In reference 73, a graphical analysis of the estimated effects by means of bar plots was performed. In reference 74, a statistical analysis was made in which an estimation of error based on negligible two-factor interaction effects was used to obtain the critical effects between levels [—1,0] and [0,4-1]. 

The single-factor second-order parameter estimates lie along the diagonal of the S matrix, and the two-factor interaction parameter estimates are divided in half on either side of the diagonal. 

The first column of Table 14.3 gives the response notation (or, equivalently, the factor combination). The next eight columns list the eight factor effects of the model the three main effects (A, B, and C), the three two-factor interactions (AB, AC, and BC), the single three-factor interaction (ABC), and the single offset term (MEAN, analogous to PJ in the equivalent linear model). 

In a similar way, the classical interaction effects AB, AC, BC, and ABC can be defined as the difference in average response between the experiments carried out at the high level of the interaction and the experiments carried out at the low level of the interaction. Again, the high level of an interaction is indicated by a plus sign in its column in Table 14.3 (either both of the individual factors are at a high level, or both of the individual factors are at a low level). The low level of a two-factor interaction is indicated by a minus sign in its column in Table 14.3 (one but not both of the individual factors is at a low level). Thus, the classical two-factor interaction effects are easily calculated ... 

Offset First-order effects Two-factor interactions Three-factor interaction Residual... 

Calculate the grand average (MEAN), the two classical main effects (A and B), and the single two-factor interaction (AB) for the two-factor two-level full factorial design shown in the square plot in Section 14.1. (Assume coded factor levels of -1 and +1). 

Suppose that the experimenter runs the 2 fractional factorial design shown in Table 2.3. With this design each main effect is aliased with the two-factor interaction composed of the other two factors that is, is aliased with x x, x is aliased with and x is aliased with x x. This can be verified by multiplying together the appropriate columns, as was done for x Xy... 

Some protection against the effect of biases in the estimation of the first-order coefficients can be obtained by running a resolution IV fractional factorial design. With such a design the two-factor interactions are aliased with other two-factor interactions and so would not bias the estimation of the first-order coefficients. In fact the main effects are aliased with three-factor interactions in a resolution IV design and so the first-order effects would be biased if there were third-order coefficients of the form xxx, in... 

If there are only p=2, or p=3 variables then a full factorial design is often feasible. However, the number of runs required becomes prohibitively large as the number of variables increases. For example, with p=5 variables, the second-order model requires the estimation of 21 coefficients the mean, five main effects, five pure quadratic terms, and ten two-factor interactions. The three-level full factorial design would require... 

One final point it should be noted that the experimenter is not constrained to use a resolution V design or to add star points for all of the factors. In particular, if it is believed that certain two-factor interactions... 

As was mentioned above for central composite designs, the experimenter can modify these designs if they believe that certain two-factor interactions can be assumed negligible. Box and Jones [20,21] show how this can be done to yield what they call a modified Box-Behnken design that requires fewer runs than the standard Box-Behnken design. 

Thus, not only will this design estimate all of the linear and quadratic terms and interactions between the design and the environmental variables, but it will also estimate all of the two-factor interactions among the design variables and among the environmental variables. It will accomplish this in only (26 + runs, compared with the 81 runs for the Taguchi design that yields less information. 

An alternative design would be to use a half-fraction of the design variables for each run of the chamber. Such a design, before randomization, is shown in Table 2.22. With this design the ABCDE five-factor interaction is confounded with the TxH whole-plot contrast. Under the assumption of negligible three-factor and higher-order interactions all main effects and two-factor interactions can be estimated as well as interactions between the design and the environmental variables. 

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,... 

To explain the concept of interaction, let us consider a two-factor interaction. This interaction occurs when the effect of the first variable obtained at the lowest level (-) of the second variable is different from the effect of the first factor at the highest level (+) of the second one. The effect of one variable is influenced by that of the other and therefore it is said that they interact . We will try to explain this with an example. Suppose that in Table 3.5 factor B is the HPLC column (level (+) = column K and level (-) = column L) and factor A is the pH of the mobile phase (level (+) = 5.2 and level (-) = 4.8). A two-factor interaction between the column manufacturer and the pH of the mobile phase occurs when the effect of the pH on the response (e.g. resolution) on column K is different from the effect of the pH on column L. The interaction is calculated as half the difference between the effect of the pH on column K and the effect of the pH on column L. The interaction is called in this example the pH by column interaction and is symbolised by pH x column or AxB or AB or BA. 

Let us now consider three-factor interactions (e.g. ABC in Table 3.5) to give a general idea how these and higher-order interaction effects (four-, five-factor interaction effects, etc.) are derived. A three-factor interaction means that a two-factor interaction effect is different at the two levels of the third factor. Two estimates for the AB interaction are available from the experiments, one for each level of factor C. The AB interaction effect is estimated once with C at level (+) (represented by Eab,C(+)) nd once... 

As is the case for the two-factor interactions, the three-factor interaction is also symmetric in all its variables the interaction effects ABC, ACB, BAC, CAB, BCA and CBA all give the same result. Higher-order effects are calculated by analogous reasoning. 

In terms of absolute size, main effects tend to be larger than two-factor interactions, which in turn tend to be larger than three-factor interactions, and so on. In the half-fraction factorial design of Table 3.9 the main effects are expected to be significantly larger than the three-factor interactions with which they are confounded. As a consequence it is supposed that the estimate for the main effect and the interaction together is an estimate for the main effect alone. 

The defining relations are then 1 = ABODE, 1 = ABCF and 1 = DEF. The resolution of the design is III since the smallest defining relation contains three terms. This means that certain main effects are confounded with two-factor interactions, e.g. D = EF = ABCE = ABCDF. 

In Table 3.16 the columns of contrast coefficients for the two-factor interactions are given. They were obtained using the above stated rules. The contrast coefficients for three- and higher-order interactions can be... 

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