Second-order interaction models

In the second-order interaction model cross-product terms have also been included. A significant estimated cross-product coefficient, bVj, shows that the influence of variable X is dependent on the settings of variable xp i.e. there is an interaction effect between these variables. Geometrically this corresponds to a twist of the response surface over the xi x3 plane. 

If also interaction effects are to be evaluated, second-order interaction models can be determined from a limited number of runs four variables in (3/4 x24) twelve runs (constant term, four linear coefficients, and six cross-product coefficients) five variables in sixteen runs (constant term, five linear coefficients, and ten cross-product coefficients). 

The second-order interaction model obtained from the experiments was... 

As a generally accepted reaction mechanism has not yet been established it is not possible to exclude any interactions among the variables. A second-order interaction model was therefore indicated and a fractional factorial design, 2s 1, was employed for obtaining estimates of the model parameters. The design is shown in Table 3. The scaling of the variables is given in a note to Table 3. 

Fig. 10. Normal probability plot of estimated model parameters after fitting a second-order interaction model to the score vector of the pyridines...

Pig. 3.4 A second-order interaction model can describe interactions between tbe experimental variables. The surfece is a twisted plane. The twist is described by the cross-product term. 

The parameters should measure the influence of the corresponding variables, i.e. the slopes of the surface, flj, as measures of linear dependencies of the variables the twists of the surface, By, as measures of interactions between variables, and the curvatures, fly, as measures or non-linear influences of the variables, and nothing else This calls for careful spacing of the variable settings in the experimental domain to determine the series of experiments used to estimate the parameters, i.e. the experimental design. These aspects will be treated in detail in the following chapters Chapters 5-7, which deal with screening experiments based on linear and second order interaction models, and Chapter 12, which describes quadratic models for optimization. 

However, this was not the only objective. It was also desired to determine the influence of the variables, as well as their interactions. Thus, a second order interaction model was assumed to give a satisfactory description, i.e. 

Full factorial designs Such designs are the best choice when the number of variables is four, or less. A full four-variable factorial design gives estimates of all main effects and two-variable interaction effects, and also an estimate of the experimental enor variance. This is obtained firom the residual sum of squares after a least squares fit of a second-order interaction model, see (Example Catalytic hydrogenation, p. 112). A full factoral design should be used if individual estimates of the interaction effects are desired. Otherwise, it is recommended first to run a half fraction 2 " (I = 1234), and then run the complementary fraction, if necessary, (see Example Synthesis of a semicarbazide, p. 135). 

To establish a complete second order interaction model, a Resolution V (or higher) design must be used. Such designs can be constructed by using four-variable interaction columns to define the "extra" variables, e.g, 2 (I = 12345), 2 (I = 12346 = 12357 = 12458 = = 13459 = 2345 10)... 

At the outset of an experimental study, the shape of the response surface is not known. A quadratic model will be necessary only if the response surface is curved. It was discussed in Chapters 5 and 6 how linear and second-order interaction models can be established from factorial and fractional factorial designs, and how such models might be useful in screening experiments. However, these models cannot describe the curvatures of the surface, and should there be indications of curvature, it would be convenient if a complementary set of experiments could be run by which an interaction model could be augmented with squared terms. 

Estimate ofthe experimental error Replication of the center point experiment gives an independent estimate, s of the experimental error variance, o, which can be used to asses the significance of the model. It can be used to evaluate the lack of fit by comparison with the residual mean square, as weU as to assess the significance of the individual terms in the model. (2) Check of curvature If a linear or a second-order interaction model is adequate, the constant term ho will correspond to the expected response, y(0), at the center point. If the difference y(0) - should be significantly greater than the standard deviation of the experimental error as determined by the r-statistic... 

When linear and second order interaction models are determined from a two-level design, the estimate of the constant term will be an alias in which the "true" intercept fig is confounded with all square coefficients. 

Run experiments by a factorial or a fractional factorial design and fit a second order interaction model. 

A least squares fit of a second order interaction model (without the constant term, as the data had been centred prior to computing the PC model) to the score values afforded the estimated coefficients shown in Table 17.4. 

To reduce the computational cost, a POSSIM (polarizable simulations with second-order interaction model) force field was later proposed, in which the calculation of induced dipoles stops after one iteration. The... 

G. A. Kaminski, S. Y. Ponomarev, and A. B. Liu, /. Chem. Theory Comput., 5(11), 2935-2943 (2009). Polarizable Simulations with Second Order Interaction Model - Force Field and Software for Fast Polarizable Calculations Parameters for Small Model Systems and Free Energy Calculations. 

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