Main effects model

We call these equations (or models), main effects models. In the next subsection we will be adding to the main effects (of treatment and the covariates), treatment-by-covariate interaction terms. 

Again let z = 0 for patients in the control group and z = 1 for patients in the test treatment group and assume that we have several covariates, say Xj, and X3. The main effects model looks at the dependence of pr(y= 1) on treatment and the covariates ... 

The method provides a model for the hazard function. As in Section 6.6, let z be an indicator variable for treatment taking the value one for patients in the active group and zero for patients in the control group and let Xj, X2, etc. denote the covariates. If we let t) denote the hazard rate as a function of t (time), the main effects model takes the form ... 

We analyzed the dyestuffs data using a log-linear main effects model for the dispersion. The dispersion effect of F is significant with a coefficient of 0.54 and a standard error of 0.19 and the effect of B is barely significant with a coefficient of —0.40 and a standard error of 0.19. If the CEF interaction is added to the dispersion model, it proves to have a significant effect with a coefficient of 0.43 and a standard error of 0.19. We wonder, though, whether many scientists would be prepared to adopt a model with a three-factor interaction affecting the variance. 

In problems of chemical or biological nature, it is more a rule than an exception that interactions between variables exist. Therefore, main effect models serve only as rough approximations, and are used typacally in cases with a very high number of variables. It is also quite often useful to try to model some transformation of the response variable. 

Classical statistical tests can be applied mainly to validate regression models that are linear with respect to the model parameters. The most common empirical models used in EXDE are linear models (main effect models), linear plus interactions models, and quadratic models. They all are linear with respect to the p>arameters. The most useful of these (in DOE context) are 1) t-tests for testing the significance of the individual terms of the model, 2) the lack-of-fit test for testing the model adequacy, and 3) outlier tests based on so-called externally studentized residuals, see e.g. (Neter et. al., 1996). 

Now, let us look at the results of the regression analyses of a linear plus interaction model (model 1), the same model without the agitation main effect (model 2), and the model with aeration only (model 3). The regression analyses are carried out using basic R and some additional DOE functions written by the author (these DOE functions, including the R-scripts of all examples of this chapter, are available from the author upon request). 

If the resolution is III, only a main effect model can be used... 

If the resolution is IV, a main effect model with half of all the pairwise interaction... 

It is the purpose of this section to review ways in which processes involving electrons are either explicitly accounted for in calculations on polymeric systems or in which a more or less rigorous abstraction from the electronic degrees of freedom into effective models of a coarser-grained nature is performed. The next level up from electrons is obviously atoms. Hence, this section deals mainly with the connection between quantum chemistry and atomistic (force field) simulations. Calculations which exclusively use quantum chemistry are not covered. This excludes, for example, all of the recent work on metallocene catalysis. 

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... 

Over the last decade, some research has indicated that (1) partition coefficients (i. e.,Kd) between solid and solution phase are not measured at true equilibrium [51,59-61], (2) the use of equilibrium rather than kinetic expressions for sorption in fate and effects models is questionable [22-24,60,61], and (3) sorption kinetics for some organic compounds are complex and poorly predictable [22 - 24,26]. This is mainly due to what has recently been discussed as slow sorp-tion/desorption of organic compounds to natural solid phase particles [107, 162-164,166-182]. The following is a summary of some important points supporting this hypothesis [1,66,67,170-183] ... 

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 Section 14.3, a coding of -1 and +1 gave linear model main effects (b, b, and fcj) that differed by a factor of Vi from the classical main effects (A, B, and C). If the coding had been -2 and +2 instead, by how much would they have differed ... 

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... 

To construct the central composite design to estimate the coefficients of the second-order model (equation (14)), usually a fractional factorial design of at least resolution V is used. In this case, if the model is valid, then all of the estimates of the main effect coefficients, p., and the interaction coefficients, p. are imbiased. An alternative to the central composite designs for estimating the coefficients of the second-order model are the Box-Behnken designs or the designs referenced in Section 2.2.5. 

It can be seen that this model contains all main effects, all quadratic terms in the design variables, all interactions among the design variables, and all interactions between the design and the environmental variables. An estimate of the pure experimental error can be obtained from the replication at the four center points. 

Going to the finer details the interaction energy does, for instance, depend on the d-band width, even in the simple Newns-Anderson model. The main effect is that the narrower the band the stronger the interaction. This is an additional reason why, in the calculations described in the previous section, the open surfaces have lower activation energies than the more close packed ones. The surface atoms in an open surface have a lower metal coordination number and since the band width is roughly proportional to the square root of the coordination number, the band width is smaller. 

When there are more than two factors, the possible interactions increase. With three factors there can he three, two-way interactions (1 with 2,1 with 3, and 2 with 3), and now one three-way interaction (a term in Xj X2 X3). The numbers and possible interactions build up like a binomial triangle (table 3.1). However, these higher order interactions are not likely to be significant, and the model can be made much simpler without loss of accuracy by ignoring them. The minimum number of experiments needed to establish the coefficients in equation 3.3 is the number of coefficients, and in this model there is a constant, k main effects and Vzk [k— 1) two-way interaction effects. 

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