All possible regressions

The analysis of a supersaturated design is usually conducted by using some type of sequential model-fitting procedure, such as stepwise regression. Abraham et al. (1999) and Holcomb et al. (2003) have studied the performance of analysis methods for supersaturated designs. Techniques such as stepwise model fitting and all-possible-regressions type methods may not always produce consistent and reliable results. Holcomb et al. (2003) showed that the performance of an analysis technique in terms of its type I and type II error rate can depend on several factors,... 

One way to identify important predictor variables in a multiple regression setting is to do all possible regressions and choose the model based on some criteria, usually the coefficient of determination, adjusted coefficient of determination, or Mallows Cp. With this approach, a few candidate models are identified and then further explored for residual analysis, collinearity diagnostics, leverage analysis, etc. While useful, this method is rarely seen in the literature and cannot be advocated because the method is a dummy-ing down of the modeling process—the method relies too much on blind usage of the computer to solve a problem that should be left up to the modeler to solve. 

Several model-building techniques were elaborated forward selection and backward elimination, both in stepwise manner, all possible regressions, etc. [8]. 

The implicit LS, ML and Constrained LS (CLS) estimation methods are now used to synthesize a systematic approach for the parameter estimation problem when no prior knowledge regarding the adequacy of the thermodynamic model is available. Given the availability of methods to estimate the interaction parameters in equations of state there is a need to follow a systematic and computationally efficient approach to deal with all possible cases that could be encountered during the regression of binary VLE data. The following step by step systematic approach is proposed (Englezos et al. 1993)... 

In this section we consider typical examples. They cover all possible cases that could be encountered during the regression of binary VLE data. Illustration of the methods is done with the Trebble-Bishnoi (Trebble and Bishnoi, 1988) EoS with quadratic mixing rules and temperature-independent interaction parameters. It is noted, however, that the methods are not restricted to any particular EoS/mixing rule. 

The denominator in Equation 4.100 is the (absolute) size of all Lasso regression parameters for a particular choice of AL (compare Equation 4.89), and the nominator describes the maximal possible (absolute) size of the Lasso regression parameters (in case there is no singularity problem this would correspond to the OLS solution). The optimal choice is at a fraction of 0.3 which corresponds to a MSEPCy of 63.4 and to a SEPCv of 7.7. 

The regression models of the partition coefficients of the sulphonamides investigated are listed in Table 7.6. The models are used to design response surfaces of the criteria presented in Section 7.2. Not all possible response surfaces are given since there are numerous possibilities. The most interesting response surfaces are shown and discussed. 

It is also possible with computers to perform many regressions sequentially and automatically. With less than 100 data points on a rapid computer, or less than 20 on a slow computer, it is possible to routinely regress all possible combinations of three wavelengths from a field of 100 or a field of 20, respectively, and select the best combination. Computer programs are also available for all... 

Polynomial regression of the relationship of A(518Oi ) versus elevation (z) measured in meters derived from modeling all possible modem starting T and RH pairs for values of A(518Op) between 0%c and -25%c results in a curve that is only slightly different from the equation reported by Currie et al. (2005) and used by Rowley and Currie (2006), as well as that of Rowley and Garzione (2007). The revision is Equation (5) ... 

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

The Bayesian approach is more than a tool for adjusting the results of the all subsets regression by adding appropriate effects to achieve effect heredity. Take, for example, the sixth model in Table 4 which consists of Al,Bl, AlDq, BlHl, BlHq, BqHq. The AlDq effect identified as part of this model does not appear in the best subsets of size 1-6 in Table 3. The Bayesian procedure has therefore discovered an additional possible subset of effects that describes the data. 

This matrix does not have the state of a diagonal matrix and consequently, all the regression coefficients are in mutual correlation. So we carmot develop a different significance test for each of the coefficients. From this point of view it is not possible to use the tj values given by relation (5.95) as the base of a procedure for the process factor arrangement ... 

Calibration models are developed to determine the relationship between calibration set spectra and the constituent value of interest for those samples. Calibration involves taking spectra from many samples varying over the measurement range and also measuring the desired parameters. A rugged chemometric model for a complex sample may require hundreds to thousands of samples taken from all possible situations, in and out of specification, that it may encounter. Samples selected for calibration must contain all of the variables affecting the chemical and physical properties of the samples to be analyzed. To characterize each source of variation, it is recommended that 15 to 20 samples be run for each variable. Application of a math treatment, such as second derivative, prepares the raw spectral data for use in a regression and subsequent development of a calibration equation. This type of treatment results in a data file that will yield more information more easily than a raw data file. 

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