Multiple regression equations

A good illustration is provided [53] by fluorenone (FL) in a group of n = 19 solvents, which possesses both donor and acceptor groups (see structural scheme of FL in Fig. 6), the multiple regression equations were obtained as ... 

Previous studies of the number of sperm men ejaculate have had medical origins and have almost universally involved samples collected via masturbation. The multiple regression equations that are the basis of the single variable analyses presented in this paper are given in Table 1. The variables of greatest interest to ethologists are discussed in later sec-... 

Table 1. Multiple regression equations for the number of sperm inseminated during copulation or ejaculated during masturbation. Also shown are the variables entered but subsequently removed from the equations because P>0.05. The equations are based on the total data sets with missing values for independent variables replaced by means as explained in Materials and Methods. Both equations are a very significant fit to the data (P<0.001) and all variables have P<0.05. However, because of pseudo-replication, the regression equations are a tool rather than ends in themselves...

The multiple regression analysis by four independent variables is shown in Fig. 12 (here, [0]= 1- [C] - [Cr] - [Ni], aand n are all dependent variables). The multiple regression equation is expressed as follows ... 

An excellent correlation coefficient, r a multiple regression equation is formed with the three... 

The experimental data in Figures 2 and 4 were used in multiple regression equations to predict foam capacity and stability of 2% to 30% suspensions adjusted to pH values of 1.5 to 11.5 (Figures 6 and 7). Observed and predicted data of the... 

Multiple regression analysis is a useful statistical tool for the prediction of the effect of pH, suspension percentage, and composition of soluble and insoluble fractions of oilseed vegetable protein products on foam properties. Similar studies were completed with emulsion properties of cottonseed and peanut seed protein products (23, 24, 29, 30, 31). As observed with the emulsion statistical studies, these regression equations are not optimal, and predicted values outside the range of the experimental data should be used only with caution. Extension of these studies to include nonlinear (curvilinear) multiple regression equations have proven useful in studies on the functionality of peanut seed products (33). 

When faced with any claim that a significant regression (or multiple regression) equation has been discovered, it is always worth asking how many potential predictors were initially considered. 

As with simple regression equations, extrap>olation of a multiple regression equation is unwise unless there is a very sound theoretical basis. It is good practice to quote the ranges of the independent variables that have occurred in the data used for calculating the regression equation, in order to discourage extrapolation. 

Baker (268) compared the responses of volunteers to EA 3580 administered as a dose per man or as a dose per unit of body weight. Four Indicators of effect were used accommodation for near vision, arm-hand steadiness, dynamic flexibility, and manual dexterity The general conclusion was that the use of the dose per unit of body weight may Increase variance, rather than control for extraneous sources of variation, when the purpose of a study is to establish the effects of a substance Itself The use of multiple-regression equations was suggested as an approach to the establishment of definitive Information on effects of chemicals. 

Partial Least Squares regression (PLS) is usually performed on a - data matrix to search for a correlation between the thousands of CoMFA descriptors and biological response. However, usually after - variable selection, the PLS model is transformed into and presented as a multiple regression equation to allow comparison with classical QSAR models. 

Table 2 shows typical data obtained when operating the engine both in dual-fuel mode and in diesel fuel mode. Again, the data are reduced to a more convenient form by multiple regression equations ... 

In another article, Wang et al. (1993) used the CVCC conductivity cell for measurement of a variety of molten cryolite melts with additives of aluminum fluoride, aluminum oxide, calcium fluoride, magnesium fluoride, and lithium fluoride. On the basis of the measured results, a multiple regression equation for the electrical conductivity of cryolite melts was derived. Influence of the bath composition on the electrical conductivity at different bath temperatures was discussed. A comparison of the measured results with the published electrical conductivity values for cryolite melts was made. The new regression equation can be used to calculate electrical conductivity of cryolite melts in modern industrial bath chemistry. 

An example of a Hansch analysis (see section III. B. 2.a) using MLR is a study on substituted tetrahydroisoquinolines with affinity for both phenylethanolamine iV-methyltrans-ferase (PNMT) and the a2-adreno-receptor " (see Figure 23.10). The multiple regression equations obtained were ... 

The chemistry of Mo makes a larger proportion of adsorbed molybdates soluble and available at alkaline pH than at acid pH. However, pH is only one of several factors that affect the availability of Mo. Assessment of Mo availability in an alkaline soil, as in any other soil, necessitates determination of the critical limits for Mo in soils and in plants based on specific soil-plant systems. The determination of the critical limits should make allowance for the dominant soil factors that affect availability and for the plant efficiency of absorption and utilization of Mo. Use of multiple-regression equations to account for the contributions of the individual factors (Sillanpaa, 1982) will make the critical limits more predictable. 

The molecular descriptors for a CoMFA analysis number in the hundreds or thousands, even for datasets of twenty or so compounds. A multiple regression equation cannot be fitted for such a dataset. In such cases. Partial Least Squares (PLS) is the appropriate method. PLS unravels the relationship between log (1/C) and molecular properties by extracting from the data matrix linear combinations (latent variables) of molecular properties that best explain log (1/C). Because the individual properties are correlated (for example, steric properties at adjacent lattice points), more than one contributes to each latent variable. The first latent variable extracted explains most of the variance in log (1/C) the second the next greatest degree of variance, etc. At each step iP and s are calculated to help one decide when enough variables have been extracted—the maximum number of extracted variables is found when extracting another does not decrease x substantially. Cross-validation, discussed in Section 3.5.3, is commonly used to decide how many latent variables are significant. For example, Table 3.5 summarizes the CoMFA PLS analysis... 

Biological activities of new compounds can be predicted by transforming the PLS result into a multiple regression equation e.g. [608, 1010]). For a comparison... 

Several Taft values can be used in one regression equation, as was done by Kutter and Hansch (1969) who studied the preventive effect of substituted benzoic acids on the combination of ovalbumin antigen with its antibody, each benzoic acid being incorporated as a hapten in the antigen. They obtained this multiple regression equation ... 

Another use of rho is to convert sigma values, derived from benzoic acid, for use in other nuclei. This was first achieved for naphthalene the results were then extended to quinoline, and then to other heterocycles [Perrin, 1965c Perrin, Dempsey and Serjeant, 1981 (their Section 7.2)]. Although rho continues to appear in formal statements of Hansch s multiple regression equation, it is usually only indirectly present, namely as a nucleus-modified sigma value. 

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