Graphics explain variability

An analysis is conducted of the predicted values for each team member s factorial table to determine the main effects and interactions that would result if the predicted values were real data The interpretations of main effects and interactions in this setting are explained in simple computational terms by the statistician In addition, each team member s results are represented in the form of a hierarchical tree so that further relationships among the test variables and the dependent variable can be graphically Illustrated The team statistician then discusses the statistical analysis and the hierarchical tree representation with each team scientist ... 

The biplot in Fig. 37.3 has been constructed from the factor scores of the 12 compounds and the factor loadings of the five physicochemical and biological variables [42,43]. (The biplot graphic technique is explained in Section 31.2.) It is... 

In 1869, C. W. Blomstrand11 attempted to explain the molecular structure of these compounds by assuming the halogen to be bivalent. This would make potassium acid fluoride K—F—F—H. It is considered more probable that fluorine is uni-, ter-, quadri-, or septa-valent, in harmony with the corresponding variable valency of the other halogens. This would make the graphic formula of potassium acid fluoride, cryolite, etc. ... 

The sign in the present instance is negative if we regard q as the number of particles greater than a given diameter d. We are not concerned with the form of either q or F(d) for the time being. The technique involved requires only that we determine the size-distribution graphically. With sedimentation methods the form of frequency function is important mathematically only in so far as it explains the relationships of the variables measured. 

The idea that a unit operation could have two or more steady states for the same values of the input variables is not only confusing in practice but somewhat hard to understand conceptually. We will try to explain the situation, first in words and then graphically. The verbal explanation of multiplicity centers around two of the necessary conditions nonlinearity and process feedback. 

Significant effects, i.e. effects that are significantly larger than could be due to experimental variability, can be identified by means of both graphical and statistical methods. The graphical method that is used most often is the normal probability plot explained in the preceding section (Fig. 6.9). The statistical tests are often based on a /-test, where the test statistic can be written as... 

Population PK/PD data is multidimensional. In an analysis of PK data, the most obvious predictor we have is time. In an analysis of PD data, we have time and drug exposure as the fundamental independent variables. What should not be forgotten, however, is that there may be other potential predictors that can explain the observed variability (e.g., body weight, sex, age, and other covariates), some of which also vary with time. Again, we must use graphical methods that can accommodate this situation. 

Fig. 13. A graphical display of the correlation result for a series of different variable combinations. The numbered circles refer to the numbering in the top row in the table to the right. Each number is a correlation analysis and the small circles in the tables indicate which variables that have been used in respectively Set 1 and Set 2. The x and y axis is similar to the axis in the significance plot as explained for Fig. 8. (Figure drawn using Po Correlation)...

The first three principal components explain more than (93.14 and 91.1%) of the total variance of the system, utilizing lignin from sugarcane straw and bagasse, respectively. This means that the 390 variables for each temperature can be reduced to only three with more than 90% confidence level. Each spectrum can be reduced to a single point, as shown by PCI x PC2 and PCI x PC3 graphics (Figs. 7 and 8), where differences can be evaluated. 

In sttmmary, the quantitative description and prediction of entropic and enthalpic processes taking place within the polymer HPLC colurtm was so far not achieved. Still, their contribution to retention volume can be graphically visualized and qualitatively explained with help of the plot log M vsV. In contrast to convention of physical chemistry, according to the accepted habit of chromatography, the independent variable, logarithm of molar mass M is plotted on the axis of ordinates. 

Graphical representation and variable definition and codification. In this case, it is not necessary to have a graphical or schematic representation. As explained earlier, a schematic representation would not help us to develop the necessary equations. 

The results confirm that a structural time series model with explanatory and intervention variables is an appropriate tool for explaining the changes in the monthly number of fatalities in Poland for the period 1998-2012, in relation to economic factors such as the industrial production index and/or the unemployment rate. A prehminary graphical analysis was conducted, which confirmed that the correlation between the munber of fatalities and the industrial production index (and the imemployment rate respectively) was positive (and negative respectively) on average. Log-log and log-lin specifications were then tested for accounting for these correlations, and three models which confirm this average relation were finally retained as statistically satisfactory and interpretable. 

Wherever one wishes to explain one or several response variables Y by a linear model in a set of correlated X variables, PLS is the method of choice. The graphical presentation of the results and of diagnostics, as well as the rather natural assumptions underlying the method, facilitate its use and interpretation. Moderately nonlinear relationships can be handled by PLS and the X matrix extended with squares or cubes of the original X columns, or by linear PLS followed by a polynomial model in the PLS scores Alternatively, the combination of PLS and NN is useful, especially for more complicated types of nonlinearity. This type of data with many and correlated variables are very common in chemistry, and PLS is being used in all types of problems, from physical and inorganic chemistry to biochemistry, environmental chemistry, and molecular biology. Still the number of areas where the PLS is routinely used is fairly small, and mainly include the areas described below. 

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