Variables correlation coefficients

On occasion you need to obtain correlation coefficients between two variables. Correlation coefficients are a way of measuring linear relationships between two variables. A correlation coefficient of 1 or -1 indicates a perfect linear relationship, and a coefficient of 0 indicates no strong linear relationship. Pearson correlation coefficients are useful for continuous variables, while Spearman correlation coefficients are useful for ordinal variables. For example, look at the following SAS code ... 

Some variables often have dependencies, such as reservoir porosity and permeability (a positive correlation) or the capital cost of a specific equipment item and its lifetime maintenance cost (a negative correlation). We can test the linear dependency of two variables (say x and y) by calculating the covariance between the two variables (o ) and the correlation coefficient (r) ... 

A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. 

By applying this method, they demonstrated that the removal of insignificant variables increases the quality and reliability of the models despite the fact that the correlation coefficient, r, always decreases, although only shghtly. For example, the characteristics of a model with six orthogonalized descriptors were r = 0.99288, s = 0.9062, F = 127.4 and the quality of this model was sufficiently improved after removal of the two least significant descriptors, to r = 0.9925, s = 0.8553,... 

The magnitude of dependencies in the variables is determined by the correlation coefficient. The correlation coefficient according to Pearson is given by Eq. (1). 

Correlation analysis reveals the interdependence between variables. The statistical measure for the interdependence is the correlation coefficient. 

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

The known models for describing retention factor in whole variable space ar e based on three-phase model and containing from three to six par ameters and variety combinations of two independent factors (micelle concentration, volume fraction of organic modifier). When the retention models are comparing or the accuracy of fitting establishing, the closeness of correlation coefficient to 1 and the sum of the squared residuals or the sum of absolute deviations and their relative values is taken into account. A number of problems ar e appear in this case ... 

Then vkt is calculated from the vX values as (-ln(l-vX)). The independent function Temperature vx is expressed as 1000 K/vT for the Arrhenius function. Finally the independent variable vy is calculated as In(vkt). Next a linear regression is executed and results are presented as y plotted against Xi The results of regression are printed next. The slope and intercept values are given as a, and b. The multiple correlation coefficient is given as c. 

Figure 2.15(a) shows the relationship between and Cp for the component characteristics analysed. Note, there are six points at q = 9, Cp = 0. The correlation coefficient, r, between two sets of variables is a measure of the degree of (linear) association. A correlation coefficient of 1 indicates that the association is deterministic. A negative value indicates an inverse relationship. The data points have a correlation coefficient, r = —0.984. It is evident that the component manufacturing variability risks analysis is satisfactorily modelling the occurrence of manufacturing variability for the components tested. 

Figure 4.8 Correlation coefficient, r, for several relationships between x and y variables...

Component assembly variability risk Component manufacturing variability risk Correlation coefficient Reliability... 

Correlation analysis quantifies the degree to which the value of one variable can be used to predict the value of another. The most frequently used method is the Pearson product-moment correlation coefficient. 

The correlation coefficient ranges between 1 and — 1. A perfect positive correlation has r=l, no correlation at all is r = 0, and a perfect negative correlation r= —1. Some examples of correlations are shown in Figure 11.5b. A measure of the significance of a relationship between two variables can be gained by calculating a value of t ... 

Correlation coefficient. In order to establish whether there is a linear relationship between two variables xx and the Pearson s correlation coefficient r is used. 

When comparisons are to be drawn among scales derived with different criteria of physical validity, we believe this point to be especially appropriate. The SD is the explicit variable in the least-squares procedure, after all, while the correlation coefficient is a derivative providing at best a non linear acceptability scale, with good and bad correlations often crowded in the range. 9-1.0. The present work further provides strong confirmation of this conclusion. 

The natural and correct form of the isokinetic relationship is eq. (13) or (13a). The plot, AH versus AG , has slope Pf(P - T), from which j3 is easily obtained. If a statistical treatment is needed, the common regression analysis can usually be recommended, with AG (or logK) as the independent and AH as the dependent variable, since errors in the former can be neglected. Then the overall fit is estimated by means of the correlation coefficient, and the standard deviation from the regression line reveals whether the correlation is fulfilled within the experimental errors. 

However, it is not proper to apply the regression analysis in the coordinates AH versus AS or AS versus AG , nor to draw lines in these coordinates. The reasons are the same as in Sec. IV.B., and the problem can likewise be treated as a coordinate transformation. Let us denote rcH as the correlation coefficient in the original (statistically correct) coordinates AH versus AG , in which sq and sh are the standard deviations of the two variables from their averages. After transformation to the coordinates TAS versus AG or AH versus TAS , the new correlation coefficients ros and rsH. respectively, are given by the following equations. (The constant T is without effect on the correlation coefficient.)... 

R r Multiple correlation coefficient. R indicates the percentage of the variability of the relative biological response that can be accounted for by the selected independent variables. 

For example, let us take a look at the data of Table 35.5a. This table shows two very simple data sets, X and Y, each containing only two variables. Is there a relationship between the two data sets Looking at the matrix of correlation coefficients (Table 35.5b) we find that the so-called intra-set (or within-set) correlations are strong ... 

The squared inter-set correlation coefficients vary from 0.21 to 0.38. Thus, only some 20% to 40% of the variance of the individual variables can be explained by one of the variables from the other data set. At a first glance these low inter-set correlations do not indicate a strong relation between the two data tables. In... 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

The first PROC CORR sends the Pearson correlation coefficients to a data set called pearson for the continuous variables Age and Weight, while the second PROC CORR sends the Spearman correlation coefficients to a data set called spearman for the categorical variables Race and Treatment Success. The correlation coefficients are found where the TYPE variable is equal to CORR in the pearson and spearman data sets. 

The correlation coefficient, which is a characteristic for the relationship between random variables, is not meaningful in calibration (Currie [1995] Danzer and Currie [1998]) and should, therefore, not be used to characterize the quality of calibration (instead of rxy the residual standard deviation sy.x should be applied see Eq. (6.19)). 

Stochastic relationship between random variables in such a way that one depends on the other. The degree of relationship may be estimated by the correlation coefficient. 

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