Linear correlation coefficients

Coefficient of variation (replicates at 50%) Pooled standard deviation (25 at 5 levels) Correlation coefficient (linear fit)... 

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) ... 

The presented algorithm was applied to 4 proteins (lysozyme, ribonuclease A, ovomucid and bovine pancreatic trypsin inhibitor) containing 51 titratable residues with experimentally known pKaS [32, 33]. Fig. 2 shows the correlation between the experimental and calculated pKaS. The linear correlation coefficient is r = 0.952 the slope of the line is A = 1.028 and the intercept is B = -0.104. This shows that the overall agreement between the experimental and predicted pKaS is good. 

A correlation coefficient of/ = +1 means that there is a completely positive linear relationship. High values on the y-axis are associated with high values on the y-axis. 

If the correlation coefficient is r = -1 high values on thejr-axis are associated with low values on the y-anis, The relationship is negatively linear. 

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. 

Once the descriptors have been computed, is necessary to decide which ones will be used. This is usually done by computing correlation coelficients. Correlation coelficients are a measure of how closely two values (descriptor and property) are related to one another by a linear relationship. If a descriptor has a correlation coefficient of 1, it describes the property exactly. A correlation coefficient of zero means the descriptor has no relevance. The descriptors with the largest correlation coefficients are used in the curve fit to create a property prediction equation. There is no rigorous way to determine how large a correlation coefficient is acceptable. 

In order to parameterize a QSAR equation, a quantihed activity for a set of compounds must be known. These are called lead compounds, at least in the pharmaceutical industry. Typically, test results are available for only a small number of compounds. Because of this, it can be difficult to choose a number of descriptors that will give useful results without htting to anomalies in the test set. Three to hve lead compounds per descriptor in the QSAR equation are normally considered an adequate number. If two descriptors are nearly col-linear with one another, then one should be omitted even though it may have a large correlation coefficient. 

The value of d obtained by linear regression is 0.96 with a correlation coefficient of 0.9985. For 2 alkylpyridines 8 is 2.030 (256), which leads to the conclusion that 2-alkylpyridines are twice as sensitive to steric effects as their thiazole analogs. 

Here again it is possible to find a linear relationship between the log (k/feo) (ko = methyl) values of 2-alkyl- and 2,4-dialkylthiazoles and between the latter value and Tafts Eg parameter (256). The value of 5 for 2,4-dialkylthiazoles is 1.472 with a correlation coefficient of 0.9994. Thus the sensitivity to substituent effects is more marked than in the case of a single substituent in the 2-position. Furthermore, the 4-position is again more sensitive than the 2-position. 

We thus get the values of a and h with maximum likelihood as well as the variances of a and h Using the value of yj for this a and h, we can also calculate the goodness of fit, P In addition, the linear correlation coefficient / is related by... 

Application of IP and NCS in conjunction with specification tolerance limits enables to substantiate acceptance criteria for linear regression metrological characteristics (residual standard deviation, correlation coefficient, y-intercept), accuracy and repeatability. Acceptance criteria for impurity influence (in spectrophotometric assay), solution stability and intermediate precision are substantiated as well. 

Liquid chromatography was performed on symmetry 5 p.m (100 X 4.6 mm i.d) column at 40°C. The mobile phase consisted of acetronitrile 0.043 M H PO (36 63, v/v) adjusted to pH 6.7 with 5 M NaOH and pumped at a flow rate of 1.2 ml/min. Detection of clarithromycin and azithromycin as an internal standard (I.S) was monitored on an electrochemical detector operated at a potential of 0.85 Volt. Each analysis required no longer than 14 min. Quantitation over the range of 0.05 - 5.0 p.g/ml was made by correlating peak area ratio of the dmg to that of the I.S versus concentration. A linear relationship was verified as indicated by a correlation coefficient, r, better than 0.999. 

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. 

In addition to analyzing the residuals, it may be desirable to determine the degree of agreement between sets of paired measurements and estimates. The linear correlation coefficient is... 

Comparisons (49) of measured concentrations of SFg tracer released from a 36-m stack, and those estimated by the PTMPT model for 133 data pairs over PasquiU stabilities varying from B through F, had a linear correlation coefficient of 0.81. Here 89% of the estimated values were within a factor of 3 of the measured concentrations. The calculations were most sensitive to the selection of stability class. Changing the stability classification by one varies the concentration by a factor of 2 to 4. 

This simply relates to how linear the relationship between the peak molecular weight of narrow polystyrene standards versus elution volume fits a straight line. This is typically measured with the linear correlation coefficient, r. ... 

The Hammett equation is said to be followed when a plot of log k against a is linear. Most workers take as the criterion of linearity the correlation coefficient r, which is required to be at least 0.95 and preferably above 0.98. A weakness of r as a statistical measure of goodness of fit is that r is a function of the slope p if the slope is zero, the correlation coefficient is zero. A slope of zero in an LEER is a chemically informative result, for it demonstrates an absence of a substituent... 

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. 

Once a linear relationship has been shown to have a high probability by the value of the correlation coefficient (r), then the best straight line through the data points has to be estimated. This can often be done by visual inspection of the calibration graph but in many cases it is far better practice to evaluate the best straight line by linear regression (the method of least squares). 

The possibility of an entropy-enthalpy relationship for the reaction was examined and found to give a correlation coefficient of only 0.727 which was however improved to 0.971 if only the external contributions to these parameters were used, i.e. these contributions arising from solvent interactions only. If compounds with substituents ortho to the amino group were excluded, this further improved to 0.996 and is likely therefore to be real [cf. the comments on p. 9). It was argued that the different amounts of desolvation of the aromatic on going to the transition state would depend upon the substituent, and that the resultant greater freedom for solvent molecules would mean decreased interaction energy or increased enthalpy so that the linear relationship follows. 

FIGURE 3.10. (a) Showing the relationship between the activation free energy Ag and the reaction free energy AG0 for the X + CH3Y- XCH3 + Y system. (6) The dependence of the "linear correlation coefficient 8 = d bg /d AG0 on AG0. 

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. 

Linear regression analysis was performed on the relation of G"(s) versus PICO abrasion index. Figure 16.10 plots the correlation coefficient as a function of strain employed in the measurement of loss modulus. The regression results show poor correlation at low strain with increasing correlations at higher strains. These correlations were performed on 189 data points. 

Inclusion of entries 1 and 2 into the initial T versus AE(r) linear regression analyses for reactions 1 and 4 did not appreciably affect the slopes, intercepts or correlation coefficients of the plots. 

The slopes, Y-intercepts and squares of correlation coefficients for the linear regression analyses of the T versus AE(ir) plots (equation 7) for reactions 1-4 for one-hour and ten-hour half life rates of decomposition to form free radical products are given in Table II. 

The best fits to the linear equation 8, for temperature differentials (from equation 7) versus reactant state steric effects, are obtained for reaction 4 (Table III). A modest correlation for equation 8 is obtained for reaction 1. Essentially no fit to equation 8 is found for reactions 2 and 3 (small correlation coefficients and small N slopes). 

If the T values of Table I are first fitted against EA values, without first fitting with EE(v) values, poor linea correlations result. For example, for tKe one-hour half-life temperatures of reactions 1 and 4, the squares of the correlation coefficients for these linear regression analyses are only 0.51 and 0.55, respectively. 

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