Straight correlation coefficient

Boque, R. Rius, F. X. Massart, D. L. Straight Line Calibration Something More Than Slopes, intercepts, and Correlation Coefficients, /. Chem. Educ. 1993, 70, 230-232. 

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. ... 

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 final key point to note about the MND, which can also be seen from Figure 1-2, is the fact that when the MND is projected onto the plane defined by any two axes of the coordinate system the data may show some correlation (as does the data in Figure 1-2). In fact, the projection onto any of the planes defined by two of the axes will have some value for the correlation coefficient between the corresponding pair of variables. The amount of correlation between projections along any pair of axis can vary from zero, in which case the data would lie in a circular blob, to unity, in which case the data would all lie exactly on a straight line. 

X-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 67-1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the X-data itself, which, as we said above, is the measure of nonlinearity. Table 67-1 also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

Using the data of figure 2.4, the plot of inp against 1/77 leads to a excellent straight line (inset of figure 2.4 correlation coefficient 0.9999) and allows the enthalpy of vaporization of ethanol to be derived as 41.8 kJ mol-1 from the slope of that line (-5024.27). 

It is important to know the effectiveness of the model for predicting values however, it is also important to know the strength of the linear relationship between the two variables (known and predicted) being studied. This is achieved using the linear correlation coefficient (Pearson s product moment correlation coefficient), r, as a descriptive measure for the strength of the linear relationship (straight line) between the two variables ... 

Visual inspection of a plot of response versus concentration will show a straight line. The correlation coefficient (r), residual sum of squares, and v-intercept should be reported. In the case of the dissolution profile for a modified release product, 20% of the stated range will need to be prepared. For example, a range of 0 to 110% of release will be required for a profile of 20 to 90% release. 

Figure 12.1 shows how well the predicted results agree with the experimental values listed in Table 12.2. The predictions consistently overestimate experimental dielectric constant values for this series of samples. The difference varies from 12 to 27% of the experimental values. Qualitatively, predictions follow the same trend as experiment. The calculated slope for the best straight-line fit through the data points is equal to 0.822 with a standard deviation of 0.088. The correlation coefficient is equal to 0.881. For several of the polyimides tested here the method... 

Try, in succession, the first-order, second-order, and third-order plots until a straight-line plot is obtained. It turns out that the first-order plot gives an excellent fit (correlation coefficient = 0.9999), so a = 1. The slope is -8.718 x 10 4min ,... 

The first-order plot gives an excellent straight line with correlation coefficient of 0.9999, so there is no Rgift in trj g p d-j jginfi i r jjJq r i h qg is -4.41 x 10-3min 1 they equal to -k/2.30, so... 

Series (1) tells us that the reaction is second-order with respect to NO, and series (2) that the reaction is first-order with respect to 02. The superiority of l/Pso versus t for series (1) is much more apparent from the graphs (the first-order plot has a very pronounced curve, whereas the second-order plot is extremely Straight) than from the correlation coefficients (0.9999 for the second versus 0.9908 for the first). We conclude, therefore, that we need the rate constant for the overall rate expression... 

When these data are plotted as in Figure 15-7,. you get a.straight line whose slope is -1 407 x 104 K and whose y lntercePt is 14.98, with a correlation coefficient of 0.9998. The slope is equal to -AHJ2.3R. Therefore,... 

The slope and the pA BH+ values obtained by application of equation 35 to spectral data for iV-benzoylthioLirca and six met a - and para-substituted V-benzoylthioureas are given in Table 24. A plot of these pA BH+ values vs a yields the straight line shown in Figure 17, the slope (Hammett p value) of which equals —0.42 (correlation coefficient 0.961). 

Like all other test statistics discussed in this book, there is a probability distribution associated with the correlation coefficient, and, as usual, the degrees of freedom associated with the test statistic need to be determined in order to assess whether or not the test statistic attains statistical significance. In this case, the value for the degrees of freedom is N minus 2. This value is related to the fact that any two points lie on a straight line, and a third is needed to determine whether or not the points all lie on the same line. Degrees of freedom are discussed in Section 7.6.1. 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

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