Obtaining 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  

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. 

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Gifford and Hanna tested their simple box model for particulate matter and sulfur dioxide predictions for annual or seasonal averages against diffusion-model predictions. Their conclusions are summarized in Table 5-3. The correlation coefficient of observed concentrations versus calculated concentrations is generally higher for the simple model than for the detailed model. Hanna calculated reactions over a 6-h period on September 30, 1%9, with his chemically reactive adaptation of the simple dispersion model. He obtained correlation coefficients of observed and calculated concentrations as follows nitric oxide, 0.97 nitrogen dioxide, 0.05 and rhc, 0.55. He found a correlation coefficient of 0.48 of observed ozone concentration with an ozone predictor derived from a simple model, but he pointed out that the local inverse wind speed had a correlation of 0.66 with ozone concentration. He derived a critical wind speed formula to define a speed below which ozone prediction will be a problem with the simple model. Further performance of the simple box model compared with more detailed models is discussed later. 

Fig. 18.5. Calibration curve obtained by the immunoassay procedure for the detection of anti-CTB using the ITO-PPB-coated optical fibers. The curve was fitted according to the equation y = A+B ln(.r), where x is the anti-CTB dilution value and y is the chemiluminescence response. The obtained correlation coefficient is R2 = 0.95.

Once the quality of the linear model has been evaluated and confirmed, a more advanced validation step is carried out termed leave-one-out cross-validation . Here, each data point (potency value) is removed once from the training set, the model is derived again for the remaining data points, and then applied to predict the value that was left out. The then obtained correlation coefficient is called cross-validated r2 or q2 ... 

Topliss was the first to investigate the risk of chance correlations in Hansch analyses in a systematic manner. Several simulations, using random numbers instead of real parameter values, revealed that for a given number of compounds the chance of obtaining correlation coefficients r larger than 0.9 not only drastically increases with the number of variables included in the equation, but also with the number of variables from which the different combinations are being selected [400, 401],... 

It is important to be consistent in the use of fugacity coefficients. When reducing experimental data to obtain activity coefficients, a particular method for calculating fugacity coefficients must be adopted. That same method must be employed when activity-coefficient correlations are used to generate vapor-liquid equilibria. 

Given that Wj. r) we finally obtain for the pair 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. 

Although the correlation coefficient r would easily be calculated with the aid of a modern calculator or computer package, the following example will show how the value of r can be obtained. 

An idea of the scale of turbulence can be obtained by measuring instantaneous values of velocities at two different points within the fluid and examining how the correlation coefficient for the two sets of values changes as the distance between the points is increased. 

Figure 5.69 Calibration curves obtained from (a) LC-ToF-MS and (b) LC-MS-MS using selected-reaction monitoring for Idoxifene in human plasma, fortified from 5 to 2000 ngml for LC-ToF-MS and 0.5 to 1000 ngml for LC-MS-MS with a triple quadrupole is the correlation coefficient, a measure of the quality of calibration (see...

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. 

It has already been shown how a correlation between road test ratings and laboratory abrasion can be obtained over a range of energies and speeds (Table 26.6). Usually, a good correlation, i.e., a high correlation coefficient and a regression coefficient near 1 are obtained only over a limited range. 

Lines in Figure 30.12 were drawn with parameters obtained when fitting data with Equation 30.3. It is fairly obvious that, outside the experimental window, data would not necessarily conform to such a simple model, which in addition cannot meet the inflection at 100% strain. All results were nevertheless fitted with the model essentially because correlation coefficient were excellent, thus meaning that the essential features of G versus strain dependence are conveniently captured through fit parameters. Furthermore any data can be recalculated with confidence within the experimental strain range with an implicit correction for experimental scatter. Results are given in Table 30.1 note that 1/A values are given instead of A. 

Data Analysis Because of the danger of false conclusions if only one or two parameters were evaluated, it was deemed better to correlate every parameter with all the others, and to assemble the results in a triangular matrix, so that trends would become more apparent. The program CORREL described in Section 5.2 retains the sign of the correlation coefficient (positive or negative slope) and combines this with a confidence level (probability p of obtaining such a correlation by chance alone). 

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). 

In QSAR equations, n is the number of data points, r is the correlation coefficient between observed values of the dependent and the values predicted from the equation, is the square of the correlation coefficient and represents the goodness of fit, is the cross-validated (a measure of the quality of the QSAR model), and s is the standard deviation. The cross-validated (q ) is obtained by using leave-one-out (LOO) procedure [33]. Q is the quality factor (quality ratio), where Q = r/s. Chance correlation, due to the excessive number of parameters (which increases the r and s values also), can. 

The study is based on four iinear hydrocarbons (in Ci, Ce to Ca) and the model uses Antoine and Clapeyron s equations. The flashpoints used by the author do not take into account all experimental values that are currently available the correlation coefficients obtained during multiple linear regression adjustments between experimental and estimated values are very bad (0.90 to 0.98 see the huge errors obtained from a correlation study concerning flashpoints for which the present writer still has a coefficient of 0.9966). The modei can be used if differences between pure cmpounds are still low regarding boiling and flashpoints. 

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