Pearson correlation coefficients

The Pearson Correlation Coefficient can be used to determine if a relationship exists between two variables—both of which are continuous in nature. Examples of safety performance data that can be analyzed can include the number of lost workdays for an injured employee and the years of experience on the job, the concentrations of hazardous materials in the air, and the number of visits to the nurses office, etc. In addition to the requirement for both variables to be continuous, the following additional 

Each variable is measured on either an interval scale or ratio scale. 

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FIGURE 3.3 Pearson correlation coefficients between fish (Gambusia) Hg concentration and MeHg concentrations in various environmental media sediment, porewater, surface water, and suspended particulate matter (SPM) from the Florida Everglades (1995-1998). 

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. 

Fig. 3. Scatterplots of the Nh residual dipolar couplings of A131A measured in 2, 4, 6, or 8 M urea (y-axes) plotted against the same couplings measured in the absence of urea (x-axis) ris the Pearson correlation coefficient. Alignment was achieved with alkyl PEG bicelles (Ackerman and Shortle, 2002.)...

One-way-ANOVA tests were made to control the quality of the data. To achieve the proposed objectives, multiple comparison of means of the four groups was made by Tukey (p<0,05) test. The Dunnett (p<0,05) test was used to compare the means of three groups of mining sites with the control group (Seaman et al. 1991). Pearson correlation coefficients were also obtained to confirm the tests results. 

The (Pearson) correlation coefficient The most common measure of dependency is the (Pearson) correlation coefficient. It holds the advantage that it is straightforward to estimate and distinguishes between positive and negative dependencies. The results obtained for the model of the... 

The classical correlation coefficient is the Pearson correlation coefficient, (rik, r) which is according to Equation 2.7 the sample covariance, standardized by the standard deviations, v and sk of the variables. 

The range of rjk is — 1 to +1 a value of +1 indicates a perfect linear relationship, a value of —1 indicates a perfect inverse linear relationship absolute values of approximately <0.3 indicate a poor or no linear relationship. The Pearson correlation coefficient is best suited for normally distributed variables however, it is very sensitive to outliers. This coefficient is the most used correlation measure as usual also throughout this book the term correlation coefficient will be used for the Pearson correlation coefficient. 

FIGURE 4.1 Univariate regression is not successful none of the single variables xt, x2, x3 (data set in Table 4.2) is useful to predict the property y. R2 is the squared Pearson correlation coefficient. 

FIGURE 4.2 Linear combinations of the x-variables (data set in Table 4.2) are useful for the prediction of property y. For the left plot, xh x2, and x3 have been used to create an OLS-model for y, Equation 4.2 for the right plot x and x2 have been used for the model, Equation 4.4. R2 is the squared Pearson correlation coefficient. Both models are very similar the noise variable x3 does not deteriorate the model. 

Correlation-. Select those 3, 5, 10, and 15 x-variables with the (absolute) highest Pearson correlation coefficient with the y-variable. 

Note Heating value in kJ/kg, others in mass %. The squared Pearson correlation coefficients, between experimental values and predicted values from leave-one-out CV and the standard error of prediction from leave-one-out CV (SEPCV, see Section 4.2.3) are given for a joint PLS2 model, and for separate PLS models developed for each variable seperately using the optimal number of components opt f°r each model. 

Pearson Correlation Coefficients between the Variables in the Cereal Data Set... 

PRESS Predicted residual error sum of squares (sum of squared prediction errors). rjk (Pearson) correlation coefficient between variables j and k r2 is the... 

In order to eliminate parameters that are correlated to each other, we calculate their Pearson correlation coefficients (25). Linearly uncorrelated parameters have Pearson correlation coefficients close to zero and likely describe different aspects of the phenotype under study (exception for non-linearly correlated parameters which cannot be scored using Pearson s coefficient). We have developed an R template in KNIME to calculate Pearson correlation coefficients between parameters. Redundant parameters that yield Pearson correlation coefficients above 0.4 are eliminated. It is important to visually inspect the structure of the data using scatter matrices. A Scatter Plot and a Scatter Matrix node from KNIME exist that allow color-coding the controls for ease of viewing. 

Lastly, it is desirable that parameters are able to discriminate between positive and negative conditions in a variety of experimental conditions. In other words they should be robust and reproducible. For this purpose, the Pearson correlation coefficient between all experimental repeats using control wells is calculated. Robust parameters have high Pearson correlation coefficients (above 0.7) in pairwise comparisons of experimental repeats. For this analysis we have developed another R template in KNIME to calculate the Pearson correlation coefficient between experimental runs. 

Textural and Rheological Characteristics of Raw and Cooked Potatoes 257 Table 9.4 Pearson correlation coefficients for rheological and selected textural properties... 

We compiled literature and our own extraction data to compare the distribution of the same solutes from water to [C4CiIm][PFg] and 48 various conventional solvents. As a measure of similarity of the extraction properties of any two solvents, we used the Pearson correlation coefficient between IgD for the same solutes. Note that a high correlation coefficient does not mean that the distribution ratios determined with the two solvents are close by absolute value rather, it means that the distribution ratios change in the same marmer from one solute to another. 

The results of the seven studies were analyzed using the statistical method of meta-analysis. Homogeneity of the studies was checked with an appropriate statistical test, while combined Pearson correlation coefficients were estimated with four different methods. The main findings were ... 

The squared Pearson correlation coefficients for the linear models built upon the Diaminopyrimidine, Pyrrolopyrazole, Pyrrolopyrimidine, and Quinazoline series across the 45 protein kinases are, respectively, in the range of = 0.82 0.95... 

The relationship between the two variables is usually characterised by the dimensionless Pearson correlation coefficient, R. A value of +l or — 1 for R indicates a strong relationship between the two variables. This approach assumes that the errors in y follow a normal distribution. R2 is called the determination coefficient and indicates what percentage of the variations in x overlap variations in y. 

Pearson correlation coefficients showed a relationship between color and extractable lipid. Lipid extraction was negatively correlated with brightness (r = -0.68,/ < 0.01), and yellowness (r = -0.67, p < 0.01), whereas positively correlated with redness (r = 0.40,/ < 0.01) color values. Thus, the extraction... 

FdUMP[10], a 10-mer of 5-fluoro-2 -deoxyuridine 5 -monophosphate (FdUMP), was found to be 338-fold more potent than 5-fluorouracil at inhibiting cell proliferation in the National Cancer Institute 60 cell-line screen. The 5-fluoro-2 -deoxyuridine derivative 75 (Figure 11) showed a Pearson correlation coefficient of 0.781 with regard to the other compounds with the highest correlation which ranked 237th among all the compounds/extracts deposited in the DTP database <2005MI4844>. 

E.g., the Galton-Pearson Correlation coefficient see G. F. Lipps, Bestimmung d. Abh ngigkeit zwischen der Merkmalen eines Gegenstandes, Ber. d. sdchs. Oes. d. Wise., 1905. Also the formal apparatus to describe extensive statistical ensembles, cf. H. Bruns, Wahrsch.-R. u. Kollektivmasslehre (Leipzig, 1906). Com-... 

Example 1 (continued). In the ecological study conducted by Kono et al., Pearson correlation coefficients were calculated for annual data on consumption of selected nutrients and food and colon cancer rate with a lag time of 20 years. A positive correlation was found between consumption of fat (r = 0.97 for men and 0.98 for women), meat (r = 0.98 for men and 0.97 for women) and alcohol (r = 0.96 for men and 0.98 for women) and cancer incidence similar results were found for cancer mortality. Negative correlations were found for consumption of miso and cereal. The high correlation found for consumption of alcohol and fat may be somewhat misleading because there were only small variations in consumption whereas there was a much larger variation in average intake of meat and cancer incidence. 

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