Pearson product moment correlation coefficient

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. 

In regression there is a dependence of one variable on another. In correlation we also consider the relationship between two variables, but neither is assumed to be functionally dependent on the other. The strength of the association or correlation between the variables is given by the correlation coefficient r, also known as the Pearson product-moment correlation coefficient -. 

For continuous data, the Pearson product moment correlation coefficient, r, is calculated. Since continuous data are used here, certain fundamental assumptions... 

Returns the Pearson product moment correlation coefficient between two data sets. 

Having outlined the random error components related to regression analysis, some comments on the correlation coefficient may be appropriate. The ordinary correlation coefficient p, also called the Pearson product moment correlation coefficient, is estimated as r from sums of squared deviations for xl and x2 values as follows using the same notation as above ... 

Correlation coefficients are used to look for relationships between two variables, and the most common correlation coefficient used is the Pearson product-moment correlation coefficient (r). When calculating correlation coefficients, the two variables must be at the interval or ratio level (2), which means that correlation coefficients cannot be used with category data that are dichotomous (mutually exclusive) and non-numerical (like animation/non-animation group, male/female, single/married/divorced, etc.). Values for the Pearson r vary from -1 to +1. Negative r-values imply negative correlations (as one variable increases, the other decreases) while positive r-values imply positive correlations (as one variable increases, so does the other and vice versa) r-values of 0 imply no relationship between the two variables. It is important to note that Pearson r-values assume linear relationships between the two variables if non-linear relationships are expected or observed, correlation ratios rj) that recognize non-linear relationships can be calculated (10). 

Correlation gives a quantitative measure of the relationship between two variables - the amount of variance from the common area between them. For data that are normally distributed, the Pearson product-moment correlation coefficient can be calculated by many commercial analysis packages (e.g. SAS, SPSS, MS Excel). The degree of correlation is indicated by a number between—1 and 1. A correlationofO indicates complete independence between the variables, and a correlation of 1 indicates a perfect increasing linear relationship. 

Common PMs include (1) the average (or mean) absolute error (or deviation), [210, - Dj ]/N, where the sum is over i and N is the number of cases (2) the average (or mean) squared error (sometimes called PRESS or SEC ), 2(0, -Di)VN (3) the root-mean-square error (RMSE), which most authors take as [2(0, - D,2 )/N]i/2 but which others take as [2(Oj - D,)2]t/2/>f. gnJ (4) Pearson product-moment correlation coefficient, or simply the correlation coefficient. This coefficient is defined as follows ... 

Choosing the best curve to fit to the data is as much art as science. Using rather complicated mathematics, one could produce an elaborate polynomial that comes as close to the observed values as desired, but such an equation rarely conveys any theoretical or qualitative understanding of the processes that cause growth. Nor is the well-known Pearson product-moment correlation coefficient of much use in choosing among the candidate... 

Fields can be utilized in virtual screening applications for assessing the similarity (alignment) or complementarity (docking) of molecules. Two similarity measures have achieved the most attention. These are the so-called Garbo- [195] and Hodgkin indexes [196] respectively. Others are Pearson s product moment correlation coefficient [169] and Spearman s rank correlation coefficient [169]. 

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 ... 

Pearson s product-moment correlation coefficient (r) is the most commonly used correlation coefficient. If both variables are normally distributed, then r can be used in statistical tests to test whether the degree of correlation is significant. If one or both variables are not normally distributed you can use Kendall s coefficient of rank correlation (t) or Spearman s coefficient of rank correlation (rs). They require that data are ranked separately and calculation can be complex if there are tied ranks. Spearman s coefficient is said to be better if there is uncertainty about the reliability of closely ranked data values. 

Pearson s product-moment correlation coefficient, often simply referred to as the correlation coefficient, r, has two interesting properties. First,... 

We make a relative analysis after the 2 round 20 classes undergraduate students data by square root inverse sine transfer with SAS9.13, the Pearson product moment correlation of coefficient r = 0.89394, P < 0.0001, which show that the 2 round survey result is high correlation in the first layers. With the same transfer of the 2 round 18 classes graduate students survey data, we get the relative analysis Pearson product moment correlation of coefficient r = 0.91645, P < 0.0001, which show that the survey result is high correlation of all the 38 classes, and our survey method have a high reliability. 

Thermal conductivity derived in Eq. 6.14 is plotted and compared with the experimental data of GSA-SDS/FMWNT composites as in Fig. 6.12. The predicted values of the composites differ by 0.003 0.002 W/m-K which is marginally small for a temperature profile from 290 to 370 K. Correlation coefficient is a numerical measure of the strength of the relationship between two random variables. The value of correlation coefficient varies from -1 to 1. A value close to +1 or -1 reveals the two variables are highly related. Pearson s product moment correlation coefficient measures the linear relations between two data sets and was determined to be 0.935 between the predicted model (Eq. 6.14) and the experimental data. 

Also known as the Pearson Correlation Coefficient, Pearson s r, or PMCC is the most widely used measure of the linear correlation between two variables. For a sample of pairs, (xj, y,), of observations or measurements of two variables, X and Y, Pearson s product-moment correlation coefficient is equal to the un-weighted covariance of the variable parrs divided by the product of the individual variable sample standard deviations. For N pairs, (xj, y,), with individual sample means, and Py, Pearson s product-moment correlation coefficient, r, is given by ... 

Organic-Inorganic Ion Relationships. Figures 1 and 2 summarize the Pearson product moments calculated for various formate-inorganic ion and acetate-inorganic ion pairings, respectively. The correlation coefficients were generally low (r values usually < 0.6) and... 

The correlation measures the relation between two or more variables and goes back to works performed in the late nineteenth century [48]. The most frequently used type of correlation is the product-moment correlation according to Pearson [49]. The Pearson correlation determines the extent to which values of two variables are linearly related to each other. The value of the correlation (i.e., the correlation coefficient) does not depend on the specific measurement units used. 

Correlation may be defined as a measure of the strength of association between two variables measured on a number of individuals, and is quantified using the Pearson product-moment coefficient of Unear correlation, usually known as the correlation coefficient. Thus the calculadon of the correlation coefficients between CaO and AI2O3 and K2O and Na20 can provide an answer to the questions asked above. 

The Pearson product-moment coefficient of linear correlation is based upon the following assumptions ... 

Sometimes geochemical data cannot strictly be used in product-moment correlation of the type described above for they do not fuliil the requisite conditions. For example, some populations are not normally distributed and others include oudiers. An alternative, therefore, to Pearson s product-moment coefficient of linear correlation is the Spearman rank coefficent of correlation, usually designated r. This type of correlation is applicable to major or trace element data measured on a ranking scale rather than the equidistant scale used in Pearson s product-moment correlation. The Spearman rank correlation coefficient is calculated as follows ... 

Cox and Clifford (1982) have proposed a way of presenting correlation coefficient data for a suite of rocks in a diagrammatic form. Their method, which is purely descriptive, uses the Pearson product-moment coefficient of correlation and is an attempt to utilize and display graphically the large amount of information contained in a correlation matrix, without resorting to plotting the enormous number of... 

Note that in data analysis we divide by n in the definition of standard deviation rather than by the factor n - 1 which is customary in statistical inference. Likewise we can relate the product-moment (or Pearson) coefficient of correlation r (Section 8.3.1) to the scalar product of the vectors (x - x) and (y - y) ... 

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