Product-moment correlation

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. 

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 ... 

The initial step in the analysis of the data generally requires the calculation of a function that can indicate the degrees of interrelationship that exist within the data. Functions exist that can provide this measure between either the variables when calculated over all of the samples or between the samples calculated over the variables. The most well-known of these functions is the product-moment correlation coefficient. To be more precise, this function should be referred to as the correlation about the mean. The "correlation coefficient" between two variables, Xj and Xj over all n samples is given by... 

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... 

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. 

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 ... 

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 (product-moment correlation, Pearson correlation) is a statistical measure for the relation between two or more sets of variables. 

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

Denatured Proteins. Gelatin as well as carboxymethylated reduced proteins is included in this group. It can be seen from Figure 10 and shown by analysis of the product-moment correlation coefficients (32) that the tritium distributions of the CM-reduced proteins are more similar... 

Secondary structure fractions jr, from X-ray diffraction and from CD are compared by two different criteria, r is the Pearson product-moment correlation, defined as... 

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 ... 

The particular advantages of the Spearman rank correlation coefficient are (1) they alone are applicable to ranked data and (2) they are superior to the product- moment correlation coefficient when applied to populations that are not normally distributed and/or include outfiers. A further advantage 1 that the Spearman rank correlation coefficient (r,) is speedy to calculate and may be used as a quick approximation for the product-moment correlation coefficient (r). 

If the sympathetic and parasympathetic components correspond to cardiac neurogenic autonomic inputs, then they should correlate with PEP and RSA, respectively. Pearson product-moment correlations were computed for the sympathetic component and PEP and for the parasympathetic component and RSA. The correlations were computed both within tasks and within subjects. 

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). 

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. 

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. 

By far, the most common method for evaluating regression models in the cheminfor-matics literature is Pearson s product-moment correlation [30], more commonly referred to as Pearson s r, or its square Pearson s r can be calculated in a number of ways one of the most straightforward is shown next. 

---