The product-moment correlation coefficient

In this section we discuss the first problem listed in the previous section - is the calibration plot linear A common method of estimating how well the experimental points fit a straight line is to calculate the product-moment correlation coefficient, r. This statistic is often referred to simply as the correlation coefficient because in quantitative sciences it is by far the most commonly used type of correlation coefficient. We shall, however, meet other types of correlation coefficient in Chapter 6. The value of r is given by  

The numerator of equation (5.2) divided by n, that is X )(yi T)1M is called the covariance of the two variables x and y. it measures their joint variation. If x and y are not reiated their covariance will be close to zero. The correlation coefficient r equals the covariance of x and y divided by the product of their standard deviations, so if a and y are not related r will also be close to zero. Covariances are also discussed in Chapter 8. 

Standard aqueous solutions of fluorescein are examined in a fluorescence spectrometer, and yield the following fluorescence intensities (in arbitrary units)  

In practice, such calculations will almost certainly be performed on a calculator or computer, alongside other calculations covered below, but it is important 

The figures below the line at the foot of the columns are in each case the sums of the figures in the table note that - x) and Z(Ti - y) are both zero. Using these totals in conjunction with equation (5.2), we have  

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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... 

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... 

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). 

This technique is concerned with quantitative analysis of data when analysing a pair of variables. Data can be displayed using a scatter diagram, and numerically, using simple correlation and regression - the product-moment correlation coefficient measures the strength of a linear relationship between the variables. If variables are measured using an ordinal scale then Spearman s rank or Kendall s tau may be used to indi-... 

In this example, is 1 - (24/336), i.e. 0.929. Theory shows that, like the product-moment correlation coefficient, can vary between -1 and +1. When K = 7, Tj must exceed 0.786 if the null hypothesis of no correlation is to be rejected at the significance level P = 0.05 (Table A.13). Here, we can conclude that there is a correlation between the sulphur dioxide content of the wines and their perceived quality. Bearing in mind the way the rankings were defined, there is strong evidence that higher sulphur dioxide levels produce less palatable wines ... 

The product-moment correlation coefficient is widely used in bmriate data analysis to measure the extent of the correlation between two variables, but it is not clear that it is necessarily appropriate to measure the extent of the similarity between two objects. Other sorts of correlation coefficient are available, such as the Spearman rank correlation coefficient which has been used by Manaut et al. as a measure of electrostatic similarity, but these have not found extensive application in similarity searching systems. Similar comments apply to probabilistic coefficients, which are calculated from the frequency distribution of descriptors in a database, and which Adamson and Bush < found to give poor results when applied to 2D chemical structures. 

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 ... 

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. 

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

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 ... 

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 ... 

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... 

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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