Sample Correlation Coefficient

The value of r is always between —1 and +1, a value of —1 indicates a perfectly linear relationship between v and y, with the value of y decreasing as v increases. A value of +1 also indicates a perfectly linear relationship between v and y, but with y increasing as v 

in order to find r, the five sums—and are required. 

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Significant lack-of-fit can be detected by various sensible methods. However, the sample correlation coefficient r does not belong to the pool of these methods to assess linearity. The sample correlation coefficient may be misleading and is, despite its widespread use, to be discouraged for two reasons First, r depends on the slope. That is, for lines with the same scatter of the points about the line, r increases with the slope. Second, the numerical value of the correlation coefficient cannot be interpreted in terms of degree of deviation from linearity. Put differently, a correlation coefficient of 0.99 may be due to random error of a strictly linear relationship or due to systematic deviations from the regression line. 

The amount of oak wood equivalent to a flavor level detectable in 50% of the tests over chance level was variable stave to stave, 100-1400 mg dry wood/liter wine. A good part of this variability was attributable to differences in extractable solids and phenols in the various oak samples. Correlation coefficients between the grams of oven-dry oak wood to give threshold flavor and the extractable solids and phenols in that wood were, respectively, —0.47 and —0.56 in 1961 and —0.42 and —0.61 in 1971. These values are significant at the 1% level for phenols and at lower levels for the solids. An average of 34% of the variation of flavoring contribution by different oak samples was explained by the relative content of extractable phenol. The more extractable phenol the wood contained the more flavor it contributed to wine, not only because of the phenols themselves but because the mixture of extractives tends to be of uniform composition. 

Figure 27.14 shows the results of AOA determination by potentio-metric and RANDOX methods in 10 wine samples. Correlation coefficient is 72%. 

In order to minimize the sample correlation while sorting the features, we select only those consecutive elements for which the sample correlation coefficients with all the elements already selected does not exceed an assumed correlation threshold. We denote the selected bits by b , n = 0,... 

Another measure of interest is the sample correlation coefficient, or Pearson s r, or Pearson s133 product-moment formula, or the linear correlation of xand y ... 

Table IV. Sample Correlation Coefficients for Experiments in the Steel Chamber...

Step 1 Estimate the population serial correlation coefficient, P, with the sample correlation coefficient, r. It requires a regression through the origin or the (0, 0) points, using the residuals, instead of y and x, to find the slope. The equation has the form... 

Correlation coefficient Pearson s product-moment coefficient of linear correlation p measures the strength of the linear relationship between two variables X and yin a population. An estimation of the sample correlation coefficient ris given by... 

The sample correlation coefficient (r) is an estimate of the population correlation coefficient (p), i.e. the correlation that exists in the total population of which only a sample has been measured. 

The theoretical concept of correlation arises in conjunction with the bivariate normal distribution function. That function has five parameters. If the two variables are X tmd Y, the peuameters are the means (/x, /Xy) and the variances (correlation coefficient, p (rho). This chapter does not deal with the theoretical bivariate (or multivariate) normal distribution. However, in practice, the sample correlation coefficient, r, is a useful measure of linear association. It is a dimensionless ratio ranging from —1.0 (perfect inverse linear agreement) through zero (orthogonal or Unearly unrelated) to +1.0 (perfect direct linear agreement). The value can be obtained from Eq. (10) and used as an index without any assertion whatever being made about distribution form. 

If a random sample of size n is selected and two observations are made (e.g. height and weight) on each member of the sample, giving n pairs of observations, which are denoted by the values xi, yi), (x2, y2), (x , y ), then they can be plotted in a graph by considering the points v and y to be coordinates of a point. This results in a scatter diagram (Figure 18.4). It is difficult to decide whether there is a linear relationship and the sample correlation coefficient (r) is a measure of that linear relationship ... 

None of the elemental data show any correlation with collagen yield. When compared to yield, the correlation coefficients are non-significant for C N (r = 0.4), %C (r = 0.4) and %N (r = 0.2). All samples with low elemental values are from layers of bone that showed poor histological preservation. While low values tend to come from bone with poor histological preservation, many samples from bone of equally poor preservation produced acceptable values. Lipid removal generally improved both the yield characteristics and elemental values. 

Run a set of standards of four or more concentration levels covering the expected range of residues. Generate a calibration curve for each analyte and obtain a linear regression with a correlation coefficient of at least 0.90 for each analyte. Do not use any sample run data if the combined regression for the standards run immediately before, during and after the samples does not meet this criterion. 

Optimizing the GC instrument is crucial for the quantitation of sulfentrazone and its metabolites. Before actual analysis, the temperatures, gas flow rates, and the glass insert liner should be optimized. The injection standards must have a low relative standard deviation (<15%) and the calibration standards must have a correlation coefficient of at least 0.99. Before injection of the analysis set, the column should be conditioned with a sample matrix. This can be done by injecting a matrix sample extract several times before the standard, repeating this conditioning until the injection standard gives a reproducible response and provides adequate sensitivity. 

Van Emon et al. ° developed an immunoassay for paraquat and applied this assay to beef tissue and milk samples. Milk was diluted with a Tween 20-sodium phosphate buffer (pH 7.4), fortified with paraquat, and analyzed directly. Fortified paraquat was detected in milk at less than 1 pgkg , a concentration which is considerably below the tolerance level of 10 pg kg Ground beef was extracted with 6 N HCl and sonication. Radiolabeled paraquat was extracted from ground beef with recoveries of 60-70% under these conditions. The correlation coefficient of ELISA and LSC results for the ground beef sample was excellent, with = 0.99, although the slope was 0.86, indicating a significant but reproducible difference between the assays. 

Elliott et al. utilized a clenbuterol immunoassay to determine clenbuterol residues in cattle tissues and fluids. The LOD was 0.25 ug for liver. Animals were dosed with medicated feed (1.6 ug kg per day), and pairs were slaughtered during the medication phase and at 14,28, and 42 days after withdrawal. Clenbuterol concentrations in liver and retina/choroid samples were confirmed by GC/MS. Correlation coefficients between the ELISA and GC/MS were = 0.92 for retina/choroid samples and... 

Calibration of an internal standard method is done by preparing standard samples of varying concentration. The same amount of IS is added to each, and the standard samples are analyzed using a developed method. The detector response, area or height, of each standard is determined, and a ratio is calculated. The graph of concentration vs. area ratio is plotted. The method is considered linear if the correlation coefficient is 0.99 or better. The response factor RF is calculated as... 

The most straightforward full-spectrum comparison approach uses a correlation or cross-correlation coefficient to construct a score. Given a reference signature spectrum y, and an unknown sample spectrum xh for channels i = 1, 2,..., the linear cross-correlation can be expressed as... 

The correlation coefficient rxy of a sample is an estimate of the correlation coefficient pxy of the population. 

For a graphical comparison of the correlation [r(Sr)] and the standard deviation of the samples used for calibration (Sr), a value is entered for the SEP (or SEE) for a specified analyte range as indicated through the standard deviation of that range (Sr). The resultant graphic displays the Sr (as the abscissa) versus the r (as the ordinate). From this graphic it can be seen how the correlation coefficient increases with a constant SEP as the standard deviation of the data increases. Thus when comparing correlation results for analytical methods, one must consider carefully the standard deviation of the analyte values for the samples used in order to make a fair comparison. For the example shown, the SEE is set to 0.10, while the correlation is scaled from 0.0 to 1.0 for Sr values from 0.10 to 4.0. 

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