Rank correlation

The use of an average value is necessary in the case of tied ranks in sample 2 (see Section 6.5). The sums of the ranks for the three methods A, B and C are 5.5, 8.5 and 10 respectively. These sums should total nk k + l)/2, where k is the number of methods (three here) and n the number of samples (four here). The rank sums are squared, yielding 30.25, 72.25 and 100 respectively, and these squares are added to give the statistic R, which here is 202.5. The experimental value of is then calculated from  

The Friedman test could alternatively be used in the reverse form assuming that the three analytical methods give indistinguishable results, the same procedure could be used to test differences between the four plant extracts. In this case k and n are 4 and 3 respectively, and the reader may care to verify that R is 270 and that the resulting x value is 9.0. This is higher than the critical value for P= 0.05, n = 3, k=A, which is 7.4. So in this second application of the test we can reject the null hypothesis, and state that the four samples do differ in their pesticide levels. Further tests, which would allow selected comparisons between pairs of samples, are then available. 

Friedman s test is evidently much simpler to perform in practice than the ANOVA method (Sections 3.8-3.10), though it does not have the latter s ability to study interaction effects (see Chapter 7). 

Seven different table wines are ranked in order of preference by a panel of experts. The best wine is ranked 1, the next best 2, and so on. The sulphur dioxide content (in parts per million) of each wine is then determined by flow injection analysis with colorimetric detection. Use the following results to determine whether there is any relationship between perceived wine quality and sulphur dioxide content. 

The first step in the calculation is to convert the sulphur dioxide concentrations from absolute values into ranks (tied ranks are averaged as described in previous sections)  

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This ranking correlates with the positions of these species in the left column of Table 18.1 Mn04 is near die bottom, Fe2+ closest to the top. 

Fig. 18.1. J-Alert rank correlation for passively the basis of the estimated human effective absorbed drugs only. The ranking of drug permeability (calculated from a 3D structure...

Fig. 18.2. J-Alert rank correlation for drugs within a J-Alert value on the basis of the affected by active processes during absorption, estimated human effective permeability The ranking of drug molecules on the Y-axis (calculated from a 3D structure by QMPRPIus). was obtained by first sorting the drugs on the The ranking on the X-axis was directly by basis of increasing J-Alert value, and then increasing fraction absorbed value.

The mechanistic simulation ACAT model was modified to account automatically for the change in small intestinal and colon k as a function of the local (pH-dependent) log D of the drug molecule. The rank order of %HIA from GastroPlus was directly compared with rank order experimental %HIA with this correction for the log D of each molecule in each of the pH environments of the small intestine. A significant Spearman rank correlation coefficient for the mechanistic simulation-based method of 0.58 (p < 0.001) was found. The mechanistic simulation produced 71% of %HIA predictions within 25% of the experimental values. 

Kendall s rank correlation, represented by r(tau), should be used to evaluate the degree of association between two sets of data when the nature of the data is such that the relationship may not be linear. Most commonly, this is when the data are not continuous and/or normally distributed. An example of such a case is when we are trying to determine if there is a relationship between the length of hydra and their survival time in a test medium in hours. Both of our variables here are discontinuous, yet we suspect a relationship exists. Another common use is in comparing the subjective scoring done by two different observers. 

A robust, nonparametric (distribution free) measure for the correlation of variables is the Spearman rank correlation (symbol pjk, in M r Spearman). It is not limited to linear relationships, but measures the continuously increasing or decreasing... 

Kendall s tau correlation r Kendall) also measures the extent of monotonically increasing or decreasing relationships between the variables. It is also a nonparametric measure of association. It is computationally more intensive than the Spearman rank correlation because all slopes of pairs of data points have to be computed. Then Kendall s tau correlation is defined as the average of the signs of all pairwise slopes. The range of r is —1 to +1 the method is relatively robust against outliers for many applications p and r give similar answers. 

Fischer (30) reported that airborne respirable endotoxin levels correlate (Spearman s coefficient of rank correlation) with the counts of gram-negative rods, with the percent of the total material processed that is Memphis cotton, and with the decrease in FEV, if these data are calculated only on mills where the carding areas are separate from the spinning areas. 

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]. 

Trace element contents in rocks intersected by drill holes WB-08-03 to WB-08-06 are associated with altered rhyolite Spearman rank correlation coefficients for Ag, Mo and S with Au are moderate to high. Correlation coefficients for Au with As, Cu, Sb and Mo in drill hole WB08-11 (vuggy quartz) are moderate to high. These element correlations suggest that mineralization may be related to an epithermal system (Panteleyev 1995). 

Ag exhibits a strong Spearman s Rank correlation with Pb, Zn, Cd, and Sb reflecting its cogenetic nature with these elements as they are controlled by free energies, dissolved metal species, and degree of saturation. 

The number of circulating blood platelet is markedly reduced by injection of a lethal dose of toxin-LR into mice ( 7 ). The time course of the decrease of the blood platelets is closely paralleled by the increase in fresh weight of the liver. The sharp rise in Spearman s rank correlation between platelet count and liver weight 30 minutes after injection indicates that thromocytopenia and hepatomegaly were almost concurrent. 

For each IC50 bin, the percentage of compounds that list the ADR was calculated as percent ADR positive. For each group of seven IC50 bins, a Spearman rank correlation coefficient was calculated for the percent ADR positive relative to the bin rank. A strong Spearman correlation indicates a dose response in the ADR association. 

III.2. Comparison with Spearman s Rank Correlation Coefficient... 

Apparently the oldest nonparametric measure of the strength of association between two factors [26], the Spearman rank correlation coefficient... 

In an alternative approach, critical values of Spearman s rank correlation coefficient, tabulated as a function of observation size and significance probability [32], are employed. The 5% critical value is 0.738 when N = 8, hence rs = 0.369 is not significant (not even at the 10% level, with critical value 0.643). 

Capabilities are available in risk assessment software for inducing rank correlations among variables with arbitrary parametric distributions (Warren-Hicks and Moore 1998 Vose 2000). Also see Vose for a discussion of the envelope method for handling dependencies. 

While the Youden plot (Figure 3) indicates that there is systematic error in the various laboratories, a test by Spearmans rank correlation coefficient (Table Vb) is ambiguous. The ranking is significant at the 0.05 level but not at the 0.01 level thus a correlation of the ranks of the laboratories such as we see here would appear less than five times in 100 by chance but more than one time in 100. I prefer to look at these results in terms... 

Third, one needs accurate static orientational correlation functions. These are now available for the first rank correlation functions for water, acetonitrile, and several other liquids [191]. 

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