Ranked data

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

Figure 5.2. Plot of ranked data from an analysis of a carbonate-bicarbonate mixture. Lines are drawn at the mean (solid) and 2s (the sample standard deviation of the data) (dotted).

The tenure data and the rank data are not exactly the same. The numbers differ by several hundred, sometimes several thousand. It is really hard to make clear numerical comparisons. What we should focus on is the fact that wherever you look, women are not advancing. You can look anywhere women are not advancing at the rates they should, and it cannot be explained by things like increasing numbers of women at lower positions, because you can subtract out those older men and women who are full professors and recalculate and you still see a problem. 

Quantile systems divide ranked data sets into groups with equal numbers of observations in each group. Specifically ... 

Scores and loadings of PC2 versus PCI of the ranked data in Table 6.4... 

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. 

When the objects of the data set are ranked, measures of distance can also be applied on the ranks r y and r,y, representing the ranks of the th variable of the object s and t, respectively. The most important distance measitres on ranked data are listed below ... 

The process of subtracting two rules in the expression (2 - 2) in Table 7.6 accounts for the removal of rules (00...00) and (11... 11) from the possible rule set, since these two rules imply a solution that suffers complete dominance in the first case and enjoys total domination in the second, and so de facto caimot be part of the Pareto domain. For example, in the case of a four-objective optimization problem, a maximum of 14 rules can be generated. It might seem reasonable in this case to present the decision-maker with a data set containing four solutions, resulting in a miitimum of two rules that will not appear in the final set. Meanwhile, if the expert s ranked data set contains five solutions, a total of 20 rules will be generated while the maximum number of possible distinct rules would be only 14, implying that some rules will be duplicated in the P and NP sets, and therefore eliminated. 

By way of illustrating this latter point, for a four-objective optimization problem, Renaud et al. (2007) used an expert s ranked data set containing seven solutions taken from the Pareto domain, resulting in a total of 42 rules. Removing the duplicate rules in each of the P and NP rule sets and identical rules appearing simultaneously in both the rule sets, only three rules remained in both rule sets (a total of 6) in one case considered, and five rules in each set (a total of 10) in the other case. Under ideal conditions, seven rules in each of the P and NP rule sets would be found. Obviously, some rules were eliminated because they appeared in both rule sets and, therefore, they will not used to rank the entire Pareto domain. 

When the values of a particular criterion associated with two solutions contained in the expert s ranked data set are close to each other, ranking of the two solutions will invariably not be based on this particular criterion. This situation may lead to a rule that will not be significant for that criterion and which may subsequently bias the final ranking process. There exist a few approaches to partly alleviate this problem ... 

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). 

The different kinds of control charts are based on two groupings of types of data attribute data and variable data. Attribute data includes classification, count, and rank data Variable data refers primarily to continuous data, but rank data are often analyzed using a variable-control chart (realizing that the arithmetic functions are not theoretically valid). Otherwise the ranks can be converted to classification data and analyzed using attribute charts. Figure 8 contains examples of each of these categories of data. 

Falahee, M. and Macrae, A. W. (1997). Perceptual variation among drinking waters The reliability of sorting and ranking data for multidimensional scaling. Food Quality and Preference, 8, 389-394. 

Unfortunately, research in the area of optimization strategies is hampered by a lack of datasets on which to test new methods. Ideally the data would comprise not only what was made in the project but what could have been made—a full rank dataset of every R-group at each position with every R-group at the other positions. Recendy, one of us published a study on such a dataset (with the full rank data available as supplementary material) [40]. A series of MMP-12 inhibitors, found by a high-through-put screening, was elaborated at two positions by 50 substituents each (Figure 8.18). 

Correlation methods for partially ranked data are useful in applications of ligand-based and stracture-based virtual screenings, hi both cases, ranked lists of compounds are produced by any number of methods, which ones are not important. The important point is that lists produced in this manner may not contain the same set of molecules (vide supra). As noted earlier, correlation methods for partially ranked data proved a means of evaluating how the two types of methods performed. This approach can also be applied to pairwise comparisons of compound lists generated by ligand- and... 

Critchlow DE. Metric Methods for Analyzing Partially-Ranked Data. Berlin Springer-Verlag 1985. 

Ordinal rank data Nominal categorical data... 

Critchlow DE (1980) Metric methods for analyzing partially ranked data. Springer, New York... 

The question remains if there is a significant difference in the ability of the different probability distribution functions to describe the distillation data. It is generally not recommended to apply null hypothesis testing to information-theoretic ranking data to determine if the best model is significantly better than any of the lower ranked models (Burnham and Anderson, 1998). Model selection is best achieved through inspection of evidence ratios and residuals. A summary of the AIC and evidence ratios of the best 10 ranked functions are presented in Table 12.24. It can be... 

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