Table of Ranks

Examples of Table of Ranks in the following topics:

• The Westernization of Russia

  • Peter's distrust of the elitist and anti-reformist boyars culminated in 1722 with the creation of the Table of Ranks: a formal list of ranks in the Russian military, government, and royal court.
  • The Table of Ranks established a complex system of titles and honorifics, each classed with a number denoting a specific level of service or loyalty to the Tsar.
  • Previously, high-ranking state positions were hereditary, but with the establishment of the Table of Ranks, anyone, including a commoner, could work their way up the bureaucratic hierarchy with sufficient hard work and skill.
  • With minimal modifications, the Table of Ranks remained in effect until the Russian Revolution of 1917.
  • The establishment of the Table of Ranks was among the most audacious of Peter's reforms, a direct blow to the power of the boyars which changed Russian society significantly.

• Rank Randomization for Association (Spearman's ρ)

  • Table 1 shows 5 values of X and Y.
  • Table 2 shows these same data converted to ranks (separately for X and Y).
  • The correlation of ranks is called "Spearman's ρ. "
  • There are also three other arrangements that produce an r of 0.90 (see Tables 4, 5, and 6).
  • Therefore, there are five arrangements of Y that lead to correlations as high or higher than the actual ranked data (Tables 2 through 6).

• Rank Randomization: Two Conditions (Mann-Whitney U, Wilcoxon Rank Sum)

  • The primary advantage of rank randomization tests is that there are tables that can be used to determine significance.
  • The probability value is determined by computing the proportion of the possible arrangements of these ranks that result in a difference between ranks of as large or larger than those in the actual data (Table 2).
  • Tables 3-5 show three rearrangements that would lead to a rank sum of 24 or larger.
  • The beginning of this section stated that rank randomization tests were easier to compute than randomization tests because tables are available for rank randomization tests.
  • Table 6 can be used to obtain the critical values for equal sample sizes of 4-10.

• Percentiles

  • Consider the 25th percentile for the 8 numbers in Table 1.
  • For a second example, consider the 20 quiz scores shown in Table 2.
  • Since the score with a rank of IR (which is 5) and the score with a rank of IR + 1 (which is 6) are both equal to 5, the 25th percentile is 5.
  • The score with a rank of 17 is 9 and the score with a rank of 18 is 10.
  • The score with a rank of IR is 3 and the score with a rank of IR + 1 is 5.

• Wilcoxon t-Test

  • Ties receive a rank equal to the average of the ranks they span.
  • Let $R_i$ denote the rank.
  • Calculate the test statistic $W$, the absolute value of the sum of the signed ranks:
  • For $N_r < 10$, $W$ is compared to a critical value from a reference table.
  • Alternatively, a $p$-value can be calculated from enumeration of all possible combinations of $W$ given $N_r$.

• Rank Correlation

  • A rank correlation is a statistic used to measure the relationship between rankings of ordinal variables or different rankings of the same variable.
  • A rank correlation is any of several statistics that measure the relationship between rankings of different ordinal variables or different rankings of the same variable.
  • A rank correlation coefficient measures the degree of similarity between two rankings and can be used to assess the significance of the relation between them.
  • $-1$ if the disagreement between the two rankings is perfect: one ranking is the reverse of the other;
  • This graph shows a Spearman rank correlation of 1 and a Pearson correlation coefficient of 0.88.

• Distribution-Free Tests

  • Order statistics, which are based on the ranks of observations, are one example of such statistics.
  • Friedman two-way analysis of variance by ranks: tests whether $k$ treatments in randomized block designs have identical effects.
  • McNemar's test: tests whether, in $2 \times 2$ contingency tables with a dichotomous trait and matched pairs of subjects, row and column marginal frequencies are equal.
  • Squared ranks test: tests equality of variances in two or more samples.
  • This image shows a graphical representation of a ranked list of the highest rated cars in 2010.

• Rank Correlation

  • A rank correlation is any of several statistics that measure the relationship between rankings.
  • In statistics, a rank correlation is any of several statistics that measure the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the labels (e.g., first, second, third, etc.) to different observations of a particular variable.
  • A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.
  • where $n$ is the number of items or individuals being ranked and $d$ is $R_1 - R_2$ (where $R_1$ is the rank of items with respect to the first variable and $R_2$ is the rank of items with respect to the second variable).
  • Evaluate the relationship between rankings of different ordinal variables using rank correlation

• Kruskal-Wallis H-Test

  • The Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal and W.
  • Assign any tied values the average of the ranks would have received had they not been tied.
  • Note that the second line contains only the squares of the average ranks.
  • where $G$ is the number of groupings of different tied ranks, and $t_i$ is the number of tied values within group $i$ that are tied at a particular value.
  • If a table of the chi-squared probability distribution is available, the critical value of chi-squared, ${ \chi }_{ \alpha ,g-1' }^{ 2 }$, can be found by entering the table at $g − 1$ degrees of freedom and looking under the desired significance or alpha level.

• Mann-Whitney U-Test

  • For each observation in sample 1, count the number of observations in sample 2 that have a smaller rank (count a half for any that are equal to it).
  • The sum of ranks in sample 2 is now determinate, since the sum of all the ranks equals:
  • where $n_1$ is the sample size for sample 1, and $R_1$ is the sum of the ranks in sample 1.
  • The smaller value of $U_1$ and $U_2$ is the one used when consulting significance tables.
  • As it compares the sums of ranks, the Mann–Whitney test is less likely than the $t$-test to spuriously indicate significance because of the presence of outliers (i.e., Mann–Whitney is more robust).