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# Ordinal 2 by 5

### Input.

Fill the values in the table. Are considered to be integer values, whole positive numbers without decimals. You are free to input a smaller table.

### Explanation.

This program provides a number of tests to determine whether there is an ordinal association between two variables answering the question if the number of observations increase in the higher values of the columns as one reaches the higher values of the rows. Ordinal variables consist of ordered categories without quantitative differences. For example, length is a quantitative variable: 2 meters is 4 times as long as fifty centimeters. However, religious strictness, ordered in the categories (1) progressive, (2) moderate and (3) conservative is an ordinal variable. 'Conservative' does not mean three times as strict as progressive, only 'stricter'. One of the most frequently used ordinal variables is the Likert-scale, in which people are asked to order their opinions from 'very much agree' to 'very much disagree', on a five or seven point scale.

The lowest category box, the 1,1 box, is the upper left box in the table. The highest score box is the bottom right box.

The program first presents an account of the numbers of pairs in the table. The different pairs form the basis of many analyses of ordinal association. Concordant pairs consist of individuals paired with other individuals who score both lower on the column and lower on the row variable. Discordant pairs consist of individuals paired with other individuals who are lower on the one, and higher on the other variable. Tied pairs are individuals paired with others who have the same score on either the rows or the columns.

The different pairs form the basis of most analysis of ordinal association. If you tick the full analysis box you get a p-value for the likelihood that there are tables with a higher number of concordant pairs. This is an exact test of ordinal association. This procedure is another derivative of the Fisher exact and works the same, by generating tables and estimating their likelihood. The ordinal test is similar to the single sided Fisher in a two by two table. The Algorithm used is the same as for Fisher 2*5; however, the algorithm is less efficient in this particular case. More than 30 cases for a 2*5, 60 for a 2*4 and 120 cases for a 2*3 table are not advised.

Kendall's Tau-a and Goodman and Kruskal's Gamma follow the exact tests. You will get the sample standard deviations and p-values for these measures. Tau-a is the difference between the number of concordant and discordant pairs divided by the total number of pairs; Gamma is the difference between the number of concordant and discordant pairs divided by the sum of concordant and discordant pairs. Gamma usually gives a higher value than Tau and is (for other reasons as well) usually considered to be a more satisfactory measure of ordinal association. The p-values are supposed to approach the exact p-value for an ordinal association asymptotically, and the program shows that they generally do that reasonably well. But, beware of small numbers: the p-values for the gamma and Tau become too optimistic!

The Kolmogorov-Smirnov Test is subsequently given. This test gives the likelihood of two orders coming from different orderings or the same ordering. Have a look at this table:

Do you agree or disagree with the following statement (proportions between brackets)
[cummulative proportions between square brackets]

Males

Females

Difference

Totally agree

10 (0.12)[0.12]

24 (0.26)[0.26]

14 (0.14)[0.14]

Agree

15 (0.18)[0.30]

15 (0.17)[0.43]

0 (0.01)[0.13]

Neither agree or disagree

19 (0.23)[0.53]

21 (0.23)[0.66]

2 (0.00)[0.13]

Disagree

18 (0.21)[0.74]

17 (0.19)[0.85]

1 (0.02)[0.10]

Totally disagree

22 (0.26)[1.00]

14 (0.15)[1.00]

8 (0.11)[0.00]

Total

84 (1.00)

91 (1.00)

7 (0.00)

The K-S test assesses if the largest proportional cumulative difference in a table has been caused by chance fluctuation or not. In this case this difference equals [0.14] (top right cell). The program echoes the Chi-square value of the expected largest proportional difference, (Chi-2= 3.673) and the p-value of the difference between the observed and the expected largest difference, with two degrees of freedom. The p-value in this example equals 0.15933, the difference in ordering between males and females may well have been caused by chance fluctuation.

The probability value presented is single-sided. The literature considers that the Kolmogorov Smirnov test has very little power with a high chance of a type II error, i.e. of not finding a difference when there is one. Unless there are serious theoretical or other reasons for using the K-S, use of the gamma is preferable.

Ridit analysis has a strong descriptive nature. A Ridit test has the neutral -no difference between two orderings- value of 0.5. This is based on the notion that if two orderings ‘A’ and ‘B’ are the same and one draws an individual ‘a’ and ‘b’ from each of these orderings the probability that ‘a’ has a higher position in her ordering as ‘b’ in his ordering equals 0.5. If, however, the observations on ‘A’ tend to be clusterd in the higher positions, and observation in ‘B’ are clustered relatively lower, the probability of individual ‘a’ having a higher position than ‘b’ will increase above 0.5 up to the maximum probability of 1.0, ‘a’ certainly being higher than ‘b’. Similarly, ‘b-s’ position will decrease below 0.5 towards 0, certainly being lower as ‘a’. A Ridit of 0.8 means then that ‘a’ has a 80% chance of having a higher position after random selection than ‘b’.

The Ridit procedure followed here is based on:

Selvin S. A further note on the interpretation of ridit analysis. American Journal of Epidemiology 1977;105:16-20. ->AJE
Fleiss JL. Ridit Analysis in Dental Clinical Studies. Journal of Dental Research 1997;58:2080-2084. ->JDR

### Technical Explanation.

The exact algorithm works with four loops, one for each degree of freedom. Three loops are 'dumb' and one, the last, loop is 'smart'. The algorithm can be extended to handle any size 2*n table, contact SISA if you require that. Speed can be a problem. For each additional row speed will decrease with an additional factor determined by the size of the additional margin. Tables of which the number of cases is larger than 120 in a 2*3 table, 60 in a 2*4 and 30 in a 2*5 table, are not advised.

A HIGH SPEED HIGH PRECISSION 2 BY 7 WINDOWS VERSION OF THIS PROCEDURE IS AVAILABLE HERE! 