Analysis of 2x3 Contingency Tables with Ordered Categories

Anastasia Ivanova	Vance Berger
University of North Carolina at Chapel Hill	National Cancer Institute
aivanova@bios.unc.edu	vb78c@nih.gov

Within the context of comparative Phase III randomized clinical trials, many efficacy endpoints are measured on an ordinal scale. That is, the data consist of a set of categories, and there is a natural ordering among these categories. The relative spacings among the categories, however, are not known. Consider a 2´3 contingency table with two treatment groups and three ordered outcome levels.

Antiemetic response data after 2 days (Fox et al., 1993).

	Level of response
	None	Partial	Complete	Total
Control	X1 = 12	X2 = 3	X3 = 7	N1 = 22
Treatment	Y1 = 3	Y2 = 7	Y3 = 12	N2 = 22
Total	T1 = 15	T2 = 10	T3 = 19	N = 44

The goal of a between-group analysis of an ordered categorical endpoint is to establish that one treatment tends to be associated with preferable outcomes compared to the other treatment. Consider permutation tests since they preserve the size of the test. The most common analysis for a 2´3 contingency table is a linear rank test, for which numerical scores are assigned to the three response levels. Without loss of generality the scores can be chosen as (0, v, 1), usually with 0 £ v £ 1. For example, if equally spaced scores are desired, then v = 0.5. If v = 0 or v = 1, then the test becomes a binomial test, as categories are combined for analysis. However, the choice of scores might have a profound influence on the p-value, and on the interpretation of the results.

We have developed several new tests for this problem. The software written in Splus implements six test that can be used to compare groups with ordered categorical response:

• three most common linear rank tests (v = 0, 0.5, 1);
• the Smirnov test (Berger, Permutt, and Ivanova, 1998), which is based on the largest difference between the empirical CDFs and has piecewise linear boundaries of the rejection and extreme regions;
• the convex hull test (Berger, Permutt, and Ivanova, 1998), which is based on directional convex hull peeling of the permutation sample space;
• the adaptive test (Berger and Ivanova, 2001), which utilizes the prior knowledge of the direction of the treatment effect (parameter v) and the level of confidence in this prior knowledge (parameter degree). For example, if the superiority of the active treatment to the control is due primarily to a shift from the middle to the best outcome v is small, if it is due to a shift from the worst to the middle outcome v is large, v=0.5 if shifts are equal.

  Downloads

  The programs are developed in S-PLUS 4.0 for Microsoft Windows. The following downloads include the source functions. Create a directory, say ORDINAL, for the program. Download and extract the appropriate file there. For example, to install the program ttestpv, source the ttestpv.s file into S-PLUS via

  source("<path>/ttestpv.s")

  where <path> is the directory in which the program files are located. Or just copy the function and paste it into S-PLUS.

  Function ttestpv plots extreme regions for linear rank tests with v=0, 0.5, 1, and the Smirnov test calculating corresponding p-values.

  Function ttestsize plots rejection regions for one-sided (nominal size alpha=.05) linear rank tests with v=0, 0.5, 1, and the Smirnov test calculating actual sizes.

  Function chull.test plots extreme and rejection regions (one-sided, nominal alpha=0.05) for the convex hull test calculating p-value and the actual size.

  Function adaptive.test plots extreme and rejection regions (one-sided, nominal alpha=0.05) for the adaptive test calculating p-value and the actual size.

  TRY IT

  See how the p-value and the size of the test are affected by the choice of the score

  > ttestpv(12, 3, 7, 3, 7, 12)

  > ttestsize(12, 3, 7, 3, 7, 12)

  Perform the comparison of treatments using the convex hull test

  > chull.test(12, 3, 7, 3, 7, 12)

  If you have PRIOR knowledge about the direction of the treatment effect, say v=0.5, and the HIGH level of confidence in this prior knowledge, say degree=100, try

  > adaptive.test(12, 3, 7, 3, 7, 12, .5, 100)

  If you have PRIOR knowledge about the direction of the treatment effect, say v=0.5, and the LOW level of confidence in this prior knowledge, say degree=5, try

  > adaptive.test(12, 3, 7, 3, 7, 12, .5, 5)