Abstract
Nonparametric rank tests for independence between two characteristics are commonly used in many social opinion surveys. When both characteristics are ordinal in nature, tests based on rank correlations such as those due to Spearman and Kendall are often used. The case where some ties exist has already been considered whereas Alvo and Cabilio (1995) have studied the case when there are missing values but no ties in the record. However, it frequently happens that the survey data may contain simultaneouslymany tied observations and/or many missing values. A naive approach is to simply discard the missing observations and then to make use of the rank correlations adjusted for ties. This approach would be less powerful as it does not fully utilize the information associated with the incomplete data set. In this article, we generalize Alvo and Cabilio’s notion of distance between two rankings to incorporate tied and missing observations, and define new test statistics based on the Spearman and Kendall rank correlation coefficients.We determine the asymptotic distribution of the Spearman test statistic and compare its efficiency with the corresponding statistic based on the naive approach. The proposed test is then applied to a real data set collected from an opinion survey conducted in Hong Kong.
Highlights
Discrete ordinal variables are very popularly seen in many social opinion surveys
For example in a public opinion survey carried out in early 1999 in Hong Kong by the Social Science Research Centre of the University of Hong Kong, it was of interest to determine whether the age of the respondents is related to the level of satisfaction of the Policy Address of the Chief Executive of the Hong Kong Special Administrative Region
The objective of this paper is to develop rank tests for testing independence between two ordinal variables which can incorporate the presence of both missing values and ties
Summary
Discrete ordinal variables are very popularly seen in many social opinion surveys. To test for the independence between any two such variables, a nonparametric test based on Spearman rank correlation is commonly used. When the data contain missing observations but no ties, Alvo and Cabilio (1995) proposed a new class of rank correlations based on the concept of distance between rankings and derived the corresponding asymptotic distributions of the test statistics. Their method could not be directly applied to the above-mentioned two-way ordinal classification problem as the data contain many ties. We recall the notion of compatibility of Alvo and Cabilio (1995) and use it, to introduce new test statistics based on the Spearman and Kendall distances when both ties and missing observations are present.
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