Testing for Association between Categorical Variables with Multiple-Response Data

Some survey questions provide respondents with a list of possible answers and instructions to "mark all that apply." This paper presents methods for determining when the responses to a mark-allthatapply question are associated with the responses to a standard question whose answers fall in one of several mutually exclusive categories. Such data originate from many sources including large-scale social, political, and economic surveys; work place studies; market research; and health care analyses among others. Recent approaches to the problem are briefly reviewed, a related alternative procedure proposed, and extensions that allow for the proper analysis of multiple-response data collected through complex sampling are suggested. 1 I n t r o d u c t i o n Loughin and Scherer (1998) consider testing for association with multiple-response data. They study a survey of 262 Kansas livestock farmers who were asked to specify their education level (1. high school or less, 2. vocational school, 3. two-year college, 4. four-year college, or 5. other) and primary sources of veterinary information (1. professional consultant, 2. veterinarian, 3. state or local extension service, 4. magazines, and 5. feed company representatives). The 262 f a r m e r s who were instructed to mark one education level and as many information sources as appl icableprovided 453 responses to the veterinary information question. The survey researchers were interested in determining if the proportion of farmers using each information source is constant across varying levels of education. In general, suppose X is a categorical random variable with I levels and Y (Y1, . . . , yj)! is a random vector of binary responses, i.e., Yj E {0, 1} Partial funding provided by the Gallup Research Center for all j = 1 , . . . , J. Suppose we have n independent observations, {(Xk, Y~)' (Xk, Yk l , . . . , YkJ)')k=l .... ,,~, from the joint distribution of X and Y. The vector Y can take any of 2 J values, in general, or any of 2 J 1 values when ( 0 , 0 , . . . , 0) / is not a valid response. For i = 1 , . . . , I and j = 1 , . . . , J; let mij denote the number of observations for which X = i and Yj = 1. In the survey of livestock farmers, I = 5, J = 5, Xk takes value i if the kth farmer reports education level i, and Ykj would be coded as 1 if the jth information source is selected by farmer k and 0 otherwise. The count ~Zij is simply the number of farmers who report education level i and indicate information source j among their sources of veterinary information. The complete livestock farmer survey data can be found in Loughin and Scherer (1998). Loughin and Scherer consider testing J Hoj against Hi " U j J _ = I H l j (1) H0 -('lj= 1 where, for j = 1 , . . . , J; Hoj: P ( X = i, Yj = 1 ) = P ( X = i)P(]~ = 1) for all i = 1 , . . . , I and Hlj : P ( X = i, Yj = 1) # P ( X = i)P(Yj = 1) for some i. The null hypothesis H0 is equivalent to "X is independent of Yj for all j = 1 , . . . , J." Agresti and Liu (1999) refer to this null hypothesis as the hypothesis of multiple marginal independence which is conveniently abbreviated MMI. The alternative hypothesis H1 is equivalent to "X and Yj are associated for at least one j E { 1 , . . . , J}." Recent approaches to testing MMI are sketched in Section 2. As indicated by Agresti and Liu (1999), a problem common to many of these proposals is a lack of invariance to the coding of 0 and 1 values in the components of Y. For example, it is possible for a procedure to suggest a departure from MMI when positive responses are coded as 1 but agreement with MMI when positive responses are coded as 0. This