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Testing for Marginal Independence Between Two Categorical Variables with Multiple Responses Robert Jeutong Outline • Introduction – Kansas Farmer Data – Notation • Modified Pearson Based Statistic – Nonparametric Bootstrap – Bootstrap p-Value Methods • Simulation Study • Conclusion Introduction • “pick any” (or pick any/c) or multiple-response categorical variables • Survey data arising from multiple-response categorical variables questions present a unique challenge for analysis because of the dependence among responses provided by individual subjects. • Testing for independence between two categorical variables is often of interest • When at least one of the categorical variables can have multiple responses, traditional Pearson chisquare tests for independence should not be used because of the within-subject dependence among responses Intro cont’d • A special kind of independence, called marginal independence, becomes of interest in the presence of multiple response categorical variables • The purpose of this article is to develop new approaches to the testing of marginal independence between two multiple-response categorical variables • Agresti and Liu (1999) call this a test for simultaneous pair wise marginal independence (SPMI) • The proposed tests are extensions to the traditional Pearson chi-square tests for independence testing between single-response categorical variables Kansas Farmer Data • Comes from Loughin (1998) and Agresti and Liu (1999) • Conducted by the Department of Animal Sciences at Kansas State University • Two questions in the survey asked Kansas farmers about their sources of veterinary information and their swine waste storage methods • Farmers were permitted to select as many responses as applied from a list of items Data cont’d • Interest lies in determining whether sources of veterinary information are independent of waste storage methods in a similar manner as would be done in a traditional Pearson chi-square test applied to a contingency table with singleresponse categorical variables • A test for SPMI can be performed to determine whether each source of veterinary information is simultaneously independent of each swine waste storage method Data cont’d • 4 × 5 = 20 different 2 × 2 tables can be formed to marginally summarize all possible responses to item pairs Professional consultant Lagoon 1 0 1 34 109 0 10 126 • Independence is tested in each of the 20 2 × 2 tables simultaneously for a test of SPMI Data cont’d • The test is marginal because responses are summed over the other item choices for each of the multiple-response categorical variables • If SPMI is rejected, examination of the individual 2 × 2 tables can follow to determine why the rejection occurs Notation • Let W and Y = multiple-response categorical variables for an r × c table’s row and column variables, respectively • Sources of veterinary information are denoted by Y and waste storage methods are denoted by W • The categories for each multiple-response categorical variable are called items (Agresti and Liu, 1999); For example, lagoon is one of the items for waste storage method • Suppose W has r items and Y has c items. Also, suppose n subjects are sampled at random Notation cont’d • Let Wsi = 1 if a positive response is given for item i by subject s for i = 1,.. ,r and s = 1,.. ,n; Wsi = 0 for a negative response. • Let Ysj for j = 1,.., c and s = 1..,n be similarly defined. • The abbreviated notation, Wi and Yj , refers generally to the binary response random variable for item i and j, respectively • The set of correlated binary item responses for subject s are • Ys = (Ys1, Ys2,…,Ysc) and Ws = (Ws1, Ws2,…,Wsr ) Notation cont’d • Cell counts in the joint table are denoted by ngh for the gth possible (W1…,Wr ) and hth possible (Y1…,Yc ) • The corresponding probability is denoted by τgh. Multinomial sampling is assumed to occur within the entire joint table; thus, ∑g,h τgh = 1 • Let mij denote the number of observed positive responses to Wi and Yj • The marginal probability of a positive response to Wi and Yj is denoted by πij and its maximum likelihood estimate (MLE) is mij/n. Joint Table SPMI Defined in Hypothesis • Ho: πij = πi•π•j for i = 1,...,r and j = 1,...,c • Ha: At least one equality does not hold • where πij = P(Wi = 1, Yj = 1), πi• = P(Wi = 1), and π•j = P(Yj = 1). This specifies marginal independence between each Wi and Yj pair • P(Wi = 1, Yj = 1) = πij • P(Wi = 1, Yj = 0) =πi• − πij • P(Wi = 0, Yj = 1) = π•j − πij • P(Wi =0, Yj = 0) = 1 − πi• − π•j + πij Hypothesis • SPMI can be written as ORWY,ij =1 for i = 1,…,r and j = 1,…,c where OR is the abbreviation for odds ratio and – ORWY,ij = πij(1 − πi• − π•j + πij)/[(πi• − πij)(π•j − πij )] • Therefore, SPMI represents simultaneous independence in the rc 2 × 2 pairwise item response tables formed for each Wi and Yj pair • Join independence implies SPMI but the reverse is not true Modified Pearson Statistic • Under the Null Yj Wi 1 0 1 πij πi• − πij π•i 0 π•j − πij 1 − πi• − π•j + πij 1-π•i π•j 1-π•j • (1,1), (1,0), (0,1), (1,1) The Statistic Nonparametric Bootstrap • To resample under independence of W and Y, Ws and Ys are independently resampled with replacement from the data set. • The test statistic calculated for the bth resample of size n is denoted by X2∗S,b. • The p-value is calculated as – B-1∑bI(X2∗S,b ≥X2S) • where B is the number of resamples taken and I() is the indicator function Bootstrap p-Value Combination Methods • Each X2S,i,j gives a test for independence between each Wi and Yj pair for i = 1,…,r and j = 1,…,c. The p-values from each of these tests (using a χ21 approximation) can be combined to form a new statistic p tilde • the product of the r×c p-values or the minimum of the r×c p-values could be used as p tilde • The p-value is calculated as – B-1∑bI(p* tilde ≤ p tilde) Results from the Farmer Data Method My p-value Authors p-value Bootstrap X2s 0.0001 <0.0001 Bootstrap product of p-values 0.0001 0.0001 Bootstrap minimum p-values 0.0047 0.0034 Interpretation and Follow-Up • The p-values show strong evidence against SPMI • Since X2S is the sum of rc different Pearson chi-square test statistics, each X2S,i,j can be used to measure why SPMI is rejected • The individual tests can be done using an asymptotic χ21 approximation or the estimated sampling distribution of the individual statistics calculated in the proposed bootstrap procedures • When this is done, the significant combinations are (Lagoon, pro consultant), (Lagoon, Veterinarian), (Pit, Veterinarian), (Pit, Feed companies & representatives), (Natural drainage, pro consultant), (Natural drainage, Magazines) Simulation Study • which testing procedures hold the correct size under a range of different situations and have power to detect various alternative hypotheses • 500 data sets for each simulation setting investigated • The SPMI testing methods are applied (B = 1000), and for each method the proportion of data sets are recorded for which SPMI is rejected at the 0.05 nominal level My Results • n=100 • 2×2 marginal table • OR = 25 Method My p-value Authors p-value Bootstrap X2s 0.04 0.056 Bootstrap product of p-values 0.042 0.056 Bootstrap minimum p-values 0.036 0.044 Conclusion • The bootstrap methods generally hold the correct size