testing for marginal independence between two categorical variables with multiple responses robert...
TRANSCRIPT
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 single-response 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
• Independence is tested in each of the 20 2 × 2 tables simultaneously for a test of SPMI
Professional consultant
1 0
Lagoon 1 34 109
0 10 126
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
• (1,1), (1,0), (0,1), (1,1)
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
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 χ2
1
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 DataMethod My p-value Authors p-valueBootstrap X2
s 0.0001 <0.0001
Bootstrap product of p-values 0.0001 0.0001Bootstrap 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