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Comparing Margins of Multivariate Binary Data Bernhard Klingenberg Assoc. Prof. of Statistics Williams College, MA www.williams.edu/~bklingen.

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Presentation on theme: "Comparing Margins of Multivariate Binary Data Bernhard Klingenberg Assoc. Prof. of Statistics Williams College, MA www.williams.edu/~bklingen."— Presentation transcript:

1 Comparing Margins of Multivariate Binary Data Bernhard Klingenberg Assoc. Prof. of Statistics Williams College, MA

2 Outline  Challenges: Associations of various degrees among binary variables Simultaneous Inference Sparse and/or unbalanced data, Test statistics with discrete support Asymptotic theory questionable  Setup: Two indep. groups Response: Vector of k correlated binary variables (multivariate binary)  Goal: Inference about k margins: Marginal Risk Differences Marginal Risk Ratios

3 Outline  Motivating Examples  From drug safety or animal toxicity/carcinogenicity studies Source:

4 Source:

5 Outline  Example: AEs from a vaccine trial (flu shot): > head(Y1) # ACTIVE Treatment n1=1971 ID HEADACHE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS > head(Y2) # PLACEBO Treatment n2=1554 ID HEADACHE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS

6 Notation and Setup  k-dimensional response vectors: Group 1Group 2  Random sample in each group: Group 1Group 2  Joint distrib. in each group depends on 2 k -1 parameters Group 1Group 2

7 Comparing Margins  Usually only interested in k margins. Group 1 Group 2  With just two (k=2) adverse events: Group 1 Group 2 NoYes No Yes Headache Pain NoYes No Yes Headache Pain

8 Comparing Margins Group1Group2Diff HEADACHE INJECTION SITE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS  Differences in marginal incidence rates between Group 1 (Treatment) and Group 2 (Control)

9 Family of Tests  j-th Null Hypothesis:  Unrestricted and restricted MLEs:

10 Comparing Margins  Estimates of marginal incidence rates and test statistics comparing Group 1 (Treatment) and Group 2 (Control) p-hat1p-hat2p-checkp-tildeWaldLocalGlobal HEADACHE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS

11 Asymptotic Test  Note:  Asymptotically, multivariate normal with covariance matrix determined by

12 Asymptotic Test  Correlation Matrix: > round(cov2cor(Sigma),2) d1 d2 d3 d4 d5 d6 d7 d d d d d d d > qmvnorm(0.95, tail="both.tails", corr=cov2cor(Sigma)) $quantile [1]

13 Asymptotic Test  Correlation Matrix: > round(cov2cor(Sigma),2) d1 d2 d3 d4 d5 d6 d7 d d d d d d d > qmvnorm(0.95, tail="both.tails", corr=cov2cor(Sigma)) $quantile [1]

14 Permutation Approach  When testing can use Permutation Approach  This assumes distributions are exchangeable (i.e. identical), much stronger assumption than under null  Need two extra conditions: i.Sequences of all 0's as or more likely to occur under group 2 (Control) ii.Sequence of all 1's as or more likely to occur under group 1 (Treatment)

15 Permutation vs. Asymptotic  Permutation vs. asymptotic distribution of Critical Value: (  = 0.05) c perm = c asympt = c Bonf = Permut. Distr. Asympt. Distr.

16 Family of Tests  Results: Raw and Adjusted P-values asymptoticexact DiffGlobalraw.Padj.P raw.P adj.P HEADACHE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS

17 Simultaneous Confidence Intervals  Invert family of tests: Confidence Region:  Simplifies to simultaneous confidence intervals if 

18 Simultaneous Confidence Intervals  Results: Inverting Score test diff LB UB HEADACHE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS

19 Simultaneous Confidence Intervals  We used (and recommend) score statistic  Could use Wald statistic instead  This is equivalent to fitting marginal model via GEE:  asympt. multiv. normal, with (sandwich) covariance matrix (same as before)  Use distribution of for multiplicity adjustment

20 Simultaneous Confidence Intervals  Results: GEE approach (= inverting Wald test) diff LB UB HEADACHE PAIN MYALGIA ARTHRALGIA MALAISE FATIGUE CHILLS

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