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

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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

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

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Source:

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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

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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

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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

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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)

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

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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

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

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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]

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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]

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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)

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

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

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

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

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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

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

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