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## Analysis of Overdispersed Data in SASJessica Harwood, M.S.Statistician, Center f

Wolf, Gary, Contributing Editor has reference to this Academic Journal, PHwiki organized this Journal Analysis of Overdispersed Data in SASJessica Harwood, M.S.Statistician, Center as long as Community HealthJHarwood@mednet.ucla.eduOutline – OverdispersionDefinition, background, in addition to causes of overdispersionConsequences of ignoring overdispersionAccounting as long as overdispersion in regression analysis in SASFor count dataFor binary dataConcluding remarksJessica Harwood CHIPTS Methods Seminar 1/8/20132Overdispersed dataAlso known as extra variationArises when count or binary data exhibit variances larger than those assumed under the Poisson or binomial distributions3Jessica Harwood CHIPTS Methods Seminar 1/8/2013

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Count dataDefinition: non-negative integer values {0, 1, 2, 3, } arising from counting rather than rankingExample: the number of days a student is absent in one school yearCommonly analyzed using Poisson distribution, e.g., Poisson regressionJessica Harwood CHIPTS Methods Seminar 1/8/20134Poisson DistributionPoisson: number of occurrences of a r in addition to om event in an interval of time or space. Poisson regression IRR (relative risk)Natural model as long as count dataDisadvantage – strong assumption: variance = meanOverdispersion: variance > mean Jessica Harwood CHIPTS Methods Seminar 1/8/20135Binary: 0 or 1Example: ever tested as long as HIV (1) or not (0)Grouped binaryExample: proportion tested as long as HIVCommonly analyzed using binomial distribution, e.g., logistic regression Binary: tested-HIV Grouped: num-tested-HIV/num-subjectsBinary DataJessica Harwood CHIPTS Methods Seminar 1/8/20136

Binomial distributionBinomial: the number of successes in a sequence of r in addition to om processes that results in one of two mutually exclusive outcomesOverdispersion: variance larger than that assumed under the binomial distributionJessica Harwood CHIPTS Methods Seminar 1/8/20137Causes of OverdispersionObserved data rarely follow statistical distributions exactlyThe variance of count variables tends to increase with the size of the countsCorrelated (ex: clustered) dataHeterogeneity among observations Large number of 0sJessica Harwood CHIPTS Methods Seminar 1/8/20138Consequences of Ignoring OverdispersionJessica Harwood CHIPTS Methods Seminar 1/8/20139

Checking as long as Overdispersion in SAS Count DataPROC MEANSvariance > mean PROC GENMODdist=negbin dispersion parameter significantJessica Harwood CHIPTS Methods Seminar 1/8/201310Example Count Data Differences in baseline depression between intervention conditions in a RCTIndependent variable: INTV – intervention condition1 = R in addition to omized to intervention condition0 = R in addition to omized to control conditionDependent variable: EPDS – Edinburgh Postnatal Depression Scale; weighted count of depressive symptoms felt in past weekJessica Harwood CHIPTS Methods Seminar 1/8/201311HISTOGRAM OF EPDSExample Count DataEPDS Score (range 0-30)0248101214Percent6Jessica Harwood CHIPTS Methods Seminar 1/8/201312

Example Count Data Check mean in addition to variance as long as overdispersionMean in addition to variance;proc means data=base mean var; var EPDS; run;Conditional mean in addition to variance;proc means data=base mean var; var EPDS; class INTV; run;–SAS Code SAS Output–Jessica Harwood CHIPTS Methods Seminar 1/8/201313Poisson regression ignore overdispersion;proc genmod data = base; model EPDS = INTV / dist=poisson; run;Negative binomial regression account as long as overdispersion;proc genmod data = base; model EPDS = INTV / dist=negbin; run;Example Count Data SAS regression analysisJessica Harwood CHIPTS Methods Seminar 1/8/201314Dispersion parameter significantly different from zero (see 95% CI): Indicates significant over- (> 0) or under- (< 0) dispersionUse negative binomial rather than PoissonExample Count Data Check as long as overdispersion: negative binomial regression in PROC GENMODJessica Harwood CHIPTS Methods Seminar 1/8/201315 Example Count DataResultsP-values quite differentDifferent conclusions regarding similarity of intervention conditions at baselineJessica Harwood CHIPTS Methods Seminar 1/8/201316Accounting as long as overdispersion in SAS: count dataNegative binomial Variance-adjustment modelsQuasi-likelihood EstimationEmpirical (aka robust, s in addition to wich) variance estimation Models as long as correlated dataZero-inflated modelsJessica Harwood CHIPTS Methods Seminar 1/8/201317Negative binomial (NB) Negative binomial distribution: variance is larger than the mean excellent model as long as overdispersed count dataNegative binomial regression relative riskDisadvantage: estimating extra parameter (dispersion)PROC GENMOD PROC COUNTREGJessica Harwood CHIPTS Methods Seminar 1/8/201318 SAS code: negative binomial regressionproc genmod data = base; model EPDS = INTV / dist=negbin; run;proc countreg data = base; model EPDS = INTV / dist=negbin; run;Jessica Harwood CHIPTS Methods Seminar 1/8/201319SAS output: negative binomial regressionCompare to Poisson regression: INTV: Estimate=-0.0974; SE=0.0218; P<.0001 Jessica Harwood CHIPTS Methods Seminar 1/8/201320SAS output: negative binomial regressionNB: Variance > mean Variance = mean + k mean2 SAS estimate of dispersion parameter k: Dispersion, -AlphaIf k significantly different from zero use NB rather than PoissonJessica Harwood CHIPTS Methods Seminar 1/8/201321

Count data: Quasi-likelihood Estimation (QLE)QLE allows as long as adjusting variance without specifying distribution exactlyVariances inflated by Deviance/DOF (GENMOD: dscale)Pearsons Chi-Square/DOF GENMOD: pscaleGLIMMIX: r in addition to om -residual-Poisson in addition to negative binomial regression ( in addition to logistic regression)Jessica Harwood CHIPTS Methods Seminar 1/8/201322QLE Example – SAS Code Use dscale as the norm!Poisson regression- no adjustment as long as overdispersion;proc genmod data=base;model EPDS = INTV/ dist=poisson;run;Poisson regression- adjust as long as overdispersion using DSCALE;proc genmod data=base;model EPDS = INTV/ dist=poisson dscale;run;Jessica Harwood CHIPTS Methods Seminar 1/8/201323QLE- st in addition to ard errors (SE) correctedJessica Harwood CHIPTS Methods Seminar 1/8/201324

QLE- Poisson vs. NBJessica Harwood CHIPTS Methods Seminar 1/8/201325proc genmod data=base;model EPSD=INTV / dist=poisson pscale;run;proc glimmix data=base; model EPSD=INTV / dist=poisson s;r in addition to om -residual-;run;QLE PSCALE- SAS CodeJessica Harwood CHIPTS Methods Seminar 1/8/201326Count Data QLE In SumUse as the norm, in Poisson or NBDSCALE better than PSCALE, especially as long as low countsJessica Harwood CHIPTS Methods Seminar 1/8/201327

Count Data: Empirical Variance EstimationEmpirical (or robust or s in addition to wich) variance estimation account as long as extra variation by using both empirical-based estimates in addition to model-based estimates in variance estimation Poisson in addition to NB regression ( in addition to logistic regression) GENMOD: REPEATED statementGLIMMIX: EMPIRICAL option Jessica Harwood CHIPTS Methods Seminar 1/8/201328Empirical Variance Estimation GENMOD REPEATEDPID = Participant ID, 1 observation per PID;proc genmod data=base; class PID; model EPDS=INTV / dist=poisson; repeated subject = PID; run;Jessica Harwood CHIPTS Methods Seminar 1/8/201329Compare to unadjusted Poisson regression: INTV: Estimate=-0.0974; SE=0.0218; P<.0001 Empirical Variance Estimation GLIMMIX EMPIRICALPID = Participant ID 1 observation per PID. MBN is small-sample bias correction;proc glimmix data=base empirical=mbn; class PID; model EPDS=INTV / dist=poisson s; r in addition to om -residual- /subject = PID; run;Jessica Harwood CHIPTS Methods Seminar 1/8/201330 Thank you very much!QuestionsJessica Harwood CHIPTS Methods Seminar 1/8/201361JHarwood@mednet.ucla.edu

## Wolf, Gary Contributing Editor

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