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Rates and ProportionsBios 662Michael G. Hudgens, [email protected]://www.bios.unc.edu/∼mhudgens2007-11-12 16:59BIOS 662 1 Rates and ProportionsOutline• Prevalence/incidence• Direct standardization• Indirect standardizationBIOS 662 2 Rates and ProportionsRates and Proportions• Cf Ch 15 text• Prevalence: proportion of people with a particular dis-ease at a fixed point in time ( π)• Rate: amount of change in a variable over a specifiedtime interval divided by the length of the time interval• Incidence: the number of new cases of disease over aperiod of time divided by the person-years at risk• Incidence is a rate, prevalence is notBIOS 662 3 Rates and ProportionsPrevalence• Random sample of size N from population of interest• n have disease (“cases”)• Estimator of prevalence:ˆp =nN=number of casessample size• CIs and tests for prevalence based onn ∼ Bin(N, π)where π is the population prevalenceBIOS 662 4 Rates and ProportionsPrevalence: Example• Ex: Random sample of 1717 injection drug users in 6major cities in U.S. found 206 were HIV positive.• Estimated prevalence of HIV among IDUsˆp = 206/1717 = .120• Large sample 95 % CI (0.105, 0.135)• Cf AJPH March 2002, page 385BIOS 662 5 Rates and ProportionsIncidence• Estimator of incidence:ˆI =number of new cases(sample size) * (time interval)• Example: incidence of diabetes among Pima Indians.N = 1728, time=6 years, new cases=346 [Ref: AJEOct 1, 2003, page 669]ˆI =3461728 ∗ 6= 0.033• Thus estimated incidence is 0.033 cases per pe rson-yearBIOS 662 6 Rates and ProportionsIncidence• Usually multiply by some number, say 1000ˆI1000= 33.3• Interpretation: estimated incidence is33.3 cases per year per 1000 personsor33.3 cases per 1000 person yearsBIOS 662 7 Rates and ProportionsIncidence• Note general formˆI1000= cno. of new casessample sizewhere c = 1000/no. of years• Sinceno. of new casessample sizeis a proportion, again we can use binomial principles forCIs and testsBIOS 662 8 Rates and ProportionsIncidence• Letˆp =nN=no. of new casessample sizesuch thatn ∼ Bin(N, π)• Note π here is distinct from earlier slide; probability ofbecoming a case in interval of follow-up• ThusˆV (ˆp) =ˆp(1 −ˆp)/NimplyingˆV (ˆI1000) = c2ˆp(1 −ˆp)/NBIOS 662 9 Rates and ProportionsIncidence CI• Approximate (1 − α)x100% CIˆI1000± z1−α/2qc2ˆp(1 −ˆp)/N• Diabetes example:ˆp =3461728= 0.20; c =10006= 166.67• 95% CI33.3 ± 3.14 = (30.2, 36.4)BIOS 662 10 Rates and ProportionsDirect Standardization• May need to adjust rates/proportions for possible con-founders, e.g., age, gender• Example: study of smoking in China (1984)Urban women: 1320 questioned, 330 current smokersRural women: 1338 questioned, 414 current smokersˆpu= 330/1320 = .25;ˆpr= 414/1338 = .31• Concern: age may be a confounderBIOS 662 11 Rates and ProportionsDirect Standardization• Three steps1. Divide samples into K categories of the potentialconfounder2. Compute the confounder category-specific propor-tions/rates3. Compute the weighted average of confounder-specificproportions/rates• Choice of weights based on standard or reference popu-lation; e.g. aggregate of samples in hand, governmentalpopulation surveyBIOS 662 12 Rates and ProportionsDirect Standardization• China smoking exampleUrban RuralAge N1in1iˆp1iN2in2iˆp2i35-39 129 8 .062 387 44 .11440-44 243 53 .218 441 138 .31345-49 478 135 .282 300 130 .43350-54 470 134 .285 210 102 .486BIOS 662 13 Rates and ProportionsDirect Standardization• Combined age distributionAge Niwi35-39 516 .19440-44 684 .25745-49 778 .29350-54 680 .256Total 2658 1BIOS 662 14 Rates and ProportionsDirect Standardization• Adjusted prevalence estimatorˆpjadj=PKi=1wiˆpjiPKi=1wi• Estimator of prevalence in the reference (i.e., standard)population based on the observed rates from the studypopulationBIOS 662 15 Rates and ProportionsDirect Standardization• Example: (Urban=1, Rural=2)ˆp1adj= (.194 ∗ .062 + ··· + .256 ∗ .285)/1 = .224ˆp2adj= 0.354• Crude difference:ˆp1−ˆp2= .25 − .31 = −.06• Age adjusted differenceˆp1adj−ˆp2adj= .224 − .354 = −.13BIOS 662 16 Rates and ProportionsDirect Standardization• World Health Organization Standard WeightsAge wiAge wi< 1 2.4 45-49 61-4 9.6 50-54 55-9 10 55-59 410-14 9 60-64 415-19 9 65-69 320-24 8 70-74 225-29 8 75-79 130-34 6 80-84 .535-39 6 > 84 .540-44 6BIOS 662 17 Rates and ProportionsDirect Standardization• China-smoking example with WHO standard:ˆp1adj= 0.196;ˆp2adj= 0.337ˆp1adj−ˆp2adj= −.141ˆp2adjˆp1adj= 1.72BIOS 662 18 Rates and ProportionsDirect StandardizationCrude Combined WHODifference -.06 -.13 -.14Ratio 1.24 1.58 1.72• Note Combined/WHO farther from null than Crude• Confounder age partially masks difference in smokingbetween urban and rural• Intuition: rural, older people s moke more; urban samplehas greater proportion of older peopleBIOS 662 19 Rates and ProportionsDirect Standardization•ˆpjadjis a weighted average of independent RVs (theˆpji’s)• Since nji∼ Bin (Nji, πji), we know thatV (ˆpji) = πji(1 − πji)/njiandˆV (ˆpji) =ˆpji(1 −ˆpji)/njiBIOS 662 20 Rates and ProportionsDirect Standardization• ThusˆV (ˆp1adj−ˆp2adj) =PKi=1w2i[ˆV (p1i) +ˆV (p2i)](PKi=1wi)2• Large sample tests and CIs are obtained from the CLTBIOS 662 21 Rates and ProportionsDirect Standardization• Revisiting the smoking exampleˆV (ˆp1adj−ˆp2adj) = 0.000295• Testing H0: π1adj= π2adj,Z =ˆp1adj−ˆp2adjqˆV (ˆp1adj−ˆp2adj)=−0.13√0.000295= −7.56• Conclude s ignificant difference in prevalence of smokingbetween rural and urban women after adjusting for ageBIOS 662 22 Rates and ProportionsStandardization• Direct standardization: Estimate rate/proportion inreference population using observed rate/proportion fromstudy population• Indirect standardization: Estimate rate/proportion instudy population using rate/proportion from referencepopulationBIOS 662 23 Rates and ProportionsIndirect Standardization• Suppose observed stratum specific prevalences– from reference population: mi/Mifor i = 1, . . . , K– from study population: ni/Nifor i = 1, . . . , K• Observed prevalence from study populationˆpstudy=PKi=1niPKi=1Ni• Expected prevalence for study populationˆpref=PKi=1Nimi/MiPKi=1NiBIOS 662 24 Rates and ProportionsIndirect Standardization• Standardized mortality ratio (SMR) or standardizedincidence ratio (SIR)ˆs


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