Epidemiology Epidemiology Kept SimpleKept Simple

Chapter 7Chapter 7

Rate AdjustmentRate Adjustment

GoalGoal

To reduce distortions and To reduce distortions and incomparability of rates when incomparability of rates when

making comparison over time and making comparison over time and among populationsamong populations

To encourage “like-to-like” To encourage “like-to-like” comparisonscomparisons

Illustrative ExampleIllustrative ExampleTable 7.2 (p. 144)Table 7.2 (p. 144)

Population APopulation A Population BPopulation B

AgeAge CasesCases PersonsPersons RateRate CasesCases PersonsPersons RateRate

YoungYoung 9999 99,0099,0000

100100 11 1,0001,000 100100

OldOld 1010 1,0001,000 1,0001,000 990990 99,0099,0000

1,0001,000

AllAll 109109 100,0100,00000 109109 991991 100,0100,0

0000 991991

Rate in Population B is 9× that of Population A

Illustrative Example Illustrative Example (cont.)(cont.)

Table 7.2 (p. 144) Table 7.2 (p. 144)

Population APopulation A Population BPopulation B

AgeAge CasesCases PersonsPersons RateRate CasesCases PersonsPersons RateRate

YoungYoung 9999 99,0099,0000

100100 11 1,0001,000 100100

OldOld 1010 1,0001,000 1,0001,000 990990 99,0099,0000

1,0001,000

AllAll 109109 100,0100,00000 109109 991991 100,0100,0

0000 991991

Within young, rates are identical

Illustrative Example Illustrative Example (cont.)(cont.)

Table 7.2 (p. 144)Table 7.2 (p. 144)

Population APopulation A Population BPopulation B

AgeAge CasesCases PersonsPersons RateRate CasesCases PersonsPersons RateRate

YoungYoung 9999 99,0099,0000

100100 11 1,0001,000 100100

OldOld 1010 1,0001,000 1,0001,000 990990 99,0099,0000

1,0001,000

AllAll 109109 100,0100,00000 109109 991991 100,0100,0

0000 991991

Within old, rates are identical

Why the apparent Why the apparent paradox?paradox?

Population APopulation A Population BPopulation B

AgeAge CasesCases PersonsPersons RateRate CasesCases PersonsPersons RateRate

YoungYoung 9999 99,0099,0000

100100 11 1,0001,000 100100

OldOld 1010 1,0001,000 1,0001,000 990990 99,0099,0000

1,0001,000

AllAll 109109 100,0100,00000 109109 991991 100,0100,0

0000 991991

Pop. A mostly old, Pop. B mostly young

And . . . rates are age-And . . . rates are age-relatedrelated

Population APopulation A Population BPopulation B

AgeAge CasesCases PersonsPersons RateRate CasesCases PersonsPersons RateRate

YoungYoung 9999 99,0099,0000

100100 11 1,0001,000 100100

OldOld 1010 1,0001,000 1,0001,000 990990 99,0099,0000

1,0001,000

AllAll 109109 100,0100,00000 109109 991991 100,0100,0

0000 991991

ConfoundingConfounding

• Explanatory factor Explanatory factor (population) (population) associated with ageassociated with age

• Extraneous factor Extraneous factor (age) associated with (age) associated with disease ratedisease rate

• Age Age confoundsconfounds relation between relation between explanatory factor explanatory factor and disease rateand disease rate

• BiasedBiased comparison comparison

Age

Population Rate

confounder

explanatory factor

disease

Strata-specific Strata-specific comparisonscomparisons

Population APopulation A Population BPopulation B

AgeAge CasesCases PersonsPersons RateRate CasesCases PersonsPersons RateRate

YoungYoung 9999 99,0099,0000

100100 11 1,0001,000 100100

OldOld 1010 1,0001,000 1,0001,000 990990 99,0099,0000

1,0001,000

You’re OK as long as you compare like-to-like

We can also adjust overall We can also adjust overall rate to compensate for rate to compensate for

confoundingconfounding• Rate adjustment methodsRate adjustment methods

• Direct adjustmentDirect adjustment• Indirect adjustmentIndirect adjustment

• Other statistical method of Other statistical method of adjustmentadjustment• Mantel-Haenszel methods Mantel-Haenszel methods • Regression modelRegression model

TerminologyTerminology• ““Rate” – Rate” – any incidence or prevalence any incidence or prevalence

(economy of language)(economy of language)• Crude rateCrude rate – rate for entire population – rate for entire population • Strata-specific rate - Strata-specific rate - rate within rate within

subgroupsubgroup• Adjusted rate – Adjusted rate – overall rate compensated overall rate compensated

for extraneous factor for extraneous factor • Two methods of adjustmentTwo methods of adjustment

• DirectDirect• IndirectIndirect

§7.2 Direct Age-§7.2 Direct Age-AdjustmentAdjustment

• Study population Study population – the population – the population rate you want to adjustrate you want to adjust

• Reference population - Reference population - external external population used as age norm, population used as age norm, • Reference population may beReference population may be

• arbitrary arbitrary • age distribution of some place at some age distribution of some place at some

time (“standard million”) time (“standard million”)

U.S. Standard Million, U.S. Standard Million, 19911991

Age rangeAge range Standard MillionStandard Million

0 – 40 – 4 76,15876,158

5 – 245 – 24 286,501286,501

24 – 4424 – 44 325,971325,971

45 – 6445 – 64 185,402185,402

65 – 7465 – 74 72,49472,494

75+75+ 53,47453,474

TotalTotal 1,000,0001,000,000

General Idea, Direct General Idea, Direct AdjustmentAdjustment

• Apply strata-specific rates from Apply strata-specific rates from study to a standard age age study to a standard age age distribution distribution

• Adjusted rate is a weighted Adjusted rate is a weighted average of strata-specific rates average of strata-specific rates (with weights from reference (with weights from reference population) population)

MethodMethod

where where NNii = population size, reference = population size, reference

population, strata population, strata ii

rrii = rate, study population, strata = rate, study population, strata II

Note: Note: caps denote reference pop. caps denote reference pop. values, while lower case denotes values, while lower case denotes study pop. valuesstudy pop. values

i

iidirect N

rNaR

Florida & Alaska Mortality Florida & Alaska Mortality Example Example (pp. 146 – 147)(pp. 146 – 147)

• Crude rates (per 100,000) Crude rates (per 100,000) • cRcRFloridaFlorida = 1026 = 1026

• cRcRAlaskaAlaska = 387 = 387

• See TABLE 7.5 for raw dataSee TABLE 7.5 for raw data

Age-Specific RatesAge-Specific Rates

ii AgeAge AlaskaAlaska FloridaFlorida

11 0 – 40 – 4 214214 238238

22 5 – 245 – 24 8080 6464

33 24 – 4424 – 44 172172 208208

44 45 – 6445 – 64 640640 809809

55 65 – 7465 – 74 25382538 22212221

66 75+75+ 83148314 68876887

Direct adjustment of Direct adjustment of Alaska rateAlaska rate

ii AgeAge RateRate

rrii

Std MillionStd Million

NNii

Product Product

NNi i × r× rii

11 0 – 40 – 4 214214 76,15876,158 16,297,81416,297,814

22 5 – 245 – 24 8080 286,501286,501 22,920,08022,920,080

33 24 – 24 – 4444

172172 325,971325,971 56,067,01256,067,012

44 45 – 45 – 6464

640640 185,402185,402 118,657,28118,657,2800

55 65 – 65 – 7474

25382538 72,49472,494 183,989,77183,989,7722

66 75+75+ 83148314 53,47453,474 444,582,83444,582,8366

1,000,0001,000,000 842,514,79842,514,7922

843000,000,1

792,514,842

i

iidirect N

rNaR

Comparing Adjusted Comparing Adjusted RatesRates

• Direct adjustment of Florida Direct adjustment of Florida mortality rate using same standard mortality rate using same standard million (Table 7.8, p. 147) derives million (Table 7.8, p. 147) derives aRaRFloridaFlorida = 784 = 784

• Recall, aRRecall, aRAlaskaAlaska = 843 = 843

• Conclude: slight advantage goes to Conclude: slight advantage goes to FloridaFlorida

The section on indirect The section on indirect adjustment (§7.3) may adjustment (§7.3) may or may not be coveredor may not be covered

§7.3 Indirect Age-§7.3 Indirect Age-AdjustmentAdjustment

• Same goal as direct adjustment Same goal as direct adjustment • Based on multiplying crude rate by Based on multiplying crude rate by

Standardized Mortality Ratio (SMR)Standardized Mortality Ratio (SMR)

A

SMR

where A = observed number of cases in study population = the expected number of cases (next slide)

Expected Number of Expected Number of Cases (Cases ())

iinR

where where

RRii = rate, reference population, strata = rate, reference population, strata ii

nnii = population size, study population, = population size, study population, strata strata ii

Recall: Recall: caps denote reference pop. caps denote reference pop. values and lower case denote study values and lower case denote study pop. valuespop. valuesThis is number of cases expected in study

population if it had reference population’s rates

Illustrative ExampleIllustrative ExampleZimbabwe & US Population (pp. 148 – Zimbabwe & US Population (pp. 148 –

9)9)

Indirect adjustment of Indirect adjustment of Zimbabwe rateZimbabwe rate

ii AgeAge US US RateRate

RRii

Zimb PopZimb Pop

nnii

Product Product

RRi i × n× nii

11 0 – 40 – 4 .0022.002299

1,899,201,899,2044

4,3494,349

22 5 – 245 – 24 .0006.000622

5,537,995,537,9922

34343434

33 24 – 24 – 4444

.0018.001800

2,386,072,386,0799

4,2954,295

44 45 – 45 – 6464

.0078.007899

974,235974,235 7,6877,687

55 65 – 65 – 7474

.0261.026188

216,387216,387 5,6655,665

66 75+75+ .0804.080466

136,109136,109 10,95110,951

RRiinnii = = 36,38136,381

381,36 iinR 72.2381,36

808,98

A

SMR

Zimbabwe SMRZimbabwe SMR

• Observed 98,808 deaths in Observed 98,808 deaths in ZimbabweZimbabwe

• Expected 36,381 (based on US rate)Expected 36,381 (based on US rate)• SMR = 98,808 / 36,381 = 2.72SMR = 98,808 / 36,381 = 2.72• Interpretation: Zimbabwe mortality Interpretation: Zimbabwe mortality

rate is 2.72× that of US after rate is 2.72× that of US after adjusting for ageadjusting for age

Indirectly Adjusted Indirectly Adjusted Rate Rate

Indirectly Adjusted Indirectly Adjusted RateRate

• Zimbabwe crude rate = 886 (per Zimbabwe crude rate = 886 (per 100,000)100,000)

• aRaRindirectindirect = (886)(2.72) = 2340 = (886)(2.72) = 2340

• c.f. to US rate of 860c.f. to US rate of 860

))(( SMRcRaRindirect

§7.4 Adjustment for §7.4 Adjustment for Multiple FactorsMultiple Factors

• Any extraneous factor can be Any extraneous factor can be adjusted foradjusted for

• Mortality rates are often adjusted Mortality rates are often adjusted for year, age, and sexfor year, age, and sex

• Principles of adjusting for potential Principles of adjusting for potential confounders apply to more confounders apply to more advanced studyadvanced study