Mapping Rates and Proportions

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• Incidence rates

• Mortality rates

• Birth rates

• Prevalence

• Proportions

• Percentages

Sample Data:Breast Cancer Incidence in Iowa

• Years: 1993-1996

• 7813 Cases (including in-situ)

• For each case: Age, county

• Source: State Health Registry of Iowa

Geography and Population

• 99 counties

• Number of women: 1,061,096 (ages 20+)

• Population available for each county by age group.

• Age groups: 20-24, 25-29, …, 80-84, 85+

• Source: 1990 census

IO W A

The 99 Iowa Counties

Poisson Data

Numerator: Number of events over time, such as incidence or mortality cancer cases.

Denominator: Population years at risk.

Rates and Relative Risksc = # cancer cases in e.g. a countyn = county populationC = # cancer cases in e.g. a stateN = state population

Rate = c/n

Relative Risk = c n

Breast Cancer Incidence, Relative Risks

Not a djuste d<0.750.75-0.850.85-0.950.95-1.051.05-1.15>1.25

Bernoulli Data (0/1 data)

Individual people with one of two traits, such as cancer vs. no cancer, late vs. early disease or two different treatments.

Numerator: The trait of interest.Denominator: All individuals.

The denominator may be a complete count or a random sample.

Proportions and Relative Risksc = # late stage cancer cases in a countyn = total number of casesC = # late stage cancer cases in stateN = total cases in state

Crude Rate = c/n

Crude Relative Risk = c n

The statistical methods used are slightly different for Poisson and Bernoulli data, but in terms of mapping, the principles are the same.

Age Adjustment

• Indirect vs. Direct Standardization

• Internal vs. External Standard

• Relative Risk vs. Rate

Age Adjustment

Area to be mapped (e.g. Johnson county, Iowa)cs = cancer cases in age group sns = population in age group s

Area used as the standard (e.g. State of Iowa)Cs = cancer cases in age group sNs = population in age group s

Notation

Indirect Standardization(relative risk)

o b serv ed in co u n ty

ex p ec ted in co u n ty

Direct Standardization(relative risk)

cases in s ta te if it h ad th e sam e

ag e - sp ec ific ra tes as th e co u n ty

o b serv ed in s ta te

Direct Standardization

The crude state rate, if thewhole state had the sameage-specific rates as thecounty.

(rate)

IndirectStandardization

RelativeRisk

DirectStandardization

Indirect vs. Direct Standardization Population Cases county state stateChildren, 0-19 1 200,000 400Young Adults, 20-69 19 600,000 2200Old Adults, 70+ 80 200,000 2400 Expected cases in county: 1.03

County Cases Children, 0-19 0 0 1Young Adults, 20-69 0 1 0 Old Adults, 70+ 1 0 0Direct Standardization 0.5 6.3 40.0Indirect Standardization 1.0 1.0 1.0

County Cases Children, 0-19 0 0 1 0 0 1Young Adults, 20-69 0 1 0 0 1 0Old Adults, 70+ 1 0 0 2 1 1 Direct Standardization 0.5 6.3 40.0 1.0 6.8 40.5Indirect Standardization 1.0 1.0 1.0 1.9 1.9 1.9

County Cases Children, 0-19 0 0 1 0 0 1 0 0 1Young Adults, 20-69 0 1 0 0 1 0 1 2 1Old Adults, 70+ 1 0 0 2 1 1 2 1 1Direct Standardization 0.5 6.3 40.0 1.0 6.8 40.5 7.3 13.1 46.8Indirect Standardization 1.0 1.0 1.0 1.9 1.9 1.9 2.9 2.9 2.9

Indirect StandardizationWith indirect standardization, estimates of rates and relative risks have lower variance. This is especially important for small areas such as counties or census tracts.

• Method of choice for maps with estimates of multiple areas, showing geographical variation.• Use internal standard.

Age a djuste d<0.750.75-0.850.85-0.950.95-1.051.05-1.15>1.25

Breast Cancer Incidence, Relative RisksAge-Adjusted, Indirect Standardization

Breast Cancer Incidence, Relative Risks Not Age-Adjusted

Not a djuste d<0.750.75-0.850.85-0.950.95-1.051.05-1.15>1.25

Indirect Standardization(relative risk)

No need to knowage-specific case counts in the county,only the total.

Direct Standardization(rate)

No need to know casecounts for the referencearea.

Direct StandardizationVery useful to compare rates for areas studiedat different times, by different people, using different data sets.

Use external standards:• 1970 United States Population Standard• 2000 United States Population Standard• European Standard• World Standard

U.S 1970 and World Standards U.S. 1970 World45-49 5,962 5,00050-54 5,464 5,00055-59 4,908 4,00060-64 4,240 4,00065-69 3,441 3,00070-74 2,679 2,00075-79 1,887 1,00080-84 1,124 50085+ 743 500

World Standard From: Waterhouse et al., Cancer Incidence in Five Continents, 1976

U.S. 1970 World0-4 8,442 12,0005-9 9,820 10,00010-14 10,230 9,00015-19 9,384 9,00020-24 8,056 8,00025-29 6,632 8,00030-34 5,625 6,00035-39 5,466 6,00040-44 5,896 6,000

Iowa Breast Cancer Incidence Rates1993-1996

Crude Rate: 136.4 / 100,000 women

Age-Adjusted, Direct Standardization

U.S. 1970 Standard Population: 106.4 / 100,000U.S. 2000 Standard Population: 129.3 / 100,000World Standard Population: 91.0 / 100,000

Conclusions

• Use indirect standardization, with an internal standard, for mapping geographical variation.

• Use direct standardization, with a few different standards, to calculate the rate for the map as a whole.

Uncertainty of Rate Estimates

In a regular map, a relative risk of 2 could mean that there are 2000 cases with 1000 expected in an urban county, or 2 cases with 1 expected in a rural county.

For the urban county, the relative risk of 2 is a good estimate of the true relative risk, but not for the rural county.

Probability Map

For a particular county, one can test whether the observed cases are significantly more than expected, providing a p-value for that county.

A map of these p-values is called a ‘probability map’.

Reference: Chownowski M. Maps Based on Probabilities. Journal of the American Statistical Association, 54:385-388, 1959.

Probability Map

m = expected number of casesc = observed number of cases

P c H em

( | )!

# cases

(Poisson Data)

IO W A

Probability Map

p<0.05 0.05<p<0.10

County ‘p-values’

County Obs Exp RR p= Dubuque 275 235 1.17 0.004Polk 892 817 1.09 0.004Clayton 77 57 1.34 0.006Mills 51 36 1.43 0.006Scott 411 368 1.12 0.012Linn 467 429 1.09 0.033Marion 97 82 1.18 0.048

Age a djuste d<0.750.75-0.850.85-0.950.95-1.051.05-1.15>1.25

p<0.05Regular vs. Probability Map

0.05<p<0.10

Warning

By chance, 5% of the counties will by chance have a ‘statistically significant’ p-value at the 0.05 level.

Need to adjust for multiple testing.

Pickle et al: United States Mortality Atlas

Dilemma

- Too little aggregation: Unstable rates.- Too much aggregation: Geographical variation in disease may not follow political boundaries.

Solution: Smoothed Maps

Mapping Rates and Proportions

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