• Colorado.gov

The following guidelines were adapted from those originally developed by the Assessment Operations Group of the Washington State Department of Health. CoHID gratefully acknowledges this group's willingness to share their hard work to benefit the citizens of Colorado. Persons interested in more technical details or have questions regarding data analysis are encouraged to consult statisticians and epidemiologists who have expertise in the analysis of public health data or to contact Health Statistics Section, Colorado Department of Public Health and Environment by phone (303-692-2160) or by email cohid@state.co.us

Rate Basics
Why are rates used in public heath assessment?
What is the difference between crude, age-adjusted and age-specific rates?
Should I calculate a rate when the number of events is small?
Will the rate be compared to other rates?

Guidelines for using and developing rates

Number of events (numerator)
Population at risk (denominator)
Crude rates
Age-adjusted rates, direct method
Age-specific rates
Comparing rates
Unstable rates due to small numbers
Methods for age adjustment

Direct method of age-adjustment
US Standard Populations for direct adjustment
Indirect adjustment

Glossary
References
Resources

A rate consists of a numerator and a denominator.

The numerator is the number of health events. This is often the same as the number of people who experience an event, but for some health conditions, one person may experience the event more than once. For example, one individual may have multiple hospitalizations for the same condition in a given year.

To measure incidence or prevalence of the condition, you usually want to count people. To measure the public health burden, you may want to count events. Actions based on the data may be different depending on whether the rate represents many individuals with only one event or a smaller number of individuals who have had many events.

It is customary to count only events that occur among the population at risk.

Guideline: Number of events (numerator)

The denominator is also known as the population at risk. Everyone in the population at risk must be eligible to be counted in the numerator if they have the event of interest. For example, in looking at female breast cancer, we cannot include men in the population at risk, because men with breast cancer would not be included in the numerator.

Guideline: Population at risk (denominator)

Once the numerator and denominator are established, how do we decide which rate is the most appropriate to use. The following questions are useful.

Why are rates used in public health assessment?

Much of public health assessment involves describing the health status of a defined community by looking at changes in the community over time or by comparing health events in that community to events occurring in other communities or the state as a whole. In making these comparisons, we need to account for the fact that the number of health events depends in part on the number of people in the community. To account for growth in a community or to compare communities of different sizes, we usually develop rates to provide the number of events per population unit.

Also, the frequency with which health events occur is almost always related to age. For example, acute respiratory infections are more common in children of school age because of their immunologic susceptibility and exposure to other children in schools. Chronic conditions, such as arthritis and atherosclerosis, occur more frequently in older adults because of a variety of physiologic consequences of aging. Mortality tends to increase rapidly after the age of 40. In fact, the relationship of age to risk often dwarfs other important risk factors. Because the relationship of age to risk is often resistant or impervious to interventions, analysts often remove the effects of differences in age structure when comparing rates across populations by calculating age-adjusted and age-specific rates.

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What is the difference between crude, age-adjusted and age-specific rates?

Crude rates

Crude rates are recommended when a summary measure is needed and it is not necessary or desirable to adjust for other factors. For example, rates of infectious diseases, such as tuberculosis and hepatitis, are usually not age adjusted, because public health officials are interested in the overall burden of disease in the total population irrespective of age.

A crude rate is calculated by dividing the total number of events in a specified time period by the total number of individuals in the population who are at risk for these events and multiplying by a constant, such as 1,000 or 100,000 [e.g., (numerator/denominator) x constant]. For example, number of deaths in King County for 1999 (numerator) divided by the population of King County in 1999 (denominator) times 100,000 (constant) gives the 1999 crude death rate per 100,000 population for King County.

Guideline: Crude Rates

Age-adjusted rates

Adjusted rates are used when comparing rates of health events affected by confounding factors. They are used when comparing different populations or for comparing trends in a given population over time. Because the occurrence of many health conditions is related to age, the most common adjustment for public health data is age-adjustment.

The age-adjustment process removes differences in the age composition of two or more populations to allow comparisons between these populations independent of their age structure. For example, a county’s age-adjusted death rate is the weighted average of the age-specific death rates observed in that county, with the weights derived from the age distribution in an external population standard, such as the U.S. population. Different standard populations have different age distributions and the choice will affect the resulting age-adjusted rate. If the age-adjusted rates for different counties are calculated with the same weights (i.e., using the same population standard), the effect of any differences in the counties’ age distributions is removed.

Currently, the National Center for Health Statistics (NCHS) age-adjusts rates using the US 1940 standard population. Other agencies use the US 1970 Standard. Beginning with 1999 data, federal agencies will age-adjust to the US 2000 Standard Population.

Age-adjusted rates should be presented when a single, summary measure is needed, but data analysts should inspect age-specific rates first. (Choi, 1999)

• Age-adjusted rates can mask important trends. For instance, while recent trends in cancer mortality show decreasing death rates for people under 24 and increasing rates for people over 65, the age-adjusted rates changed very little. (Anderson, 1998)
• Age-adjusted rates can over- or under-estimate differences. For instance, when age-specific rates of the populations being compared do not show a consistent relationship (i.e., the trend is not in the same direction for all age-specific rates or the ratio of age-specific rates is different for different age groups), the relationship of age-adjusted rates can vary with the choice of a standard population. If the pattern is not consistent, the use of age-specific rates, rather than age-adjusted rates, is recommended.

Guideline: US Standard Populations for Direct Adjustment
Guideline: Age-adjusted Rates, Direct Method

When the number of events is relatively small, the age-specific rates needed to calculate an age-adjusted rate by the direct method are unstable. This may result in unstable age-adjusted rates when using the direct method of age-adjustment. Additionally, since the age-adjusted rate calculated by the direct method provides a somewhat arbitrary summary statistic that depends on the choice of a standard, it may not provide the best summary measure in explaining health status to communities. An alternative approach is the development of ratios developed using indirect adjustment.

Guideline: Indirect Adjustment

Age-specific rates

Because age-adjusted rates can mask important trends or over- or under-estimate differences, age-specific rates are used for comparing age-defined subgroups when rates are strongly age-dependent. Age-specific rates are also used when specific causal or protective factors or the prevalence of risk exposures are different at different ages. For example, at highest risk for head injury are males 15-24 years of age (related to motor vehicle occupant injuries) and those 75 or older (mainly due to falls). Restricting the age range in the development of a rate is sometimes called an age-limited rate.

Guideline: Age-Specific Rates

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Should I calculate a rate when the number of events is small?

Rates based on small numbers of events can fluctuate widely from year to year for reasons other than a true change in the underlying frequency of occurrence of the event.

Guideline: Small Numbers

Will the rate be compared to other rates?

When calculating rates, the numerator and denominator (i.e., events and population) must be defined consistently over time and place. Areas where public health professionals are most likely to find inconsistencies include:

• The definition of the health event: Because analysts may classify diseases differently, it is useful to compare technical definitions of the number of events before comparing rates. For example, coronary heart disease is defined differently in Healthy People 2000 than in information from the National Center for Health Statistics; there are differences in the definitions of several cancer sites between the National Center for Health Statistics and the National Cancer Institute.
• The coding scheme: Coding schemes that may appear to collect the same information often contain subtle differences. For example, starting with 1999 deaths, state and federal agencies will code cause of death according to the International Classification of Diseases,10^th Revision (ICD-10). Differences in coding practices between ICD-9 and ICD-10 will affect mortality rates and trends for many disease categories. It is anticipated that hospitalization data will continue to be coded according to the ICD-9-Clinical Modification (CM) until 2001 or 2002 before converting to an ICD-10-CM. Differences in the coding of deaths versus hospitalizations needs to be considered when planning assessment projects that combine data from these two sources.
• Data collection and definitions: Data collection processes or definitions for the number of events or the population at risk may change over time.
• Geographies: Data collection may vary from one geographic area to another. This issue is generally more relevant to the number of events than the population at risk and more relevant to some types of events than others.

In addition to the previous issues, when comparing age-adjusted rates, the standard population must be the same for all rates to be compared. Different national, state and local agencies may use different standard populations when age-adjusting. International agencies also usually use different standard populations than those used in the United States.

When comparing age-specific rates, if the age categories are relatively large, it is important to consider the possibility of residual confounding by age. For example, if the proportions of very old individuals in the group "65 years and over" are different in two populations being compared, differences in rates may be a reflection of the difference in the age distribution of the populations. When age categories are relatively wide, consider developing age-specific rates using smaller age groups or age-adjust within the broader age group.

Guideline: Comparing Rates

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How do I know whether two rates are different?

Surveillance data, even if based on complete counts, may be affected by chance. If variation in the occurrence of the disease is random and not affected by differential diagnosing, reporting, or other systematic differences, confidence intervals (CIs) may be calculated to facilitate comparisons over time or between geographic locations (e.g. counties).

• When the number of events is small in relation to the population at risk (i.e., the event is rare), calculation of 95% CIs based on the Poisson probability distribution is recommended. (95% CIs correspond to a p-value of 0.05. If you are making many comparisons, remember that approximately 5% of the comparisons may be statistically significant due to chance alone.) For crude rates, the CI can be developed by using a table or computer software to determine the CI around the number of events and developing a rate based on the high and low confidence limits. For age-adjusted rates, use the method developed by Fay and Feuer, 1997. In general, if CIs for two separate rates overlap, there is no statistically significant difference between the two rates.

Narrow CIs for rates indicate with greater certainty that the calculated rate is a reliable approximation of the true rate, while wide CIs signal greater variability and less certainty that the calculated rate is a good estimation of the true rate.

Confidence intervals around rates account for random fluctuation but not bias. Bias is also known as systematic error. Bias can occur, for example, when reporting or measuring practices vary by geographic region, time period, or the person making the report. For example, if a large proportion of a county’s hospitalizations occur in hospitals that are not included in the statewide hospitalization database (such as, in military and veteran’s hospitals or out of state), the hospitalization rate for that county will be biased downward.

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Guidelines for Using and Developing Rates

Number of events (numerator)

• In developing a rate, it is customary to count only events that occur among the population at risk
• For conditions where one person may be counted more than once, note whether you are counting events or people.
• Tables containing rates should provide precise definitions of the number of events (e.g., specify ICD codes) so that people using the information can calculate comparable rates.
• When reporting rates, the data source for the number of events should be specified.

Population at risk (denominator)

• Everyone in the population at risk must be eligible to be counted in the numerator if they have the event of interest. For example, in looking at female breast cancer, we cannot include men in the population at risk, because men with breast cancer would not be included in the number of events.
• When reporting rates, the data source for the population at risk should be specified.

Crude rates

• Crude rates are recommended when a summary measure is needed and it is not necessary or desirable to adjust for age.
• Choose a constant (e.g., rate per 1,000 or rate per 100,000) that is compatible with commonly reported rates for the topic (e.g., birth information is generally reported per 1,000 live births; death information is commonly reported per 100,000 population.)

Age-adjusted rates, direct method (for method, see age-adjustment method)

• Age-adjusted rates are recommended when making comparisons in the rates of age-related health events between different populations or for comparing trends in a given population over time.
• Age-adjusted rates are essential for events that vary with age (e.g., cancer deaths), when comparing populations with different age distributions.
• Age-adjusted rates should be used only for the purpose of comparison. Because an age-adjusted rate is based on an external standard population, it does not reflect the absolute frequency of the event in a population.
• Age-adjusted rates should be presented when a single, summary measure is needed, but data analysts should inspect age-specific rates first. (Choi, 1999)
• Use of the US 2000 standard population is recommended in current analyses, unless there is a need to compare to data that have been adjusted to another standard. In the latter case, you must use the standard population used in the comparison data. See US Standard Populations.

Age-specific rates

• The development of age-specific rates is recommended before developing age-adjusted rates to determine whether the populations being compared show a consistent relationship among age-specific rates. If the pattern is not consistent, use of age-specific rates, rather than age-adjusted rates, are recommended.
• Age-specific rates are recommended for comparing age-defined subgroups between or within populations when rates are strongly age-dependent.
• Age-specific rates are recommended when specific causal or protective factors or the prevalence of risk exposures are different at different ages.
• In defining sub-groups for age-specific rates, select age ranges appropriate to the condition of interest.

Comparing rates

• Only compare rates when the numerator and denominator (i.e., events and population) are defined consistently over time and place. Look for
  • Consistency in definition of event
  • Consistency in coding scheme
  • Consistency over time
  • Consistency among geographies
• If comparing age-adjusted rates, compare rates that have been adjusted to the same standard population.
• When comparing age-specific rates, if the age categories are relatively large, it is important to consider the possibility of residual confounding by age.

Unstable rates due to small numbers

Rates based on small numbers of events can fluctuate widely from year to year for reasons other than a true change in the underlying frequency of occurrence of the event.

• Calculation of rates is not recommended when there are fewer than five events in the numerator, because the calculated rate is unstable and exhibits wide confidence intervals.
• Small counts should be included, where possible, even if the rates are not reported, so that the counts can be combined into larger totals (for example, three or five year averages) which would be more stable.

Methods for age adjustment

Direct method of age adjustment

Multiply the age-specific rates in the target population by the age distribution of the standard population.

Where m is the number of age groups, d_i is the number of cases (events or people) in age group i, P_i is the population in age group i, and s_i is the proportion of the standard population in age group i. This is a weighted sum of Poisson random variables, with the weights being (s_i / P_i).

US Standard Populations for Direct Adjustment

Currently, the National Center for Health Statistics (NCHS) age-adjusts rates using the US 1940 standard population with eleven age groups. These groups are: less than 1 year, 1-4 years and nine 10-year age groups beginning at age 5. The National Cancer Institute (NCI) uses the US 1970 standard population with eighteen 5-year age groups.

Starting with 1999 deaths, the estimated U.S. population in 2000 will become the standard population for age-adjusting death rates and cancer incidence rates. This will affect the size of age-adjusted rates, since the new standard will have a higher concentration of the middle-aged and older population (see Anderson, 1998, for the population pyramids for the two populations). Generally, the magnitude of age-adjusted death rates will increase for causes of death with higher rates in older people, and decrease for causes of death with higher rates in younger people. The age-adjusted mortality rate for total deaths will also be higher with the new population standard. The NCHS will use the same age groups as in the 1940 standard. The NCI may continue to use the eighteen 5-year age groups.

Below are the US 1940, 1970 and 2000 standard populations.

Indirect Adjustment

When the number of events in a community is small, or when developing statistics for use in communities concerned about the number of events, compare the observed number of events to the expected number, using indirect age-adjustment or age- and sex-adjustment.

• To develop a count of expected events using indirect adjustment, apply the age-specific rates in a larger population (e.g., the State of Colorado) to the number of people in the age-specific group in the population of interest, and total the results for all age groups. Ideally, the larger population should be large enough that the rates in that population are stable (i.e., exhibit little random variation).
• Compare the observed number of events (usually abbreviated "O") and the expected number obtained using indirect adjustment (usually abbreviated "E"). Generally, if the confidence interval (CI) around O does not include E, the observed number of events is statistically significantly different from the expected. (Breslow and Day, 1987) The Poisson confidence limits around O can be obtained from standard tables or can be calculated using several software packages. This method assumes that E is developed from stable rates (i.e., the standard population is large enough that there is little random variation in the rates for that population.) A more conservative estimate of statistical significance (and the estimate that should be used when the rates in the larger population are not as stable as one would like) is to develop CIs around both E and O. If the CIs do not overlap, the difference between O and E is statistically significant.
• A more precise method of determining whether the observed number of events is different from the expected is to develop a ratio of O to E (O/E) and conducting a statistical test to determine whether the ratio is statistically significant. Most biostatistics textbooks provide methods for conducting the statistical test. Fisher and van Belle (1993) provide an approximation for large samples only. Rosner (1990) provides methods for small and large samples.

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Glossary

Rate: A rate is a measure of the frequency of an event per population unit. The use of rates, rather than raw numbers, is important for comparison among populations, since the number of events depends, in part, on the size of the population.

Numerator: In calculating rates, the numerator is the number of events in a specified population.

Denominator: In calculating rates, the denominator is the number of people a specified population. Everyone in the denominator must be eligible to be counted in the numerator. The denominator is often called the "population at risk."

Crude rate: A crude rate is calculated by dividing the total number of events in a specified time period by the total number of individuals in the population who are at risk for these events and multiplying by a constant, such as 1,000 or 100,000 [e.g., (numerator/denominator) x constant].

Age Adjustment: Age-adjustment is the process by which differences in the age composition of two or more populations are removed, to allow comparisons between these populations in the frequency with which an age-related health event occurs.

Age-adjusted rate (direct adjustment): An age-adjusted rate adjusted by the direct method is "…the rate that would occur if the observed age-specific…rates were present in a population with an age distribution equal to that of a standard population." (Anderson, 1998)

Age-specific or age-limited rate: An age-specific rate is a rate in which the number of events and population at risk are restricted to an age group (e.g., the birth rate for women age 15 to 19; death rate for people age 45 to 64).

Standard population: The standard population refers to the choice of populations used in developing age-adjusted rates.

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References

Anderson RN, Rosenberg HM. Age standardization of death rates: Implementation of the Year 2000 Standard. National Vital Statistics Reports; vol. 47 no. 3. Hyattsville, Maryland: National Center for Health Statistics, 1998.

Breslow NE and Day NE. Statistical Methods in Cancer Research: Volume II – The Design and Analysis of Cohort Studies. New York: Oxford University Press, 1987.

Choi BCK, de Guia NA, and Walsh P. Look before you leap: Stratify before you standardize. American Journal of Epidemiology, 149: 1087-1096, 1999.

Fay MP and Feuer EJ. Confidence intervals for directly standardized rates: A method based on the gamma distribution. Statistics in Medicine, 16: 791-801, 1997.

Fischer LD and van Belle G. Biostatistics: A Methodology for the Health Sciences. New York: John Wiley and Sons, Inc., 1993.

Kuller LH. Age-adjusted death rates: A hazard to epidemiology? [Editorial] Annals of Epidemiology, 9(2): 91-2, 1999.

Last JM [ed]. Dictionary of Epidemiology, 3^rd Edition. New York: Oxford, 1995.

Rosner, BA. Fundamentals of Biostatistics, 3^rd Edition. Boston: PWS-Kent Publishing, 1990.

Selvin S. Statistical Analysis of Epidemiologic Data, 2^nd Edition. New York: Oxford University Press, 1996.

Sorlie PD, Thom TJ, Manolo T, Rosenberg HM, Anderson RN and Burke GL. Age-adjusted death rates: Consequences of the year 2000 standard. Annals of Epidemiology, 9:93-100, 1999.

Guidelines For Using and Developing Rates in Public Health Assessment (Word Document)

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