## Concept Description

Last Updated: 2001-05-03

### Introduction

Rates of events, typically by geographical regions, are frequently computed in epidemiologic and health services research. There are many different types of rates which can be computed and compared.

### A. Rates - Definitions

• Rate - calculated by dividing the number of events (e.g. deaths among persons aged 0-64) by the population at risk of experiencing the event (e.g. total population aged 0-64) in the defined "study population" over a particular unit of time (e.g. one year).

• Crude (overall) rate - the actual number of events over all confounding variable(s) (e.g. age groups, gender) divided by the total (eg. over all age groups, gender) population. A crude rate is a summary of the specific rates.

• Specific rate - Specific the number of events within a specified set of confounding variable(s) (e.g. age group & gender) in a "study population" divided by the total population within the set of confounding variable(s). For example, hospitalizations among males, aged 60-64 divided by the population aged 60-64.

• Standard population - the reference population used to standardize rates; it can be the total population of the areas, or, when comparing multiple sets of data (i.e., multiple years of events) it would be one specific set of data (i.e., one of the years.) The choice of the reference population is dependent upon the study question.

• Standardized rate - a summarized rate which essentially adjusts a crude rate by accounting for one or more confounding variables. Direct and indirect rates are both examples of standardized rates.

• Direct rate - the expected number of events per 1000 people if the study population had the same distribution of confounding groups (i.e., sex and age groups) as did the standard population.

• Indirect rate - the expected number of events per 1000 people if the study population had the same rate of events for each confounding group as did the standard population.

• Standard Mortality / Morbidity Ratio (SMR) - the ratio of the number of events observed in the study population to the number of events that would have been expected in the study population, if the study population had the same rate of events for each confounding group as did the standard population.

Note: the choice of 1000 as a base is arbitrary, it could be any number for which we wish to express the rate as "so many events per that many people". Generally the number is also chosen so that the rate is expressed as a whole number (e.g. 1.245 per 1,000 rather than 0.1245 per 100).

### B. Rates of Rare Events

When computing rates of rare events, keep in mind that the method of grouping the data could change the rates significantly.
Example: When calculating premature mortality rates, using 5-yr age groups (i.e., 1-4, 5-9, 10-14, etc.) can lead to a large number of strata with 0 deaths, even when multiple years of mortality data are used.
Solution: Use larger age groups (e.g., 0-14, 15-24, etc.) to group data. The smaller the population being studied, the larger the age grouping should be. Age groups should be representative of the data you are working with. If you are unsure as to whether your age groups need to be refined, try running a histogram of the population groups.

### C. Comparisons of Rates

Methods:

Carriere and Roos (1994),(1997) have developed methods that may be used to compare standardized rates of events.

The methods use a non-parametric approach, and may be applied to binary and non-binary events, recurrent and non-recurrent events, regardless of the incidence rate of the event and regardless of whether the data are person-based.

• A non-parametric approach means that we need not assume that the data originate from a known distribution, i.e., normal or binomial.

• A binary event is an event that either occurs or does not occur. For example, a person is either admitted to the hospital or is not. A non-binary event is an event which cannot be classified on a yes/no basis; Length of stay or number of discharges for example.

• A non-recurrent event is an event that can occur only once, e.g., an appendectomy. A recurrent event is an event which can occur more than once, e.g., a myocardial infarction.

• Person-level data is individual-level data. The data have not been summarized.

### Multiple versus single records per person

• If there are multiple records per person, the event is recurrent and may be studied as a binary event (e.g. at least one admission for asthma) or as a non-binary event (e.g. number of admissions for asthma, or number of days in hospital).

• If there is only a single record per person, the event may be recurrent but only one record per person is chosen (randomly or by using first.phin for example) and may be studied as a binary event (i.e., at least one admission for myocardial infarction) or as a non-binary event (i.e., length of stay).

• The event for the single record may be non-recurrent and may be studied as a binary event (i.e., an appendectomy) or as a non-binary event (i.e., length of stay).

### Comparing Rates:

Definitions for the list below:

• Overall Test - an overall test answers a general question of 'Is there a difference?'

• Specific Test - If the answer to an overall test is 'Yes, there is a difference', a specific test will answer the question 'Where exactly is the difference?

A. Comparing rates for different areas (See POP Rate Macro Documentation below for an example and relevant SAS code for each test described) (internal access only) .
Test 1 - An overall test to determine if all sub-areas have the same rate. A test that is significant indicates that "not all the rates are the same." BUT, it does not identify which rate(s) differ.

Test 2 - Overall test to determine if the rates for each area are the same as:
a. the rate for the total area.
If the test is significant, then "at least one area has a rate that differs from the rate for the total area." BUT, it does not identify which area(s) have a different rate. See Test 3a for the follow-up test.

b. any pre-specified rate.
If the test is significant then "at least one area has a rate that differs from the pre-specified rate" BUT, it does not identify which area(s) have a different rate. See Test 3b for the follow-up test.

Test 3 -Specific test to determine which rates differ from:
a. the rate for the total area (follow-up to Test 2a)
If at least one area has a rate which is different from the rate for the total area this test will identify the area that differs. Each area's rate must be compared with the rate for the total area (after adjusting for multiple comparisons).

b. any pre-specified rate (follow-up to Test 2b)
If at least one area has a rate which is different from the pre-specified rate this test will identify the area(s) that differ(s). Each area's rate must be compared with the pre-specified rate (after adjusting for multiple comparisons)
B. Comparing rates for different groups
Test 4 - Test if the rates for each area are the same for two groups (overall test).
Test 5 - Test which areas have rates which are different for the two groups (specific test).
Test 6 - Test if the differences in rate for each area are the same for two groups.

C. Comparing the degree of variation in rates
Test7 - Compare degrees of variation in the rates.
a. for two groups.
b. for two regions.
These comparisons may be carried out using a T 2 statistic, which follows a chi-square distribution, or using a ratio of two T 2 statistics, which follows an F distribution.

If the rates for each area are not the same, they can be further examined by testing for a linear trend.

### The POP_RATE Macro

The pop_rate macro developed by MCHP computes the crude (C), directly standardized (D), and indirectly standardized (I) rates for events. A description of the macro is available as a Pop_Rate pdf file (internal access only) .

The tests described above can be carried out as follows:

• To test if the rates for the various areas are equivalent, examine the P-value (PC, PD, or PI, depending on which type of rate is being considered.) If the P-value is small, smaller than alpha, the rates are not equivalent, i.e., at least one rate is different from another.

• The 2nd test is not computed by the POP_RATE macro, however, using the output produced by the macro, the test-statistic may be computed as described in Carriere and Roos (1994).

• To determine which rates differ from the pre-specified rate, first carry out a Bonferroni adjustment (adjusts the level of significance) for multiple comparisons ( Carriere and Roos (1997). ), then set this value to be the confidence level on which the confidence intervals are based. The resulting confidence interval (LC,UC),(LD,UD), or (LI,UI) depending on which type of rate is being considered) may be examined for each area to see if the pre-specified rate falls inside. An area whose interval includes the pre-specified rate has a rate which does not differ from the pre-specified rate, while an area whose interval does not include the pre-specified rate has a rate which differs from the pre-specified rate.

The 4th, 5th, 6th and 7th tests (points B & C above) are not computed by the POP_RATE macro. However, by creating new areas, one for each group in each area, and using the output produced by the macro, the test statistics may be computed as described in Carriere and Roos (1994).

The test for linear trend is not computed by the POP_RATE macro. However, using the output produced by the macro, the test statistic may be computed.
Note: The default for the POP_RATE macro is to use a logarithmic transformation when calculating the T 2 statistics. When calculating, by hand, test-statistics using output produced by the POP_RATE macro, you should use the logarithmic transformation as described in Carriere and Roos (1994), (1997) .

• Carriere KC and Roos LL.(1994) Comparing Standardized Rates of Events. American Journal of Epidemiology; 140: 472-82.
• Lilienfeld AM. (1976). Foundations of Epidemiology. New York: Oxford University Press.
• Manitoba Centre for Health Policy and Evaluation (1996). Population Health Information System Rates Macro. Version 6. University of Manitoba.

## References

• Carriere KC, Roos LL. A method of comparison for standardized rates of low-incidence events. Medical Care 1997;35(1):57-69. [Abstract] (View)
• Last JM. A dictionary of epidemiology. 3rd edition. New York, NY: Oxford University Press; 1995. 0-0.(View)

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