# About Age Adjusted Rates, 95% Confidence Intervals and Unstable Rates

## What is age adjustment?

Age adjustment is a statistical process applied to rates of disease, death, injuries or other health outcomes that allows communities with different age structures to be compared.

## Why do we do age adjustment?

Almost all diseases or health outcomes occur at different rates in different age groups. Most chronic diseases, including most cancers, occur more often among older people. Other outcomes, such as many types of injuries, occur more often among younger people. The age distribution affects what the most common health problems in a community will be. One way of examining the patterns of health outcomes in communities of different sizes is to calculate an incidence or mortality rate, which is the number of new cases or deaths divided by the size of the population. In chronic diseases and injuries, rates are usually expressed in terms of the number of cases or deaths per 100,000 people per year.

A community made up of more families with young children will have a higher rate of bicycle injuries than a community with fewer young children. A community with a larger number of older individuals will have higher rates of cancer than one with younger individuals. This is true even if the individuals in the two communities have the same risk of developing cancer or being injured. Epidemiologists refer to this as “confounding”. Confounding happens when the measurement of the association between the exposure and the disease is mixed up with the effects of some extraneous factor (a confounding variable). Age adjustment is a statistical way to remove confounding caused by age.

## How is age adjustment done?

Age confounding occurs when the two populations being compared have different age distributions and the risk of the disease or outcome varies across the age groups. The process of age adjustment by the direct method changes the amount that each age group contributes to the overall rate in each community, so that the overall rates are based on the same age structure. Rates that are based on the same age distribution can be compared to each other without the presence of confounding by age. Adjustment is accomplished by first multiplying the age-specific rates of disease by age-specific weights. The weights used in the age adjustment of cancer data are the proportion of the standard US population within each age group. The weighted rates are then summed across the age groups to give the age-adjusted rate.

## Example of age adjustment.

This is demonstrated on the cancer mortality rates for all sites of cancer among men in New York State in 2010. The crude (unadjusted) cancer mortality rate is 184.6 deaths per 100,000 men. The age-adjusted rate is 190.6 deaths per 100,000 men. The weights used in the age adjustment of the data are the proportion of the 2000 US standard population within each age group.

Age group Number of Deaths (a) Population (b) Rate per 100,000 (c=(a ÷ b) x 100,000) Weight (d) Weighted Rate (c x d)
00 3 120,593 2.5 0.013818 0.0
01-04 15 472,352 3.2 0.055316 0.2
05-09 10 593,355 1.7 0.072532 0.1
10-14 18 617,659 2.9 0.073031 0.2
15-19 24 696,154 3.4 0.072167 0.2
20-24 43 714,642 6.0 0.066478 0.4
25-29 51 683,675 7.5 0.064530 0.5
30-34 53 633,109 8.4 0.071045 0.6
35-39 88 610,826 14.4 0.080762 1.2
40-44 202 661,829 30.5 0.081852 2.5
45-49 469 708,239 66.2 0.072117 4.8
50-54 911 689,400 132.1 0.062718 8.3
55-59 1,433 595,465 240.7 0.048454 11.7
60-64 1,976 505,042 391.3 0.038794 15.2
65-69 2,100 354,467 592.4 0.034264 20.3
70-74 2,192 259,658 844.2 0.031772 26.8
75-79 2,439 199,573 1222.1 0.026998 33.0
80-84 2,489 151,434 1643.6 0.017843 29.3
85+ 2,822 123,878 2278.0 0.015509 35.3
Total 17,338 9,391,350 184.6   190.6

## What is a confidence interval?

A confidence interval is a range around a measurement that conveys how precise the measurement is. Ninety-five percent confidence intervals are provided for the age-adjusted rates in Cancer Incidence and Mortality in New York State. These are analogous to the margins of error that are provided for news polls.

In the simplest terms, an age-adjusted breast cancer incidence rate of 124.5 cases per 100,000 women with a confidence interval of 122.5 - 126.6 cases per 100,000 means that there is a 95 percent chance that the rate was between 122.5 and 126.6 cases per 100,000 women. Conversely, there is a 5 percent chance that the rate was lower than 122.5 or higher than 126.6.

The precise statistical definition of the 95 percent confidence interval is that if the measurement were conducted 100 times, 95 times the true value would be within the calculated confidence interval and 5 times the true value would be either higher or lower than the range of the confidence interval.

## What are confidence intervals used for?

Confidence intervals are often used in research requiring that a measurement be taken on a sample of the population, such as a survey. In that case, the confidence intervals are a way to measure sampling error and are related to the size of the sample taken (the number of people surveyed). The cancer incidence and mortality rates derived from the Cancer Registry and vital records are not samples and are, therefore, not subject to sampling error. The rates are, however, subject to what is termed “random error”, which arises from random fluctuations in the number of cases over time or between different communities. The 95 percent confidence intervals are an easily understood way to convey the stability of the rates. A stable rate is one that would be close to the same value if the measurement were repeated, i.e., if the rate did not vary greatly from one year to the next. An unstable rate is one that would vary from one year to the next due to chance alone. Wider confidence intervals in relation to the rate itself indicate instability. For example, if the rate is 5 cases per 100,000 persons, but the 95 percent confidence interval is plus or minus 2.5 cases per 100,000 persons, then the rate is relatively unstable. In one year, you might have a rate of 3 cases per 100,000 and in the next year have 6 cases per 100,000. This would be a 100 percent increase in the rate, but would still be within the range of the confidence intervals.

On the other hand, narrow confidence intervals in relation to the rate tell you that the rate is relatively stable, and you would not expect to see large fluctuations from year to year. If differences are observed between stable rates (those with narrow confidence intervals), then it is likely that the differences represent true variations, rather than random fluctuations in the number of cases.

## How are confidence intervals calculated?

Confidence intervals are calculated based on the standard error (se) of the rate. The standard error, in turn, is based on the rate and the number of cases or deaths. For this report, 95 percent confidence intervals for age-adjusted rates are calculated using Fay and Feuer method with Tiwari et al. modification.

## Why are rates based on fewer than 20 cases considered unstable?

Annual rates based on fewer than 20 cases or five-year average rates based on fewer than 4 cases per year are considered unstable because they have a large relative standard error (RSE). The RSE is the standard error as a percent of the measure itself. For age-adjusted incidence and mortality rates, the RSE is equal to 1÷ √cases. A RSE of 50 percent indicates that the standard error is half the size of the rate.

The RSE of an incidence or mortality rate is based on the number of cases or deaths, unlike the standard error and confidence intervals, which are based on both the number of cases and the size of the population.

Examples:

Suppose there are 20 testicular cancer deaths among males in New York State excluding New York City every year. The rate of testicular cancer deaths is 0.4 per 100,000 males, and the standard error is 0.09. The relative standard error is 22 percent.

If there are about 20 prostate cancer deaths among males in Orange County every year, the rate is 21.9 deaths per 100,000 males; the standard error is 4.8. Again, the relative standard error is 22 percent.

So, even though the standard error for the testicular cancer rate is much smaller than the standard error for the prostate cancer rate, both have the same magnitude relative to the rate itself. This means that when there are only a few cases or deaths, small changes in the number of cases or deaths have a much bigger effect than small changes in a large number of cases or deaths. In other words, going from 10 deaths to 20 deaths reduces the RSE from 32 percent to 22 percent, while going from 60 deaths to 70 deaths reduces the RSE from 13 percent to 12 percent. It is somewhere around 20 deaths that the curve seen below in Chart 1 starts to level out. Hence, rates based on fewer than 20 deaths, in the steep end of the curve shown, are highly variable and, for that reason, are unreliable.

Chart 1. Relative Standard Error of an Incidence or Mortality Rate as a Function of the Number of Cases or Deaths