# Standardized Mortality Ratio

Standardized Mortality Ratio (SMR) is a ratio between the observed number of deaths in an study population and the number of deaths would be expected, based on the age- and sex-specific rates in a standard population and the age and sex distribution of the study population. If the ratio of observed:expected deaths is greater than 1.0, there is said to be "excess deaths" in the study population.

A closely related construct, indirectly standardized rates, is also described in this Web page.

Contents |

1 Using SMR 3 Statistical Significance of SMR 4 Indirectly Standardized Rates 5 Confidence Intervals for ISR References |

# Using SMR

The SMR is used to compare the mortality risk of a study population to that of a standard population. It is especially applicable where the two populations have dissimilar age distributions, and in cases where direct age standardization may not be appropriate because the study population is small. The Standardized Mortality Ratio (SMR) can help us tease out the age differences to better understand the relative mortality risk. SMR is expecially useful in a small population, where direct age adjustment is not feasible (i.e., when there are fewer than 25 deaths in the study population)

The formula is simple. There is just a little work involved in calculating the expected number of deaths.

As with any age-adjusted rates, indirectly age standardized rates should be viewed as relative indexes,
and used for comparison of populations. They are not actual measures of mortality risk, and do not
convey the magnitude of the problem.

To read more about the Standardized Mortality Ratio, see Lilienfeld & Stolley (1994), Curtin & Klein (1995) and Fleiss (1981).

## Statistical Significance of SMR

How can you know whether an SMR of 1.28 indicates that there are significantly more deaths than what is expected? Conceptually, if the observed number of deaths is equal to the expected number, the SMR would have a value of 1.0. So the statistical test for the significance of SMR is whether it is different from 1.0. To gauge statistical significance of SMR, we must first calculate the 95% confidence interval for the SMR. If the 95% C.I. excludes the value, "1.0," it may be considered statistically significant.

As with other similar statistics, the 95% Confidence Interval is equal to 1.96 times the standard error of the estimate. The standard error for the SMR is

As you can see, in our example, the 95% confidence interval of the SMR

Often, you will see the SMR expressed after multiplying it by 100. If you see it this way,
then an SMR of 100 indicates that observed=expected, and an SMR over 100 indicates "excess
deaths." You can also think of this as a percentage, where 1.28 x 100 = 128, indicating that the observed
deaths were 128% of expected.

# Indirectly Standardized Rates

Once the SMR is known, it is a small step to calculate the indirectly age-standardized rate: one simply multiplies the crude rate of the standard population by the SMR (Curtin & Klein).For example, let's say that the 2006 crude all-cause death rate in a state was 757.5 deaths per 100,000 population, and the crude rate for a given was 1364.6. To calculate the indirectly age- and sex-standardized death rate for that county, the crude death rate in standard population (757.5) is multiplied by the SMR for that county (1.28), yielding an indirectly standardized rate for that county of 969.6. 969.6 is still higher than the state rate, but the effects of that county's age distribution have been removed.

It should be noted that indirect age standardization was applied in our example, but because there
were more than 25 health events in the study population (there were 61 deaths in that during
the measurement period), direct age standardization would also have been appropriate.

The same SMR logic and method of calculation may be applied to other health events. When SMR is applied
to deaths, it is called the Standardized Mortality Ratio, but when it is applied to non-fatal health
events, it is called the Standardized Morbidity Ratio.

## Confidence Intervals for Indirectly Standardized Rates (ISR)

For indirectly standardized rates based on events that follow a Poisson distribution and for which the ratio of events to total population is small (<.3) and the sample size is large, the following two methods can be used to calculate confidence interval (Kahn & Sempos, 1989).

### (1) When the number of events >20:

Where...

- SMR = observed deaths in the index area/expected deaths in the index area
- e = expected deaths in the index area = SUM(Rsi x Pi)
- Rs = the crude death rate in the standard population
- Rsi = the age-specific death rate in age group i of the standard population ( number of deaths / population count]
- Pi = the population count in age group i of the small area
- K = a constant (e.g., 100,000) that is being used to communicate the rate

### (2) When the number of events <=20:

Where...

- LL is the lower confidence interval limit, and
- UL is the upper confidence interval limit.

Strictly speaking, it is not valid to compare one indirectly-standardized rate with another. However,
the amount of bias will be small in most cases. See Rothman & Greenland (1998)

## References

1. Lilienfeld, DE and Stolley, PD.

*Foundations of Epidemiology*, 3rd Ed. Oxford University Press, 1994.

2. Curtin, LR, Klein, RJ.

*Direct Standardization (Age-Adjusted Death Rates)*. Statistical notes; no.6. Hyattsville, Maryland: National Center for Health Statistics. March 1995.

3. Fleis, JL.

*Statistical methods for rates and proportions*. John Wiley and Sons, New York, 1973.

4. Rothman, Kenneth J. and Greenland, Sander (1998)

*Modern Epidemiology*(2nd Ed.). Philadelphia, PA: Lippincott.

5. Harold A. Kahn and Christopher T. Sempos (1989)

*Statistical Methods in Epidemiology*. New York: Oxford University Press.