# 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 population size of the study population by the same age/sex groups. 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 2 Calculating 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 an 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, or when lack of age-specific mortality rates precludes calculation of directly-age-standardized mortality rates.

Let's look at an example. In 2017, the all-cause death rate in New Mexico in 2017 was 888.1 deaths per 100,000 population. All other things being equal, we should expect the same death rate in De Baca County, New Mexico. BUT all other things are NOT equal. In addition to a potential difference in risk factors, in 2017 the percentage of the population over age 65 was...
• 27% in De Baca County, compared with
• 17% statewide.

In an older population, we would expect a higher death rate, and De Baca County's death rate is higher: 1,882.4 all-cause deaths per 100,000 population (compared with 888.1 in New Mexico, overall). Was the risk of mortality greater in De Baca County than in the rest of the state in 2017? The Standardized Mortality Ratio (SMR) can help us control for the age differences to better understand the relative mortality risks in De Baca County and New Mexico, overall. 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).

# Calculating SMR

In this example, we're using the state of New Mexico as the standard population, and De Baca County as the study population. The formula is simple. There are just a few steps involved in calculating the expected number of deaths.

1. The first step in calculating the SMR is to calculate the age- and sex-specific death rates in the standard population (column C).
(You can run an NM-IBIS mortality query to get these crude death rates.)
2. Next, enter the population size estimate for each age-sex group in Column D.
(You can run an NM-IBIS population estimates query to get these population estimates.)
3. Calculate the number of expected deaths in each age-sex group in Column E. using the formula: (C x D)/100,000.

4. 5. Sum across all the age- and sex-specific expected deaths to get the expected number of deaths for the study population. In the above example, the expected number of deaths was 26.5.
6. The observed number of deaths, that is, the actual number of deaths in De Baca County during the same time period, was 35. Now we can see that De Baca County had a higher-than-expected number of all-cause deaths during 2017.
7. Using the formula Observed/Expected, we get an SMR of 1.32. An SMR greater than 1.0 indicates that there were "excess deaths" compared to what was expected.

Observed De Baca County deaths: 35
Expected De Baca County deaths: 26.5

SMR

= O/E
= 35 / 26.5
= 1.32 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 1.32 indicates that there are significantly more deaths than were 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.

95% C.I. = 1.96 x s.e.SMR

The standard error for the SMR is calculated as follows:

s.e.SMR = (Square Root of O) / E

= (Square Root of 35) / 26.5
= 5.92 / 26.5
= 0.22339

1.96 x s.e.SMR = 0.438 (95% confidence interval, plus or minus)
1.32 - 0.438 = 0.88 (lower limit of 95% confidence interval)
1.32 + 0.438 = 1.76 (upper limit of 95% confidence interval) As you can see, in our example, the 95% confidence interval of the SMR does include the value "1.0," indicating that the observed number of deaths in De Baca County is not statistically significantly higher than the expected number of deaths. 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 1.32 x 100 = 132, indicating that the observed deaths were 132% 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).

In our example, the 2017 crude all-cause death rate in New Mexico was 888.1 deaths per 100,000 population, and the crude rate in De Baca County was 1,882.4. To calculate the indirectly age- and sex-standardized death rate for De Baca County, the crude death rate in standard population (888.1) is multiplied by the SMR for De Baca County (1.32), yielding an indirectly standardized De Baca County rate of 1,172.3. The indirectly age-standardized rate for De Baca County (1,172.3) is still higher than the state rate (888.1), but the effect of De Baca County's older population has 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 35 deaths in De Baca County 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.