Indirect Standardization And Standardized Ratios

Why Indirect Standardization Is Useful

A target population may have stratum-specific counts too small to estimate stable rates, even when its total count is informative. Indirect standardization compares the target’s observed total with the count that would be expected if its stratum-specific exposures experienced external reference rates. The strata might be age groups, for example, and the reference rates might come from a larger population.

The method answers a specific question: How does the target’s observed total compare with the total expected at the reference rates, given the target’s own exposure distribution? It does not, by itself, produce a rate standardized to a common population structure. That distinction is important when interpreting or comparing standardized ratios.

Definition: Let $i=1,\ldots,k$ index mutually exclusive, exhaustive strata. In stratum $i$, let $T_i$ be the target population’s exposure, $O_i$ its observed count, and $r_i^0$ an external reference rate expressed per the same exposure unit and time scale. The reference-expected count is $E_i^0=T_i r_i^0$, and the total expected count is $E^0=\sum_{i=1}^k T_i r_i^0$. When $E^0>0$, the standardized mortality ratio (SMR), or standardized incidence ratio (SIR), is $O/E^0$, where $O=\sum_{i=1}^k O_i$.

The name depends on the event being counted: mortality for an SMR, incidence for an SIR. The calculation is otherwise the same. An SMR or SIR of 1 means the observed total equals the reference-expected total; a value above 1 means it is larger, and a value below 1 means it is smaller. For example, a value of 1.20 means that the observed count is 120% of the expected count, or 20% higher. It is a ratio of counts, not a rate with units.

Calculating Expected Counts and the Ratio

Each reference rate is applied only to the target exposure in its own stratum. This preserves the target’s actual distribution across strata while asking what would be expected under the reference rates. The resulting expected counts are added before the observed-to-expected ratio is formed.

Worked Example: An SMR With Two Age Strata

A target population contributes 8,000 person-years in a younger stratum and 2,000 person-years in an older stratum. The external mortality rates are 0.002 and 0.020 deaths per person-year, respectively. The target records 28 deaths in the younger stratum and 52 in the older stratum. Calculate and interpret its SMR.

Expected deaths in the younger stratum: $8{,}000(0.002)=16$.

Expected deaths in the older stratum: $2{,}000(0.020)=40$.

Therefore $E^0=16+40=56$. The observed total is $O=28+52=80$, so

\operatorname{SMR}=\frac{80}{56}=\frac{10}{7}\approx 1.429.

The observed death count is about 1.43 times the count expected at the reference rates for this target’s age-specific person-time, or about 43% higher. This comparison alone does not establish why the counts differ, nor does it imply that the target’s age-specific mortality rates are each 43% above the reference rates.

When the Ratio Is a Weighted Average of Rate Ratios

To clarify what an SMR or SIR summarizes, suppose the target has a true rate $\lambda_i$ in stratum $i$, so its mean count at fixed exposure is $T_i\lambda_i$. The ratio of the target’s mean total count to its reference-expected total can be expressed using stratum-specific rate ratios, but only when those ratios are defined wherever the target contributes exposure.

Theorem: Suppose $E^0=\sum_i T_i r_i^0>0$ and $r_i^0>0$ for every stratum with $T_i>0$. Then $$ \frac{\sum_i T_i\lambda_i}{E^0} = \sum_{i:T_i>0} \frac{T_i r_i^0}{E^0}\frac{\lambda_i}{r_i^0}. $$ The weights $T_i r_i^0/E^0$ are nonnegative and sum to one. Thus the ratio of mean counts is a weighted average of the stratum-specific rate ratios, with weights proportional to reference-expected counts.

Proof. For each stratum with $T_i>0$, positivity of $r_i^0$ gives $T_i\lambda_i=T_i r_i^0(\lambda_i/r_i^0)$. If $T_i=0$, the target mean-count contribution is zero. Hence

\frac{\sum_i T_i\lambda_i}{E^0} = \frac{\sum_{i:T_i>0}T_i r_i^0(\lambda_i/r_i^0)}{E^0} = \sum_{i:T_i>0}\frac{T_i r_i^0}{E^0}\frac{\lambda_i}{r_i^0}.

The weights sum to $\sum_i T_i r_i^0/E^0=1$, and each is nonnegative. This proves the result. $\square$

The theorem concerns mean counts, not a claim that an observed SMR equals a weighted average of known stratum-specific rate ratios. In applications, $\lambda_i$ is generally unknown, and the observed ratio $O/E^0$ also varies with the observed counts.

Worked Example: Checking the Weighted-Average Identity

Use the two-stratum data from the first example. The target’s observed stratum-specific rates are $28/8{,}000=0.0035$ and $52/2{,}000=0.026$ deaths per person-year. Relative to the reference rates, these observed rate ratios are $0.0035/0.002=1.75$ and $0.026/0.020=1.30$.

The reference-expected counts are 16 and 40, so the weights are $16/56=2/7$ and $40/56=5/7$. Their weighted average of the observed rate ratios is

\frac{2}{7}(1.75)+\frac{5}{7}(1.30) =0.50+0.928571\ldots =1.428571\ldots.

This equals the observed SMR $80/56=1.428571\ldots$. The calculation works because both reference rates are positive in strata with target exposure. In general, if the stratum-specific rates are estimated by $O_i/T_i$, substituting them into the identity reproduces $O/E^0$.

Zero Reference Rates and the Required Condition

A zero reference rate requires care. If a stratum has positive target exposure and zero reference rate, its reference-expected count is zero. The overall expected count may still be positive because of other strata, so an SMR or SIR can be calculated. But a target rate ratio $\lambda_i/r_i^0$ in that stratum is undefined when $r_i^0=0$. The weighted-average theorem therefore cannot include that stratum as a rate-ratio term.

Worked Example: Why a Zero Reference Rate Cannot Be Ignored

Suppose the target contributes 1,000 person-years in stratum 1 and 500 in stratum 2. The reference rates are 0.002 and 0 deaths per person-year. Suppose the target’s true rates are 0.003 and 0.002 per person-year.

The reference-expected counts are $1{,}000(0.002)=2$ and $500(0)=0$, giving $E^0=2$. The target’s mean counts are $1{,}000(0.003)=3$ and $500(0.002)=1$, giving a mean total of 4. Thus the ratio of mean counts is $4/2=2$.

The stratum 1 rate ratio is $0.003/0.002=1.5$, but the stratum 2 rate ratio $0.002/0$ is undefined. The only positive reference-expected-count weight is on stratum 1; using it alone gives 1.5, not 2, because it omits the target mean count from stratum 2. The overall ratio is calculable, but it is not a weighted average of the stratum-specific rate ratios in this situation. The weighted-average identity requires a positive reference rate for every stratum with positive target exposure, unless any excluded stratum contributes zero target mean count and both totals are explicitly restricted to the included strata.

A Proportional-Rates Interpretation

A useful special case assumes that the target’s true rate in every stratum is a common multiple of the reference rate. This gives the ratio a simple population-level interpretation. The observed SMR or SIR remains a random quantity, however, and need not equal that common multiple in a particular data set.

Theorem: Suppose $r_i^0>0$ whenever $T_i>0$, and the target rates satisfy $\lambda_i=\theta r_i^0$ in all strata with positive exposure. Then the ratio of the target’s mean total count to the reference-expected total is $\theta$. If, in addition, the stratum counts are independent Poisson random variables with means $T_i\lambda_i$, then $E[O/E^0]=\theta$.

Proof. Using the proportionality assumption,

\frac{\sum_i T_i\lambda_i}{E^0} = \frac{\sum_i T_i\theta r_i^0}{E^0} = \theta\frac{\sum_i T_i r_i^0}{E^0} =\theta.

Under the additional Poisson assumption, linearity of expectation gives $E[O]=\sum_i E[O_i]=\sum_i T_i\lambda_i=\theta E^0$. Since $E^0$ is fixed and positive, $E[O/E^0]=E[O]/E^0=\theta$. This proves both claims. $\square$

Worked Example: A Common Rate Multiplier

A target contributes 10,000 person-years in stratum 1 and 1,000 in stratum 2. The reference rates are 0.001 and 0.010 deaths per person-year. Suppose the target’s true rate in each stratum is 1.2 times its reference rate.

The reference-expected counts are $10{,}000(0.001)=10$ and $1{,}000(0.010)=10$, so $E^0=20$. The target rates are 0.0012 and 0.012. Their mean counts are $10{,}000(0.0012)=12$ and $1{,}000(0.012)=12$, for a mean total of 24. The ratio of mean counts is $24/20=1.2$, as the theorem predicts.

If the observed total happens to be 25, the observed SMR is $25/20=1.25$, not 1.20. The common multiplier describes the rates and the ratio of mean counts under the assumption; random observed counts need not match that value exactly.

Why Indirect and Direct Standardization Differ

Direct standardization applies target stratum-specific rates to a chosen common set of standard-population weights. Indirect standardization instead applies external stratum-specific rates to the target’s own exposures and divides the target’s observed total by the resulting expected total. Thus the direct method uses a common stratum distribution, while the indirect ratio’s effective weights are proportional to the target-specific expected counts $T_i r_i^0$.

The difference is visible even in the first example. Suppose the chosen direct-standardization weights are one-half for each age stratum. The target’s observed rates are 0.0035 and 0.026, so its directly standardized rate is

$$ 0.5(0.0035)+0.5(0.026)=0.01475 $$

deaths per person-year. Applying the same equal weights to the reference rates gives $0.5(0.002)+0.5(0.020)=0.011$. The ratio of those directly standardized rates is $0.01475/0.011\approx1.341$, whereas the SMR is about 1.429. These are different calculations with different weights, so there is no contradiction.

In particular, SMRs or SIRs for two target populations need not be directly comparable as if they were standardized rates on a shared stratum distribution: their expected-count weights can differ because their exposures differ. Multiplying an SMR by a reference crude rate is sometimes used to report an indirectly standardized rate, but that rate depends on the chosen reference population and its crude rate. It does not generally remove the target-specific weighting issue.

Common Mistakes and Practical Checks

Using the wrong exposure for a reference rate. Check that each $T_i$ is expressed in the same unit and time scale used to define $r_i^0$. For a rate per person-year, the exposure must be person-years.
Averaging stratum rates without weights. The indirect calculation sums expected counts $T_i r_i^0$; it does not take an unweighted average of the reference rates.
Calling an SMR a standardized rate. An SMR or SIR is a dimensionless observed-to-expected ratio. A directly standardized rate has rate units and uses chosen common weights.
Interpreting an SMR as a causal effect. The ratio is a descriptive comparison under the stated reference rates and strata. By itself, it does not identify a cause for a difference.
Dropping a zero-reference-rate stratum from the numerator’s interpretation. A stratum with positive target exposure and a positive target rate contributes to the target’s mean count even if its reference-expected count is zero. The overall ratio may remain calculable, but the weighted-average rate-ratio identity does not apply as stated.

Key takeaway: Compute each stratum’s expected count as target exposure times the matching external rate, sum those expected counts, and divide the observed total by that sum. Interpret the result relative to the target’s own exposure structure; do not treat it as a directly standardized rate or assume that it is generally comparable across targets.

Check Your Understanding

Try these questions using the definitions and conditions above.

A target has exposures of 3,000 and 1,000 person-years in two strata with reference rates 0.004 and 0.015. What are the stratum-specific and total expected counts?
If the observed total is 30 and the total expected count is 24, calculate the SIR and state its interpretation in context.
What condition on the reference rates is needed for the theorem expressing the ratio of mean counts as a weighted average of stratum-specific rate ratios?
Explain why an SMR for one target population need not be directly comparable to an SMR for another target population.
If target rates are a common multiple $\theta$ of all relevant reference rates, what is the ratio of mean total count to reference-expected count? Is every observed SMR exactly equal to $\theta$?