# Direct Standardization

This function calculates directly standardized rates (DSR) with approximate confidence intervals.

DSR is simply a weighted mean event rate for a study population, using the group/stratum sizes of a reference population as the weighting scheme. Standardized or adjusted rates are summary index measures for the purpose of comparison only; their magnitude has no intrinsic value.

The choice of a reference or standard population is important; it must relate to the population under study naturally.

Please note that standardization is not a substitute for individual comparisons of stratum-specific rates.

This method is unreliable with small numbers; there should be at least 25 events observed overall and at least one event in each stratum. If the number of events is small, consider aggregating strata.

Direct standardization is not appropriate if there is not a consistent relationship between stratum-specific rates in different populations being compared.

There are a lot of pitfalls in using directly standardized rates; if you have any doubts then please consult with an Epidemiologist and/or Statistician.

Data input

• Number of events for each group from the index/study population
• Person-time for each group from the index/study population (e.g. size of each group if just one year and all subjects were followed up)
• Group sizes or weights from a reference/standard population
• Group/stratum labels, e.g. age bands

Note than an alternative binomial method is provided for situations where your observed rates are too large for the Poisson distribution to be used, namely one or more rates r are not so small that 1-r can be considered almost equal to 1.

Technical validation

Approximate confidence intervals for the DSR are calculated firstly by Chiang's normal approximation to Poisson rate sums (Chiang, 1961; Keyfitz, 1966; Breslow and Day, 1987; Armitage and Berry, 1994) and secondly by an improved approximation adjusted for the total number of observed events (Dobson et al., 1991).

- where v is the approximate (Chiang) variance, wi is the reference weight for the ith stratum, ri is the observed study rate for the ith stratum, Ni is the reference population size for the ith stratum, yi is the number of events observed in the ith stratum of the study population, ni is the person-time for the ith stratum of the study population, zα/2 is the (100 * α/2) the centile of the standard normal distribution, Y is the total number of events observed, Yl and Yu are the exact lower and upper confidence limits for the Poisson count Y and ICI l to u is the improved confidence interval due to Dobson et al. For large rates, the binomial variance is used, where r(1-r) is substituted for r in the variance formula above.

Example

Test workbook (Rates worksheet: Age Bands, Index Events, Index Group Sizes, Reference Sizes).

The following data relate to stroke deaths for males from a hypothetical medium-size US State. The reference population is the 1940 US Standard Million.

 Age group Deaths Person-Time (thousands) Reference Size/Weight Under 1 1 38 15343 1-4 0 150 64718 5-14 1 322 170355 15-24 2 344 181677 25-34 8 443 162066 35-44 21 379 139237 45-54 46 256 117811 55-64 103 189 80294 65-74 254 136 48426 75-84 371 57 17303 85 and over 212 12 2770

To analyse these data in StatsDirect you must select direct standardization from the rates section of the analysis menu. Note that annual mortality rates are often expressed as rates per 100,000 population or units of person time (i.e. 100,000 person years); so a multiplier of 100,000 should be selected for the scaling of rates in the output - you are prompted to provide this.

For this example:

Directly Standardized Rates

Rates are expressed per 100,000 units of person time:

 Index events Index PT Index rate Reference size Weight 1 38000 2.631579 15343 0.015343 0 150000 0 64718 0.064718 1 322000 0.310559 170355 0.170355 2 344000 0.581395 181677 0.181677 8 443000 1.805869 162066 0.162066 21 379000 5.540897 139237 0.139237 46 256000 17.96875 117811 0.117811 103 189000 54.497354 80294 0.080294 254 136000 186.764706 48426 0.048426 371 57000 650.877193 17303 0.017303 212 12000 1766.666667 2770 0.00277

 Index rate Exact 95% confidence interval 2.631579 0.066626 to 14.662219 Under 1 year 0 0 to 2.459253 1-4 years 0.310559 0.007863 to 1.730324 5-14 years 0.581395 0.07041 to 2.1002 15-24 years 1.805869 0.779646 to 3.558282 25-34 years 5.540897 3.429903 to 8.46985 35-44 years 17.96875 13.155383 to 23.967794 45-54 years 54.497354 44.482507 to 66.093892 55-64 years 186.764706 164.500721 to 211.201647 65-74 years 650.877193 586.323368 to 720.597325 75-84 years 1766.666667 1536.8427 to 2021.164948 85 years and over

Total events = 1019

Rates are expressed per 100,000 units of person time:

Crude rate = 43.809114

Any rates (binomial model)

Approximate standard error of R = 1.050864

Approximate 95% confidence interval = 30.89621 to 35.01552

Small rates (Poisson model)

Approximate standard error of R = 1.053213

Approximate 95% confidence interval = 30.891605 to 35.020125

Improved approximate (Dobson) 95% confidence interval = 30.923031 to 35.085216