# Directly Standardized Rates

## Overview

The dsr() function computes crude and directly standardized event rates with a variety of user defined options.

The dsrr() function can be used to compare directly standardized event rates through rate differences or ratios.

Standard errors for directly standardized rates are calculated using Chiang’s (1961) method.

## Pre-analysis

The dsr() and dsrr() functions expect aggregated event counts and unit-times by

• standardization variables (e.g. age, sex, etc).
• subgroup variables, if any.

If you are working with individual level data, you will need to aggregate it beforehand. The R package dplyr can do this quite easily.

Note: For the standard or reference population, the data must be aggregated by the standardization variables and the unit-time variable name must labeled pop. See example below.

## Example Data

To illustrate the functions of the package, we will recreate the analysis of calculating and comparing directly standardized mortality rates for the states of Alaska and Florida.

Additional details of the original analysis and data can be found here.

#Alaska death counts and person-Years by Age and Sex
df_a <- data.frame(age=rep(c('00-14','15-34','35-54','55-74','75+'),2),
sex=rep(c('m','f'),c(5,5)),
death=c(37,68,206,369,556,78,181,395,555,479),
fu=c(81205,93662,108615,35139,5491,77203,85412,100386,32118,7701),
)

#Florida death counts and pearson-years by Age and Sex
df_f <- data.frame(age=rep(c('00-14','15-34','35-54','55-74','75+'),2),
sex=rep(c('m','f'),c(5,5)),
death=c(1189,2962,10279,26354,42443,906,1234,5630,18309,53489),
fu=c(1505889,1972157,2197912,1383533,554632,1445831,1870430,2246737,1612270,868838),
state='Florida'
)

#Merge state data together
df_all <- rbind(df_a, df_f)

knitr::kable(df_all, caption='State Specific Counts')
State Specific Counts
age sex death fu state
00-14 m 1189 1505889 Florida
15-34 m 2962 1972157 Florida
35-54 m 10279 2197912 Florida
55-74 m 26354 1383533 Florida
75+ m 42443 554632 Florida
00-14 f 906 1445831 Florida
15-34 f 1234 1870430 Florida
35-54 f 5630 2246737 Florida
55-74 f 18309 1612270 Florida
75+ f 53489 868838 Florida

#Standard population person-years by Age and Sex
df_pop <- data.frame(age=rep(c('00-14','15-34','35-54','55-74','75+'),2),
sex=rep(c('m','f'),c(5,5)),
pop=c(30854207,40199647,40945028,19948630,6106351,
29399168,38876268,41881451,22717040,10494416)
)

knitr::kable(df_pop, caption='US Person-Years')
US Person-Years
age sex pop
00-14 m 30854207
15-34 m 40199647
35-54 m 40945028
55-74 m 19948630
75+ m 6106351
00-14 f 29399168
15-34 f 38876268
35-54 f 41881451
55-74 f 22717040
75+ f 10494416

## Directly Standardized Rates with dsr()

Directly standardized Mortality Rates for Alaska and Flordia are presented here. A 95% confidence interval is requested using the gamma method. A rate multiplier of 1000 (i.e. per 1000) is also specified.

library(dsr)
my_results <- dsr(data=df_all,
event=death,
fu=fu,
subgroup=state,
age, sex,
refdata=df_pop,
method="gamma",
sig=0.95,
mp=1000,
decimals=4)
#> Joining, by = c("age", "sex")

knitr::kable(my_results)
Subgroup Numerator Denominator Crude Rate (per 1000) 95% LCL (Crude) 95% UCL (Crude) Std Rate (per 1000) 95% LCL (Std) 95% UCL (Std)
Alaska 2924 626932 4.6640 4.4964 4.8362 8.0693 7.7504 8.3979
Florida 162795 15658229 10.3968 10.3463 10.4474 7.7342 7.6959 7.7726

## Directly Standardized Rate Ratios with dsrr()

A rate ratio comparing the directly standardized mortality rate of Alaska to Florida is requested. A 95% log-normal confidence interval is computed.

my_results <- dsrr(data=df_all,
event=death,
fu=fu,
subgroup=state,
age, sex,
refdata=df_pop,
refgroup="Florida",
estimate="ratio",
sig=0.95,
mp=1000,
decimals=4)
#> Joining, by = c("age", "sex")

knitr::kable(my_results)
Comparator Reference Std Rate (per 1000) Rate Ratio (RR) 95% LCL (RR) 95% UCL (RR)
Alaska Florida 8.0693 1.0433 1.0031 1.0835
Florida Florida 7.7342 1.0000 0.9930 1.0070

## Directly Standardized Rate Difference with dsrr()

A rate difference comparing the directly standardized mortality rate of Alaska to Florida is requested. A 95% normal confidence interval is computed.

my_results2 <- dsrr(data=df_all,
event=death,
fu=fu,
subgroup=state,
age, sex,
refdata=df_pop,
refgroup="Florida",
estimate="difference",
sig=0.95,
mp=1000,
decimals=4)
#> Joining, by = c("age", "sex")

knitr::kable(my_results2)
Comparator Reference Std Rate (per 1000) Rate Difference (RD) 95% LCL (RD) 95% UCL (RD)
Alaska Florida 8.0693 0.3351 0.0108 0.6594
Florida Florida 7.7342 0.0000 -0.0541 0.0541

## References

• Chiang C. Standard error of the age-adjusted death rate. US Department of Health, Education and Welfare: Vital Statistics Special Reports 1961;47:271-285.

• Elandt-Johnson, R. C., and Johnson, N. L. (1980). Survival Models and Data Analysis. New York: John Wiley & Sons.

• Fay, M.P., & Feuer, E.J. (1997). Confidence intervals for directly standardized rates: a method based on the gamma distribution. Statistics in Medicine,16, 791-801.