# Age Adjusted Rates in Epidemiology

Author

Raju Rimal

Published

December 2, 2022

In general, rates means how fast something is changing usually over time. In epidemiology uses it to describe how quickly a disese occurs in a population. For example, 35 cases of melanoma cases in 100,000 person per year convey a the sense of speed of spread of disease in that population. Incidence rate and mortality rate are two examples that we will discuss further below.

In epidemiology, rate measures the frequency of occurance of an event in a given population over certain period of time1.

Let us use melanoma as a outcome in the following discussion. Here, we can calculate a crude incidence rate as,

$\mathcal{R} = \frac{\textsf{no. of melanoma cases}}{\textsf{no. of person-year}} \times \textsf{some multiplier}$

In the case of mortality rate, we can replace the numerator of above expression by the number of melanoma deaths.

## Age-specific rate

Weather to understand a broder prespecitve or to compare across population, these rates are often analyzed stratified by sex and age. This also helps to remove the confounding effection of these factors. The incidence/mortality rate per age-group is usually referred to as Age-specific rates where rates are computed for each age-groups. This is often desirable since factor age has a strong effect on mortality and incidence of most disease especially the cronic one.

Many research articles, however presents the age-adjusted rates. Age-adjusted rates are standardized (weighted) using some standard population age-structure. For example, many european studies on melanoma uses european standard age distribution. While any reasonal studies have also used world standard population. Cancer registry in their reports sometimes uses age-structure of that country in some given year. For instance, Norway2 and Finland3 have used the their population in 2014 as standard population in their recent cancer report while Australia have used 2001 Australian population4.

Standardized (adjusted) rates makes comparison between the population possible. Figure @ref(fig:age-std) shows the difference in the age distribution between world population and European population. Following table are some of the standard population often used in the study. Further on standard popuation see seer.cancer.gov5.

European and World standard population
Age Group World Europe Nordic
0-4 12000 8000 5900
5-9 10000 7000 6600
10-14 9000 7000 6200
15-19 9000 7000 5800
20-24 8000 7000 6100
25-29 8000 7000 6800
30-34 6000 7000 7300
35-39 6000 7000 7300
40-44 6000 7000 7000
45-49 6000 7000 6900
50-54 5000 7000 7400
55-59 4000 6000 6100
60-64 4000 5000 4800
65-69 3000 4000 4100
70-74 2000 3000 3900
75-79 1000 2000 3500
80-84 500 1000 2400
85+ 500 1000 1900
US 2000 standard population
Age Group Std.Population
00 13818
01-04 55317
05-09 72533
10-14 73032
15-19 72169
20-24 66478
25-29 64529
30-34 71044
35-39 80762
40-44 81851
45-49 72118
50-54 62716
55-59 48454
60-64 38793
65-69 34264
70-74 31773
75-79 26999
80-84 17842
85+ 15508

## Calculating age-standardized rate

The age-standardized rate (ASR) is calculated as,

$\text{Age.Std. Rate} = \frac{\sum_i\mathcal{R}_i\mathcal{w}_i}{\sum_i\mathcal{w}_i}$

where, $$\mathcal{w}_i$$ is the weight corresponding to $$i^\text{th}$$ age-group in the reference population.

Let’s explore further with an example from melanoma cases from Australia.

## Example

The following example have used the Austrailian cancer data with 5-year age-group6 after filtering melanoma cases from 1982 to 2018. The dataset has yearly count and age-specific incidence rate of melanoma for men and women.

Let us use the above European standard population to find the yearly age-standardized incidence by sex.

data <- fread("melanoma.csv")
std_pop <- popEpi::stdpop18
data[AgeGroup %in% c("85-89", "90+"), AgeGroup := "85+"]
setnames(std_pop, c("AgeGroup", "World", "Europe", "Nordic"))
std_pop[AgeGroup == "85", AgeGroup := "85+"]
std_pop[, c(2:4) := lapply(.SD, prop.table), .SDcols = 2:4]
asp_data <- merge.data.table(data, std_pop[, .(AgeGroup, World)], by = "AgeGroup")
asp <- asp_data[, .(AgeAdjRate = sum(ASR * World)), by = .(Year, Sex)]
asp[, tail(.SD, 8), by = Sex] %>%
dcast.data.table(
formula = Sex ~ Year,
kableExtra::kbl(caption = paste(
"Age-standardized incidence rate of melanoma",
"Australia from 2011 to 2018"
)) %>%
kableExtra::kable_styling(full_width = FALSE)
Age-standardized incidence rate of melanoma Australia from 2011 to 2018
Sex 2011 2012 2013 2014 2015 2016 2017 2018
Females 28.915 29.6905 30.3050 30.5225 30.8590 32.161 31.5110 32.404
Males 41.433 42.3845 43.1275 42.8415 43.7275 45.309 45.2745 45.040

Now, let us compare the age-specific rates (crude rates) and age-standardized rates with a plot,

### Compare the age-adjusted rates by sex

ggplot() +
geom_line(
data = data,
aes(
x = Year,
y = ASR,
group = AgeGroup,
color = "Crude"
)
) +
geom_line(
data = asp,
aes(
x = Year,
group = 1,
)
) +
geom_text(
data = data[Year == max(Year)],
check_overlap = TRUE,
size = rel(2),
color = "#0f0f0f",
aes(
x = Year + 2,
y = ASR,
label = AgeGroup
)
) +
scale_x_continuous(breaks = scales::breaks_extended(8)) +
scale_y_continuous(breaks = scales::breaks_extended(8)) +
scale_color_manual(NULL, values = c("firebrick", "grey")) +
facet_grid(cols = vars(Sex)) +
theme_minimal() +
theme(
panel.border = element_rect(fill = NA, color = "darkgrey"),
legend.position = c(0, 1),
legend.justification = c(0, 1)
) +
labs(
x = "Diagnosis year",
y = paste(
"per 100,000 person year",
sep = "\n"
)
)

## Discussion

Figure Figure 1 shows that the incidence of melanoma has larger difference in men between the age-groups than in women and men also have a sharp increase in older age group. In addition, the Figure Figure 2 shows that males have higher age-adjusted incidence of melanoma than women in Australia and this trend is increasing over time with rapid increase before 1983 before a drop.

Age-adjusted rates are useful for comparing rates between population but it cannot give the interpretation required for comparing within a population or over a time period in that population. This is one of the reason, cancer registry uses the internal (population structure of their own population) to compute the age-adjusted rates.

## Footnotes

1. https://www.cdc.gov/csels/dsepd/ss1978/lesson3/section1.html↩︎

2. https://www.kreftregisteret.no/globalassets/cancer-in-norway/2021/cin_report.pdf↩︎

3. https://cancerregistry.fi/reports-and-publications/annual-report-on-cancer-in-finland/↩︎

4. https://www.aihw.gov.au/reports/cancer/cancer-in-australia-2021/summary↩︎

5. https://seer.cancer.gov/stdpopulations/stdpop.19ages.html↩︎

6. https://www.aihw.gov.au/reports/cancer/cancer-data-in-australia/data↩︎