The rate a population would have with the age structure of a standard population, for one or more groups, with Fay and Feuer's gamma confidence interval.
Arguments
- events, pop
Numbers of events and of people, one per age group (and group).
- std
The standard population, one count per age group (its shares are the weights).
- grp
Optional labels: one rate per group, each over the same age groups in the same order.
- per
Rate per this many people (default 100,000).
- conflvl
Confidence level of the Fay-Feuer intervals (default 0.95).
Value
A data frame with group, events, rate,
lower, upper, and the attributes var (the variances,
on the per-person scale) for rategap.
Details
With \(d_i\) events and \(n_i\) people in age group \(i\) and
standard weights \(w_i\) (the standard population's shares), the rate is
\(y = \sum w_i d_i / n_i\), its variance \(v = \sum (w_i / n_i)^2 d_i\),
and with \(w_M = \max_i w_i / n_i\) the limits are \(\frac{v}{2y}
\chi^2_{2y^2/v}(\alpha/2)\) and \(\frac{v + w_M^2}{2(y + w_M)} \chi^2_{2(y
+ w_M)^2/(v + w_M^2)}(1 - \alpha/2)\) (Fay and Feuer 1997). The interval is
conservative (coverage at least nominal in Fay and Feuer's simulations) and
reduces to the exact Garwood interval for a single age group: suited to the
few events of small communities. With no events the lower limit is 0. Rates
are per per people. Age groups are matched to std by position:
order the rows of each group by age, in the order of std.
The choice of standard matters: Statistics Canada's comparisons of First Nations, Metis and Inuit mortality standardise to the Indigenous population's own age structure (Kumar and Tjepkema 2019), which is younger than Canada's.
References
Fay MP, Feuer EJ (1997). Confidence intervals for directly
standardized rates: a method based on the gamma distribution. Statistics in
Medicine 16(7):791-801.
doi:10.1002/(sici)1097-0258(19970415)16:7<791::aid-sim500>3.0.co;2-#
Kumar MB, Tjepkema M (2019). Suicide among First Nations people, Metis and Inuit (2011-2016): Findings from the 2011 Canadian Census Health and Environment Cohort (CanCHEC). Statistics Canada 99-011-X2019001.
Examples
# made-up counts for one small community, by five age groups, to show the
# computation
std <- c(30000, 25000, 20000, 15000, 10000) # a younger standard population
agestd(
events = c(4, 9, 11, 6, 3),
pop = c(9000, 8000, 6000, 4000, 2000),
std = std
)
#> ╭── Directly age-standardised rate ─────────────────────────────────╮
#> │ Groups 1 │
#> │ Events 33 │
#> │ Rate per 100,000 │
#> │ Interval gamma (Fay and Feuer 1997) │
#> │ │
#> │ group events rate 95% CI │
#> │ ----- ------ ----- --------------- │
#> │ all 33 115.6 [79.24, 165.00] │
#> │ │
#> │ all: rate 115.6 [79.24, 165.00]. │
#> │ │
#> │ Each rate is what the group's age-specific rates give for the │
#> │ standard population's ages, so the groups compare without them. │
#> │ Compare only rates standardised to the same population. │
#> ╰───────────────────────────────────────────────────────────────────╯