Divides each area's observed count by its expected count to give the
standardised incidence ratio (SIR), and adds an exact Poisson confidence
interval. A ratio of one is the overall experience; above one is an
excess and below one a deficit. Use it after expcnts
to say which areas differ from the overall rate by more than chance.
Arguments
- obs
Numeric vector of observed counts (non-negative).
- expd
Numeric vector of expected counts (positive), for example from
expcnts; the same length asobs, or length one.- area
Optional vector of area labels, added as the first column. Default
NULL.- conflvl
Number in (0, 1), default
0.95: the two-sided confidence level of the interval.
Value
A data frame with one row per area: area (when given),
observed, expected, sir, lower, upper
(numeric), and logical excess (the interval lies above one),
deficit (the interval lies below one) and significant
(either).
Details
With observed count \(O\) and expected \(E\), \(SIR = O / E\).
The interval for \(1 - \alpha\) coverage is
$$\left[\frac{G^{-1}(\alpha/2;\, O)}{E},\ \frac{G^{-1}(1-\alpha/2;\,
O+1)}{E}\right]$$ where \(G^{-1}(p; k)\) is the gamma quantile with shape
\(k\); the lower limit is 0 when \(O = 0\). This is the exact Poisson
interval, from the relation between the Poisson and gamma distributions, and
so is identical to stats::poisson.test()'s. It is the interval to use
here because the counts that matter are small: a normal approximation on an
observed count of three is not an interval, it is a decoration.
Each area's interval is a separate test; with many areas some will
exclude one by chance (see scanadj). A row with a missing
observed or expected count is dropped whole (its area label with it), after
the lengths are compared.
Errors: negative, infinite or non-numeric counts; an expected count of
zero or below; lengths that differ (a single expd is recycled).
A conflvl outside (0, 1) is an error.
References
Lawson, A. B. Using R for Bayesian Spatial and Spatio-Temporal Health Modeling. Chapman and Hall/CRC, Chapter 1.
Examples
stdinc(obs = c(30, 12, 3), expd = c(20, 14, 5),
area = c("North", "South", "East"))
#> ╭── Standardised incidence ratios ──────────────────────────────────────╮
#> │ Areas 3 │
#> │ Observed 45 │
#> │ Expected 39 │
#> │ Overall ratio 1.154 │
#> │ Interval exact Poisson, at the level asked for (default 95%) │
#> │ Above one 1 │
#> │ Below one 0 │
#> │ │
#> │ area observed expected sir interval finding │
#> │ ----- -------- -------- ------ -------------- ------- │
#> │ North 30 20 1.5000 [1.012, 2.141] excess │
#> │ South 12 14 0.8571 [0.443, 1.497] │
#> │ East 3 5 0.6000 [0.124, 1.753] │
#> │ │
#> │ A ratio above 1 means more cases than the area's population would │
#> │ give at the overall rates. An excess or a deficit is an interval │
#> │ wholly above or below 1 (the significant column is either). │
#> │ │
#> │ Each interval is a separate test: with many areas some exclude 1 by │
#> │ chance (scanadj() allows for that). │
#> ╰───────────────────────────────────────────────────────────────────────╯
# An observed count of three carries almost no information, and the
# interval says so rather than hiding it.
stdinc(3, 5)
#> ╭── Standardised incidence ratios ──────────────────────────────────────╮
#> │ Areas 1 │
#> │ Observed 3 │
#> │ Expected 5 │
#> │ Overall ratio 0.6 │
#> │ Interval exact Poisson, at the level asked for (default 95%) │
#> │ Above one 0 │
#> │ Below one 0 │
#> │ │
#> │ observed expected sir interval finding │
#> │ -------- -------- ------ ---------------- ------- │
#> │ 3 5 0.6000 [0.1237, 1.7530] │
#> │ │
#> │ A ratio above 1 means more cases than the area's population would │
#> │ give at the overall rates. An excess or a deficit is an interval │
#> │ wholly above or below 1 (the significant column is either). │
#> │ │
#> │ Each interval is a separate test: with many areas some exclude 1 by │
#> │ chance (scanadj() allows for that). │
#> ╰───────────────────────────────────────────────────────────────────────╯
# a 90 percent interval, and a zero count (lower limit exactly 0)
stdinc(c(0, 40), c(2, 25), conflvl = 0.90)
#> ╭── Standardised incidence ratios ──────────────────────────────────────╮
#> │ Areas 2 │
#> │ Observed 40 │
#> │ Expected 27 │
#> │ Overall ratio 1.481 │
#> │ Interval exact Poisson, at the level asked for (default 95%) │
#> │ Above one 1 │
#> │ Below one 0 │
#> │ │
#> │ observed expected sir interval finding │
#> │ -------- -------- ----- -------------- ------- │
#> │ 0 2 0.000 [0.000, 1.498] │
#> │ 40 25 1.600 [1.208, 2.083] excess │
#> │ │
#> │ A ratio above 1 means more cases than the area's population would │
#> │ give at the overall rates. An excess or a deficit is an interval │
#> │ wholly above or below 1 (the significant column is either). │
#> │ │
#> │ Each interval is a separate test: with many areas some exclude 1 by │
#> │ chance (scanadj() allows for that). │
#> ╰───────────────────────────────────────────────────────────────────────╯
try(stdinc(c(3, 4), c(5, 0)))
#> Error : stdinc: `expd` must be positive