Estimates the trend in a series of counts (arrests, admissions, incidents per year) as a rate ratio per period: 0.85 means the count falls by 15% a year. It fits a Poisson log-linear model and, when the counts vary more than a Poisson model allows, widens the interval. An offset carries the denominator when the exposure varies, so the trend is then in the rate rather than in the count.
Arguments
- y
Numeric vector of counts (non-negative whole numbers), in period order.
- x
Numeric vector of periods, default
NULL(the position). The rate ratio is per unit ofx.- offset
Optional numeric vector of exposures, one per period – a population, a number of admissions, a number of days. Positive. The trend is then in the rate rather than in the count.
- conflvl
Number in (0, 1), default 0.95: the confidence level of the rate-ratio interval.
Value
A list of class c("rmbl_count_trend", "list") with
rateratio (per period), lower, upper,
p_value (test of no trend), logslope (\(\beta_1\)),
se (its standard error, scaled when overdispersed),
fitted (fitted counts), dispersion, overdispersed
(logical), n, data (list of the x and y
used) and method.
Details
The model is \(\log \mu_t = \beta_0 + \beta_1 (x_t - \bar x) + \log o_t\), with \(o_t\) the offset (1 when none), fitted by iteratively reweighted least squares (at most 50 iterations). The rate ratio is \(e^{\beta_1}\), with a Wald interval \(e^{\beta_1 \pm z\, \mathrm{se}}\).
The dispersion is reported because a Poisson fit assumes it is one. It
is the Pearson statistic \(\sum (y - \mu)^2/\mu\) divided by
\(n - 2\). When it exceeds 1.5 the quasi-Poisson interval is returned
instead: the standard error is multiplied by the square root of the
dispersion and the \(t\) distribution on \(n - 2\) degrees of freedom
replaces the normal, and overdispersed is set.
Missing values are dropped. Errors: negative or fractional counts;
x and y of different lengths; infinite x; every count
zero; an offset of the wrong length or not positive; fewer than three
periods; a conflvl outside (0, 1). When the counts are zero at
one end of the series only (c(0, 0, 5)) the likelihood rises without
bound and there is no finite trend: a warning says so, the rate ratio is
Inf (or 0) and the interval and p-value are NA. There is a
plot() method for the result.
References
Bilder, C. R. and Loughin, T. M. Analysis of Categorical Data with
R, 2nd edn. Chapman and Hall/CRC, on the quasi-likelihood treatment of
an overdispersed Poisson fit: the variance is scaled by an estimated
dispersion, which widens the interval while leaving the point estimate
alone. That is the behaviour reported here through dispersion and
overdispersed.
Examples
# A count falling by about 15% a year.
set.seed(3)
y <- stats::rpois(8, lambda = 200 * 0.85^(0:7))
fit <- cnttrnd(y)
round(fit$rateratio, 3)
#> [1] 0.869
c(fit$lower, fit$upper)
#> [1] 0.8444785 0.8942357
# With a varying denominator the trend is in the rate.
cnttrnd(c(20, 25, 30), offset = c(1000, 1500, 2500))$rateratio
#> [1] 0.7708656
# Counts that swing far more than Poisson noise get the wider interval.
wild <- cnttrnd(c(10, 50, 5, 60, 8, 70))
wild$dispersion
#> [1] 26.27417
wild$method
#> [1] "quasi-Poisson (dispersion above 1.5)"
try(cnttrnd(c(1.5, 2, 3)))
#> Error : `y` must be non-negative whole numbers