Trend in a series: rate ratio, Mann-Kendall and Theil-Sen

Measures the trend in a short series of counts (admissions, incidents or placements per year) three ways. The Poisson log-linear model \(\log \mu_t = b_0 + b_1 (x_t - \bar x) + \log e_t\) gives the rate ratio per unit of x, exp(b1) (0.85: a fall of 15% a period), with the exposure expo as offset so that the trend is in the rate; when the Pearson dispersion (over n - 2) is above 1.5 the interval and p-value are the quasi-Poisson ones (standard error times the square root of the dispersion, the t distribution on n - 2 degrees of freedom). The fit takes the steps of stats::glm(). Mann-Kendall's rank test and the Theil-Sen slope (the median of the pairwise slopes, with Sen's distribution-free interval) make no distributional assumption; the p-value is exact over every ordering of the observed values up to 8 periods (ties kept) and normal with the tie and continuity corrections beyond. Negating the series negates S, so rises and falls are treated alike (proof("trndrr")). yoy holds the change on the period before, and with seas the totals by season are tested for equality (chi-square).

Usage

trndrr(y, x = NULL, expo = NULL, seas = NULL, alt = "both", level = 0.95)

Arguments

y: The counts, in period order: numbers from 0, not all 0, at least 3.
x: NULL for the periods 1, 2, ..., or the periods: strictly increasing numbers (years, say).
expo: NULL or the exposure of each period (population, person-days): positive numbers.
seas: NULL or the season of each count (month, day of week): labels, two or more.
alt: The alternative of the Mann-Kendall test: "both", "up" or "down".
level: The confidence level: one number strictly between 0 and 1.

Value

An object of class trndrr, a list: rr, the rate ratio row (rr, lo, hi, p, the log slope b and its se, the dispersion disp, over when the quasi-Poisson interval was used); mk (s, its variance var, tau, p, exact); sen (slope, int, lo, hi, clamp); yoy (x, y, the change chg and per cent change pct on the period before); seas, NULL or a list of tab (season, n, expd), stat, df and p; fitted, the Poisson fitted counts; n, alt and level.

Details

When every count falls in the first or the last period the likelihood rises without bound: the rate ratio is 0 or Inf, with NA for the rest of its row, and a warning. Sen's interval needs enough periods: clamp is TRUE when an end had to be the extreme slope.

References

Sen, P. K. (1968). Estimates of the regression coefficient based on Kendall's tau. Journal of the American Statistical Association, 63(324), 1379-1389. doi:10.1080/01621459.1968.10480934

Hamed, K. H. and Rao, A. R. (1998). A modified Mann-Kendall trend test for autocorrelated data. Journal of Hydrology, 204(1-4), 182-196. doi:10.1016/S0022-1694(97)00125-X

Examples

y <- c(402, 377, 390, 368, 331, 310, 305)
trndrr(y, 2017:2023)
#> trndrr: 7 periods
#> rate ratio per period 0.9523448 (95% 0.9337437 to 0.9713165), p 1.223921e-06
#> Mann-Kendall S -19, p 0.002777778; Theil-Sen slope -17.75
# a rate: counts over a growing population
trndrr(c(20, 25, 30, 33), expo = c(1000, 1500, 2500, 2600))$rr
#>          rr        lo       hi          p          b         se      disp  over
#> 1 0.8494552 0.7109716 1.014913 0.07234571 -0.1631601 0.09079901 0.4268193 FALSE
# equal totals by quarter?
trndrr(c(5, 9, 7, 4, 6, 10, 8, 5), seas = rep(1:4, 2))$seas
#> $tab
#>   season  n expd
#> 1      1 11 13.5
#> 2      2 19 13.5
#> 3      3 15 13.5
#> 4      4  9 13.5
#> 
#> $stat
#> [1] 4.37037
#> 
#> $df
#> [1] 3
#> 
#> $p
#> [1] 0.2241485
#>