Trends and change in short count series

A published series is usually short: five to ten annual counts. Three questions recur. Is there a monotone trend? How fast is the count changing per period? Did something change at one point? Each has a method suited to short series, and each is wrong to answer with a least-squares line on five points.

A monotone trend: Mann–Kendall and Sen’s slope

The Mann–Kendall test counts, over all pairs of periods, how many later values exceed earlier ones minus how many fall below. It is distribution-free and exact for small n, so it is informative where a regression’s standard error would be nearly meaningless. Sen’s slope is the median of all pairwise slopes, a robust rate of change.

y <- c(402, 377, 190, 268, 331)        # five years of placements
trndtst(y)
#> $S
#> [1] -4
#> 
#> $tau
#> [1] -0.4
#> 
#> $p_value
#> [1] 0.4833333
#> 
#> $slope
#> [1] -21.375
#> 
#> $intercept
#> [1] 419.75
#> 
#> $slopelo
#> [1] -187
#> 
#> $slopeup
#> [1] 78
#> 
#> $slopecicla
#> [1] TRUE
#> 
#> $n
#> [1] 5
#> 
#> $var_S
#> [1] 16.66667
#> 
#> $alternative
#> [1] "two.sided"
#> 
#> $conflvl
#> [1] 0.95
#> 
#> $method
#> [1] "Mann-Kendall, exact over all 120 orderings"

A monotone series is detected even at n = 5:

trndtst(c(1, 2, 3, 4, 5))
#> $S
#> [1] 10
#> 
#> $tau
#> [1] 1
#> 
#> $p_value
#> [1] 0.01666667
#> 
#> $slope
#> [1] 1
#> 
#> $intercept
#> [1] 0
#> 
#> $slopelo
#> [1] 1
#> 
#> $slopeup
#> [1] 1
#> 
#> $slopecicla
#> [1] TRUE
#> 
#> $n
#> [1] 5
#> 
#> $var_S
#> [1] 16.66667
#> 
#> $alternative
#> [1] "two.sided"
#> 
#> $conflvl
#> [1] 0.95
#> 
#> $method
#> [1] "Mann-Kendall, exact over all 120 orderings"

One aberrant period does not create a trend:

trndtst(c(100, 100, 100, 100, 900))$p_value
#> [1] 0.4

From a data frame, with a direction when one is hypothesised in advance:

d <- data.frame(year = 2019:2023, n = y)
trndtst(d, value = "n", prd = "year")$slope
#> [1] -21.375
trndtst(d, value = "n", prd = "year", alt = "decreasing")$p_value
#> [1] 0.2416667

The rate of change: a Poisson trend

For counts, the natural model is a Poisson log-linear trend: log E[y] = a + b * t, so exp(b) is the rate ratio per period, the multiplicative change from one period to the next. cnttrnd() fits it with an interval:

set.seed(3)
y <- stats::rpois(8, lambda = 200 * 0.85^(0:7))   # falling ~15% a year
fit <- cnttrnd(y)
round(fit$rateratio, 3)
#> [1] 0.869

With a varying denominator the trend is in the rate, via an offset:

cnttrnd(c(20, 25, 30), offset = c(1000, 1500, 2500))$rateratio
#> [1] 0.7708656

Counts here rise while the rate per 1,000 falls, which is the finding.

A step change

A trend test asks “is it drifting?”; a step-change test asks “did it jump once?”. The best single split point is found by maximising the between-segment difference, and its significance is judged by permutation, which accounts for having searched over every possible split:

stepchg(c(100, 104, 98, 60, 63, 58))
#> $breakafter
#> [1] 3
#> 
#> $index
#> [1] 3
#> 
#> $before
#> [1] 100.6667
#> 
#> $after
#> [1] 60.33333
#> 
#> $difference
#> [1] -40.33333
#> 
#> $statistic
#> [1] 2440.167
#> 
#> $p_value
#> [1] 0.1
#> 
#> $nperm
#> [1] 720
#> 
#> $method
#> [1] "exact over all 720 orderings"

Pure noise still yields a best split, and it is not significant:

set.seed(2)
stepchg(stats::rnorm(12))$p_value
#> [1] 0.6223

Two survey periods

Survey tables often give only two periods, each with its confidence interval. The change between them has an interval of its own. healchg() gives it for every group and indicator at once. It lists wellness indicators first, reading change as Indigenous scholars of mental health read it, as a process of healing (Stewart 2008). Perceived mental health of adults, 2015/2018 and 2019/2022, by Statistics Canada (table 13-10-0924-01):

h <- indiex("health")
h <- h[h$age == "Total, 18 years and over" &
         h$indicator == "Perceived mental health, very good or excellent", ]
healchg(h, "group", "indicator", "period", "estimate", "lower", "upper",
               from = "2015/2018", to = "2019/2022", qual = "quality")
#>            group                                       indicator side from   to
#> 1  First Nations Perceived mental health, very good or excellent <NA> 59.8 45.8
#> 2          Métis Perceived mental health, very good or excellent <NA> 61.0 52.8
#> 3          Inuit Perceived mental health, very good or excellent <NA> 49.1 47.8
#> 4 Non-Indigenous Perceived mental health, very good or excellent <NA> 70.6 61.5
#>   change      lower      upper usewithcau
#> 1  -14.0 -17.687818 -10.312182      FALSE
#> 2   -8.2 -11.925923  -4.474077      FALSE
#> 3   -1.3  -9.974243   7.374243      FALSE
#> 4   -9.1  -9.702080  -8.497920      FALSE

A pace, projected

Some counts accumulate toward a fixed number: the 94 Calls to Action of the Truth and Reconciliation Commission, for instance. trcprog() takes the pace so far as a Poisson rate with its exact interval, and gives the year all would be complete at that pace. That is a description of the pace, not a forecast.

trcprog()
#>   completed remaining years     rate    ratelo   rateup   cmplby cmplbyearly
#> 1        13        81  7.04 1.846591 0.9832319 3.157727 2066.865    2048.651
#>   cmplbylate
#> 1   2105.381

Choosing between them

Question	Tool	Reports
Is the series moving in one direction?	`trndtst()`	Mann–Kendall p-value, Sen’s slope with interval
By what factor per period?	`cnttrnd()`	rate ratio with interval, optional offset
Did it change once, and when?	`stepchg()`	split point, segment means, permutation p-value
Year over year, with publication rounding	`yoy()` (its own vignette)	exact intervals on each change

All of them work on the five to ten values a published table gives, and none of them assumes the counts are large.

The trend in a picture

plot() draws the counts with the fitted trend.

plot(fit)

References

Mann (1945). Nonparametric tests against trend. Econometrica 13(3). Kendall (1975). Rank Correlation Methods. 4th ed. Sen (1968). Estimates of the regression coefficient based on Kendall’s tau. JASA 63(324).