Sequential detection of emerging variants

Applies a sequential probability ratio test (SPRT) or CUSUM procedure to lineage frequency data, determining when accumulated evidence is sufficient to declare a variant "emerging" rather than sampling noise. Controls the false alarm rate while minimising detection delay.

Usage

alert_threshold(
  data,
  method = c("sprt", "cusum"),
  alpha = 0.05,
  beta = 0.1,
  delta_0 = 0,
  delta_1 = 0.03,
  threshold = 5
)

Arguments

data: An lfq_data object.
method: Detection method: "sprt" (default) for the sequential probability ratio test, or "cusum" for the cumulative sum control chart.
alpha: False alarm probability. Default 0.05.
beta: Missed detection probability. Default 0.10.
delta_0: Null hypothesis growth rate (no emergence). Default 0 (frequency is stable).
delta_1: Alternative hypothesis growth rate (emergence). Default 0.03 (3 percent per-week increase on logit scale).
threshold: CUSUM decision threshold. Default 5.0. Only used when method = "cusum".

Value

A tibble with columns lineage, date, statistic (log-likelihood ratio or CUSUM value), alert (logical), direction (emerging/declining/ stable), and confidence (1 - alpha).

Details

SPRT (Wald, 1945) computes the log-likelihood ratio between the alternative (lineage is growing at rate \(\delta_1\)) and the null (frequency is stable). The test stops when the cumulative log-ratio crosses the upper boundary \(B = \log((1-\beta)/\alpha)\) (declare emerging) or the lower boundary \(A = \log(\beta/(1-\alpha))\) (declare stable).

CUSUM accumulates deviations from expected frequency under the null: \(S_t = \max(0, S_{t-1} + (x_t - k))\) where \(x_t\) is the observed frequency change and \(k\) is the allowance (half the shift to detect). An alert is raised when \(S_t > h\).

References

Wald A (1945). Sequential tests of statistical hypotheses. Annals of Mathematical Statistics, 16(2), 117–186.

Examples

# \donttest{
sim <- simulate_dynamics(n_lineages = 3,
  advantages = c("A" = 1.5, "B" = 0.8),
  n_timepoints = 15, seed = 1)
alerts <- alert_threshold(sim)
alerts
#> # A tibble: 3 × 6
#>   lineage date       statistic alert direction    confidence
#>   <chr>   <date>         <dbl> <lgl> <chr>             <dbl>
#> 1 A       2026-07-20     0.222 FALSE inconclusive       0.95
#> 2 B       2026-07-20    -0.193 FALSE inconclusive       0.95
#> 3 ref     2026-07-20    -0.193 FALSE inconclusive       0.95
# }