Permutation test for a single candidate structural change point

Tests whether the observed reduction in pooled MSE from the best candidate split in find_partition_point() is larger than what a single homogeneous (unsplit) growth process would produce by chance, properly accounting for the multiple comparisons implicit in scanning the whole candidate grid: the null distribution is built from the grid-maximum MSE reduction in each of B parametric-bootstrap replicates simulated from the unsplit null fit, following the same logic as the permutation test of ecp::e.divisive (Matteson & James, 2013) but specialised to parametric growth curves rather than a general multivariate distributional distance. For computational tractability, the null replicates' grid search uses a single growth model (split_mse_single()) rather than the full three-model ensemble used for the reported fit; this is an approximation, disclosed here rather than left implicit.

Usage

permutation_test_partition(
  t,
  y,
  min_obs = 12L,
  B = 99L,
  null_model = NULL,
  beta_fixed = 1.5,
  use_ensemble_null = TRUE
)

Arguments

t, y

Time index and cumulative counts of the open window.

min_obs

Minimum segment length (as in find_partition_point()).

B

Number of parametric-bootstrap null replicates.

null_model

Growth model used for the (cheaper) null-replicate grid searches; defaults to whichever of the three models best fits the unsplit window.

beta_fixed

Passed to the growth-curve fits.

use_ensemble_null

Logical. If TRUE, the null replicates are generated from, and re-fit with, the full three-model ensemble (matching the observed statistic exactly, at roughly three times the cost); if FALSE (default), the cheaper single-model proxy of split_mse_single() is used. We found the single-model proxy under-powered in simulation (it can fail to confirm genuine change points detected by the full ensemble); use_ensemble_null = TRUE is recommended whenever the extra cost is affordable.

Value

A list with p_value, observed_reduction (MSE_nosplit minus MSE at the best split), null_reductions (the B simulated null maxima) and d_hat (the best split, as in find_partition_point()).