Test a circular sample for unimodality against bimodality

A parametric-bootstrap likelihood-ratio test of a single von Mises distribution (unimodal) against an asymmetric two-component von Mises mixture (bimodal). The null hypothesis is unimodality; a small p-value is evidence of a second mode.

Usage

test_unimodality(hd, group_col = NULL, angle_col = "heading", n_boot = 199L)

Arguments

hd: Data frame with a heading column in radians.
group_col: Column to group by. NULL (default) tests the whole data frame as one group.
angle_col: Heading column name. Default "heading".
n_boot: Number of parametric-bootstrap replicates. Default 199.

Value

Tidy data frame with columns group_col (if supplied), statistic (the likelihood-ratio statistic), p_value, n, and test. Groups with fewer than ten finite angles, or where the von Mises fit fails, yield NA statistics.

Details

The mixture likelihood-ratio statistic has a non-standard null distribution (a boundary / non-identifiability problem), so its p-value cannot be read from a \(\chi^2\). Instead it is obtained by a parametric bootstrap: n_boot samples are simulated from the fitted von Mises, both models are refitted to each, and the observed statistic is compared with the resulting null distribution. This is deliberately not a critical-bandwidth (Silverman) test: the critical bandwidth's null distribution is not valid for circular data (Fisher & Marron 2001), whereas this bootstrap is well calibrated.

The test is computationally intensive – every bootstrap replicate refits the mixture – so it is best run with a modest n_boot. Set the RNG seed with set.seed for reproducible p-values. It reuses the fitters behind circ_model_select, which offers a complementary information-criterion view of the same unimodal / bimodal comparison.

References

Silverman, B.W. (1981). Using kernel density estimates to investigate multimodality. JRSS B 43(1), 97–99. Fisher, N.I. & Marron, J.S. (2001). Mode testing via the excess mass estimate. Biometrika 88(2), 499–517.