Non-parametric circadian rhythm analysis • zeitR

Overview

Non-parametric circadian rhythm analysis (NPCRA) characterises the 24-hour rest-activity pattern without assuming a sinusoidal waveform. It was introduced by Van Someren et al. (1997, 1999) and has become the standard approach for actigraphy-based circadian assessment in both healthy and clinical populations.

zeitR implements compute_npcra(), which returns five core variables from a single function call.

The five variables

Interdaily stability (IS)

IS quantifies how consistent the 24-hour activity pattern is across days. It is computed as the ratio of the variance of the average hourly activity profile to the overall variance of the raw signal:

$\text{IS} = \frac{n}{p} \cdot \frac{\sum_{h=1}^{p} \left( \bar{x}_h - \bar{x} \right)^2} {\sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2}$

where $n$ is the total number of epochs, $p$ is the number of epochs per day, $\bar{x}_h$ is the mean activity at hour-of-day $h$ , and $\bar{x}$ is the grand mean.

Range: 0–1. Higher values indicate a more stable, clock-driven rhythm. Typical values in healthy adults are 0.6–0.8.

Intradaily variability (IV)

IV measures the fragmentation of the rest-activity rhythm — how often and how abruptly the signal transitions between rest and activity:

$\text{IV} = \frac{n \sum_{i=2}^{n} (x_i - x_{i-1})^2} {(n-1) \sum_{i=1}^{n} (x_i - \bar{x})^2}$

Range: $\geq 0$ . Higher values indicate a more fragmented rhythm. Values above 1.0 are often seen in older adults and patients with neurological disorders.

Relative amplitude (RA)

RA captures the contrast between the most active and least active periods of the day:

$\text{RA} = \frac{M10 - L5}{M10 + L5}$

Range: 0–1. Higher values indicate a stronger amplitude of the circadian rhythm.

L5 and M10

L5 is the mean activity level during the least active 5-hour window in the 24-hour averaged profile. M10 is the mean activity during the most active 10-hour window. Both are computed over the 24-hour average profile (not raw epochs), making them robust to day-to-day variability.

The onset times (L5_onset, M10_onset) give the start time of the respective windows in HH:MM format.

Computing NPCRA

compute_npcra() accepts either a zeitr_recording (from read_actigraphy()) or any data frame with datetime and activity columns.

rec   <- read_actigraphy("recordings/P001.txt", tz = "America/Sao_Paulo")
npcra <- compute_npcra(rec)
npcra

## # A tibble: 1 x 10
##   participant_id    IS    IV    RA    L5 L5_onset   M10 M10_onset n_days n_epochs
##   <chr>          <dbl> <dbl> <dbl> <dbl> <chr>    <dbl> <chr>      <dbl>    <int>
## 1 P001            0.72  0.43  0.89  12.3 02:30    84.7  11:00       7.0     10080

Interpretation example

For participant P001:

IS = 0.72 — a moderately stable rhythm, consistent with a healthy adult with regular working hours.
IV = 0.43 — low fragmentation; the rest-activity transitions are relatively smooth.
RA = 0.89 — strong circadian amplitude; there is a large contrast between the active and rest phases.
L5 onset = 02:30 — the least active window is centred around 02:30, as expected for a typical sleeper.
M10 onset = 11:00 — the most active window peaks at mid-morning.

Customising window widths

The 5-hour and 10-hour windows are defaults. They can be adjusted:

# Narrower windows — more sensitive to sharp peaks
compute_npcra(rec, L5_hours = 4, M10_hours = 8)

Note that IS and IV are independent of the window parameters; only L5, M10, and RA are affected.

Using raw data frames

If you have activity data in a plain data frame (e.g. imported from another tool), compute_npcra() works directly:

df <- data.frame(
  datetime = seq(
    as.POSIXct("2021-05-27 00:00:00", tz = "UTC"),
    by = 60,
    length.out = 10080
  ),
  activity = c(...)  # your activity counts
)

compute_npcra(df, epoch_s = 60)

The epoch_s argument is optional — if omitted it is estimated from the median inter-epoch interval in datetime.

Batch NPCRA across a study

For multiple participants, study_summary() calls compute_npcra() for every recording in a zeitr_study and returns a single summary tibble. See vignette("study-analysis") for a full walkthrough.

References

Van Someren, E. J. W., Lijzenga, C., Mirmiran, M., & Swaab, D. F. (1997). Long-term fitness training improves the circadian rest-activity rhythm in healthy elderly males. Journal of Biological Rhythms, 12(2), 146–156. https://doi.org/10.1177/074873049701200206

Van Someren, E. J. W., Swaab, D. F., Colenda, C. C., Cohen, W., McCall, W. V., & Rosenquist, P. B. (1999). Bright light therapy: Improved sensitivity to its effects on rest-activity rhythms in Alzheimer patients by application of nonparametric methods. Chronobiology International, 16(4), 505–518. https://doi.org/10.3109/07420529908998724

Marler, M. R., Gehrman, P., Martin, J. L., & Ancoli-Israel, S. (2006). The sigmoidally transformed cosine curve: a mathematical model for circadian rhythms with symmetric non-sinusoidal shapes. Statistics in Medicine, 25(22), 3893–3904. https://doi.org/10.1002/sim.2466