Statistical methods for detecting and comparing periodic data and their application to the nycthemeral rhythm of bodily harm: A population based study

Abstract

Background: Animals, including humans, exhibit a variety of biological rhythms. This article describes a method for the detection and simultaneous comparison of multiple nycthemeral rhythms.

Methods: A statistical method for detecting periodic patterns in time-related data via harmonic regression is described. The method is particularly capable of detecting nycthemeral rhythms in medical data. Additionally a method for simultaneously comparing two or more periodic patterns is described, which derives from the analysis of variance (ANOVA). This method statistically confirms or rejects equality of periodic patterns. Mathematical descriptions of the detecting method and the comparing method are displayed.

Results: Nycthemeral rhythms of incidents of bodily harm in Middle Franconia are analyzed in order to demonstrate both methods. Every day of the week showed a significant nycthemeral rhythm of bodily harm. These seven patterns of the week were compared to each other revealing only two different nycthemeral rhythms, one for Friday and Saturday and one for the other weekdays.

Figure 1

Normalized crime rates and its Fourier approximations. Black dots show the relative frequency of 11402 crimes of bodily harm committed in the years 2002 to 2005 in Middle Franconia, Bavaria/Germany during the 168 hours of a week, starting Sunday at 9:00 a.m. Nycthemeral rhythms are visible. Solid line show the Fourier approximation of relative number of crimes versus cumulative time in hours, starting at 9:00 a.m. Light gray line shows the Fourier approximations of normalized crime rates for d1 to d5 (Sunday 9:00 a.m. to Friday 9:00 a.m.); the dark gray line for d6 and d7 (Friday 9:00 a.m. to Sunday 9:00 a.m.). A difference of these two rhythms is a shift of the maxima from 10:00 p.m. to 1:00 a.m. Furthermore the maxima of the second rhythm are higher than those of the first.

Figure 2

Distribution of crimes of bodily harm on the seven days of a week. Distribution of the 11402 crimes of bodily harm committed in the years 2002 to 2005 in Middle Franconia, Bavaria/Germany on the seven „days“ of a week, with 95% confidence intervals. d1 is the 24 hour timespan starting at Sunday 9:00 a.m. and ending at Monday 9:00 a.m. and so on.

Figure 3

Periodogram of incidents per hour. The periodogram is applied to the 35064 hours of the four year sampling period. The period of 168 hours (one week), 24 hours (one day) and their corresponding harmonic frequencies are tagged with circles. All high peaks of the spectral density coincide with these frequencies. The other shown periods have relatively small density.

Figure 4

Quantil-quantil-plot of residuals. Quantil-quantil-plot of residuals of d6 and d7 against standard normal quantiles (black cross). The gray line joins the first and the third quartile. The absence of large deviations between the black crosses and the gray line implies a normal distribution of the residuals.