Statistics for Sleep and Biological Rhythms Research: Longitudinal Analysis of Biological Rhythms Data

Abstract

This article is part of a Journal of Biological Rhythms series exploring analysis and statistical topics relevant to researchers in biological rhythms and sleep research. The goal is to provide an overview of the most common issues that arise in the analysis and interpretation of data in these fields. In this article, we address issues related to collection of multiple data points from the same organism or system at different times, since such longitudinal data collection is fundamental to the assessment of biological rhythms. Rhythmic longitudinal data require additional specific statistical considerations, ranging from curve fitting to threshold definitions to accounting for correlation structure. We discuss statistical analyses of longitudinal data including issues of correlational structure and stationarity, markers of biological rhythms, demasking of biological rhythms, and determining phase, waveform, and amplitude of biological rhythms.

Introduction

Biological rhythms are periodic oscillations occurring at scales from intracellular dynamics to whole organism or system behaviors. Whether the biological signal is endogenously generated or a response to exogenous factors (e.g., circadian vs. diurnal rhythms) does not affect the advice given here which focuses on analysis and inference in general. In this article, we will discuss analyses of longitudinal data in which there are multiple observations from a single experimental organism or preparation (e.g., cell culture), fitting curves to these data, and masking of data. After each section, we will highlight key points. Recognizing the potential pitfalls in statistical analysis is crucial for continued advancement of the biological rhythms fields, including any clinical applications. Inappropriate analyses and statistics may compromise conclusions and waste time, money, and resources. As in Part 1 of this series, we will not recommend specific commercial programs or websites for analysis. For demonstration purposes, we will consider various statistical issues in the context of simulated data from a hypothetical research group that has created a new mouse strain harboring a deletion of a novel gene, bayz, that they hypothesize is involved in sleep and circadian physiology. Consulting with a statistician throughout the experimental design and analysis for appropriate treatment of the issues described below is recommended.

Longitudinal data

Collecting longitudinal data

An early step in conducting experiments with longitudinal data is determining the timing of the samples, as this will affect the analysis plan and ability to detect differences in the longitudinal signal. The sampling rate must be at least twice, and preferably more than twice, the frequencies of interest. For example, in experiments in which the purpose is to determine if there is a 24-hr rhythm, there must be samples at least every 12 hours for multiple cycles to determine the period of the rhythm. If the purpose is to distinguish small differences in period (e.g., between a period of 24.0 hrs and 24.2 hrs) and sampling were only every 12 hours, then >60 days of data would be needed. In addition, such sampling, would not enable determination of the waveform of the rhythm; for that purpose, many more points would be needed. More frequent samples are required if the single is “noisy”, or depending on the type of noise and the signal-to-noise ratio.

Many analyses assume that the data are collected in evenly-spaced intervals. If this is not the case, by design or because of data collection problems, then adjustments to the analysis plan should be made, such as interpolation.

Mixed-effects modeling

Most methods taught in introductory statistical courses assume the values of individual data points are random, independent, and identically distributed (i.i.d.). When there are multiple data points from a single experimental organism, however, these assumptions are not valid, because (i) there is additional information about each data point (i.e., we have other time points from the same organism) and (ii) the data points may not be independent (i.e., they may be correlated with the previous data points). For example, if the hypothetical bayz mice are weighed daily, then if animal A is heavier than animal B at the start of a study, this relationship is likely to remain so on subsequent days. In this example, the baseline differences that are independent of the experimental paradigm represent additional information that needs to be considered in statistical analysis. Similarly, the daily weights of each animal represent non-independent correlated measures, since they are related to the previous days’ weights. Specific statistical methods designed to analyze such data should be used.

One approach to analyzing repeated measures data is to use “mixed-effects” models; these models quantify both random effects (i.e., differences that arise among the organisms unrelated to the intervention or condition) and fixed effects (i.e., differences from the experimental intervention or condition). An alternative approach is to use generalized estimating equations (GEE). Mixed-effects models are suitable when sample sizes are relatively small (<50 in general) and when researchers are interested in both the group average of the response and the individual specific effect, while a GEE approach is more robust to the correlation structure (i.e., it maintains validity over different types of correlations between the data points) and is often used when researchers are more interested in the group average of the response.

When mixed-effects models are used, the correlation structure (i.e., the relationship between adjoining points) should be considered so that an appropriate statistical approach can be taken.

Specifically, one should determine if the current value is independent of previous values, correlated with the previous value, or correlated with multiple previous values. For example, body weight is likely to be related to previous values. If there is a relationship with previous values, it could be linear (e.g., half the previous value or a moving average of multiple previous values), a decaying exponential weighting of multiple previous points, or some other function. Knowing the correlation structure may itself be a challenge, and depends on prior knowledge, as well as having a large enough sample size; a related issue was discussed in the section on understanding the distribution of a data set in Part 1 of this series.

As an example, the sleep efficiency of six bayz and six control animals was calculated across 24 hours in a LD-12:12 cycle (Fig 1A). In this dataset, there is inter-individual variability in the sleep efficiency within both Control and bayz groups. This is an example of an animal-specific “random” effect. In Figure 1B, black and red lines show the means and standard deviations calculated for the sleep efficiency at each hour across the day if the procedure does not take into account the fact that repeated measures were taken from each animal. By comparison, grey and pink lines illustrate the result when a mixed model is used to analyze the data. Note that the differences in mean values between groups is smaller and the error ranges around each point are much smaller.

Figure 1. Sleep efficiency in bayz and Control animals.

Panel A: simulated individual sleep efficiency data across 24 hours in a LD-12:12 cycle. Panel B: means and standard deviations of sleep efficiency for each hour when each time point is (incorrectly) treated as if it is independent (open symbols) or as part of a mixed longitudinal analysis (closed symbols).

Mixed-effects model analysis of this hypothetical bayz sleep efficiency data finds a significant effect of time and the interaction of group and time. This should be interpreted to mean that the pattern of sleep efficiency differs over time, and the way in which it differs with time is different between the groups. Using a mixed-effects model to analyze the data also means that statistical tests at each time point do not need to be performed individually. Note that when multiple tests are performed on a data set (instead of longitudinal-based methods), the probability deemed significant must be adjusted. For example, when the Bonferroni correction for multiple comparisons is used, if there are 24 tests performed (one per each hour of data), a p-value threshold of “significance” would be p=0.05/24, rather than p=0.05. To avoid increasing the probability of rejecting one or more true null hypothesis due to multiple testing, a researcher should never “cherry-pick” a particular time point from a longitudinal dataset for post-hoc statistical analysis without applying multiple testing correction methods. In certain experimental designs, you may want to perform statistical comparisons at a meaningful time point, such as the experiment end point, but this should only be done if this comparison was specified as an experimental hypothesis before the data were collected.

Key points.

• Longitudinal data are likely to exhibit within-individual correlations between the repeated measurements. Statistics that assume independence of data points thus should not be used (e.g., comparing at each time point). Instead, mixed effects models, GEE, or similar approaches are appropriate.

• Mixed-effects models can be used to include both the effect of the individual organism (i.e., the random effect) and the effects of the interventions (i.e., the fixed effect) on the data. These approaches “fit” combined data (from all organisms), and are statistically stronger than fitting each organism’s data set and then averaging across data sets.

Stationarity of data

Another issue to consider with longitudinal data is the stationarity of the data. Stationarity refers to whether the characteristics of the data change during the time frame of analysis. In circadian research, some measurements are often binned together (such as 1 or 2 hour bins), as shown in the sleep efficiency example in Figure 1. Doing so ignores any temporal structure in the data at time scales within each bin. Thus, it is common to refer to stationarity in reference to a specific time frame. For example, sleep efficiency over 24 hours is not stationary, and binning in that time frame may be appropriate for some, but not all, research questions. Changes in stationarity may occur with an intervention or an undefined physiological event. When this occurs, new parameter values (e.g., a new phase relationship or period) may be required for describing the data after this change. Plotting the data before analyzing it is important (see Part 1 of this series) as a first step. There are also other statistical techniques for quantifying stationarity.

More detailed suggestions about analyses of longitudinal data, including stationarity concerns and mixed-effects modeling, will be the topic of another article in this series.

Key points.

• Longitudinal data sets should be checked for stationarity – i.e., no change in the structure of the data set within the time frame of analysis.

Markers of biological rhythms

In studies of organism-level biological rhythms, the status of the endogenous pacemaker cannot usually be determined directly. Therefore, a marker of the pacemaker is required. Both the marker itself (e.g., assay results of melatonin concentrations) and the estimate of some aspect of the marker (e.g., time of onset of secretion) have their own error estimates and potential limitations. The researcher should consider how the measures themselves combine with the analysis method to impact the results and interpretation.

There may be multiple accepted methods for using a given marker to infer properties of the underlying pacemaker. For example, there are multiple accepted methods for using melatonin as a marker of the human circadian pacemaker: crossing of a threshold (e.g., dim light melatonin onset or DLMO), which in itself has different definitions including fixed threshold or 25% of fitted peak); peak of fit (typically using a parametric function to fit the data) (Van Someren et al., 2007), and physiologically-based differential equations that describe the pacemaker and secretion of melatonin (Breslow et al., 2013). Threshold crossing methods introduce choices into the analysis, including the how the threshold was chosen and what to do if the threshold is crossed more than once (Klerman et all, 2012). The variety of marker metrics used may make it difficult to compare across studies and conditions. When there is no commonly accepted marker method, we encourage researchers to justify their choices and acknowledge the limitations of the choice. For example, we reported a comparison of multiple melatonin analysis methods on a variety of data sets and showed that the different methods may produce different “outliers” and may not produce expected results under some conditions (e.g., missing or low amplitude data) (Klerman et al., 2012).

Masking of biological rhythms is a particularly challenging pitfall to analyses. Masking occurs when the marker or variable is dependent on both the endogenous biological pacemaker and other environmental or biological signals that are exogenous to the pacemaker. Some markers are heavily masked: for example, in rodents, sleep timing is influenced by the circadian pacemaker and by light/dark conditions. In humans, sleep timing can be influenced by the circadian pacemaker, work and social constraints, and ingested substances. The effects of endogenous and exogenous signals on the marker may not be known, and there may be nonlinear interactions of the influences on the marker (e.g., the amount of masking by an environmental factor may depend on oscillator phase)(Klerman et al., 1999).

As an example, Figure 2 shows testing of one linear demasking technique applied to core body temperature data from a non-entrained blind individual who was studied under both constant routine (CR) and forced desynchrony protocol conditions. The CR enables determination of circadian phase from core body temperature or other data under conditions designed to minimize masking and to evenly distribute over time any masking effects (e.g., through constant sleep/wake state, constant activity level, constant light level)(Duffy, 1993). This forced desynchrony protocol included a sleep/wake cycle of 28 hours to which the circadian pacemaker could not entrain, thereby forcing desynchrony of sleep/wake times and the underlying circadian rhythms. Therefore, core body temperature data were available at many combinations of circadian phase and length of time awake/asleep. When daily data from this individual during the forced desynchrony protocol were linearly demasked (using four different methods) and the calculated phases compared with the value calculated using CR data from the beginning and end of the protocol (Phase from CR or P-CR, which are not masked), the demasked phase estimates were within 1 hour of the P-CR values for only 26% of the days. While only 36% of the endogenous circadian phases (calculated using P-CR) occurred during scheduled sleep episode, 41–74% of the demasked circadian phases were during those times. Even when P-CR was during a sleep episode, only 25–40% of the demasked phase estimates were within 1 hour of P-CR (Klerman et al., 1999).

Figure 2. Example of failure of demasking of core body temperature using data from non-entrained blind volunteer.

Raster plot (double plotted) comparing constant routine data to ambulatory data for a non-entrained blind participant. The double-plotted straight lines are the best-fit lines through Temperature minima from the constant routine temperature data (P-CR). The diamonds represent the times of temperature minima when sinusoid-based analysis applied to raw ambulatory temperature to demask it. The hatched line represents nocturnal scheduled sleep time. Arrows indicate the dates of CRs. Note: MEST = Middle European Summer Time. From Klerman et al. JBR 1999.

Experiments should be designed to minimize or control for the effects of masking, but this is not always feasible (e.g., in the real world), so analytic methods are needed to demask the data (i.e., isolate the endogenous component of the marker). These analytic methods need to be tested and the appropriate conditions for their use given. Since in circadian rhythms data (and shown in Figure 2) there are frequently non-linear interactions of circadian and other influences, demasking techniques based on linear addition or subtraction may not yield physiological results (Klerman et al., 1999). Therefore, we cannot recommend any current demasking technique for determining endogenous circadian phase that has not been tested under conditions in which circadian phase is known (i.e., known from another method).

Key points.

• Many different markers exist for studying biological rhythms. The features (e.g., sensitivity to conditions or values) of the marker must be considered in developing an analysis plan.

• For some markers, there are multiple accepted methods for analysis. When there is no commonly accepted analysis method, we encourage researchers to justify their choices and acknowledge the limitations.

• Masking presents a great challenge in biological rhythms data. Experiments should be designed to minimize or control for masking. Sophisticated models are needed to demask data, because the respective effects - singly and in combination - of endogenous and exogenous signals on the marker are generally not known a priori.

• Currently there are no good ways to demask human core body temperature data, since the effect of an activity on temperature changes with circadian phase.

Determining waveform and amplitude

For biological rhythms, metrics of interest are usually the period (or frequency) of a cycle and some measure of a time (i.e., phase) within each cycle. To obtain these metrics, the data may be fit with a function. There are two general options for doing this: fit with an already defined parametric function (e.g., sinusoid, line, quadratic) or fit without any pre-defined function (nonparametric), such as a smoothing spline. The latter typically requires a very large dataset to have reasonable confidence in the shape of the non-parametric function. One non-parametric data method includes “folding” or averaging the data at different periodicities (e.g., 23.1, 23.2, 23.3,.... 24.8, 24.9, 25.0 hrs), calculate the variance at each fit and then use the fit with the lowest variance. This is the basis of the “periodogram.” An example fit to the Control/Bayz temperature data is in Figure 3.

Figure 3.

Fit curves to ~30 days of Bayz temperature data when the data are “folded” at periods from 23.0 to 25.0 hours. Blue line indicates “folded” data waveform corresponding to best fit period of 24.1hr. The relatively low resolution (i.e., few possible data point values) of the temperature data collection makes the graph appear jagged.

Another common method for fitting rhythmic data is with a periodic parametric function, such as a sinusoid. The choice of parametric function will shape the results. Therefore, the first step before any formal data analysis occurs should be to plot and visually assess the raw data for general shape, outliers, missing data, and invalid data (see Part 1 of this series).

Cosinor analysis with multiple harmonics (e.g., 24.2, 12.1, 8.0, 6.0 hr) is often used to analyze data for shape, amplitude and phase. Such frequency decomposition is the basis of Fourier analyses. Fourier transformation (which is implemented in the Fast Fourier Transform (FFT) method) decomposes a signal into the relative contributions of many different frequencies (in units of cycles per time). The process assumes stationarity of the signal (discussed above) within the time bin used to apply the FFT. Some analysis methods include baseline-correction, band-pass filtering, or detrending (e.g., removing a linear slope or non-linear drift from the data) before performance of frequency decomposition. The assumptions underlying these methods should be reviewed, since some slow frequencies may actually represent important physiology and in such cases detrending should only include much slower frequencies to avoid filtering potentially useful information.

An important consideration for spectral analysis methods is the choice of window. Since Fourier transforms assume an infinite duration time signal or a time signal that is an exact integer number of wavelengths, edge effects can be introduced when studying real data sets of finite duration or subsets of a time series. Windows help to reduce the artifactual effects of performing a Fourier transform on a non-integer number of wavelengths by smoothing out the edges. The choice of window should be dictated by the effect of that window in the frequency range of interest, as different window functions have different “roll-off” rates for affecting frequencies on either side of the main frequency. This can cause “spectral leakage”, which is when power is introduced at a new frequency by the windowing method. The Hanning window is commonly used, and is a good first choice if little is known about the signal, as it has a high roll-off rate and good frequency resolution. The Hamming window is often a useful alternative – it has a lower roll-off rate, but gives better cancellation of frequencies near the main frequency peak, which can allow for better frequency resolution.

In circadian rhythms results, cosinor analyses with relatively few harmonics are usually used: cosinors with fundamental only (~24 hrs), fundamental plus 1 harmonic (~24 and ~12 hours) or fundamental plus 2 harmonics (~24, ~12, and ~8 hours). Before using the results from the cosinor analysis to determine amplitude or phase, how well the cosinor analysis fits the data should be considered. Fitting a cosine with 1 or 2 harmonics to a square-shaped data set (e.g., actigraphy data across the day) would also not be expected to yield a good fit and therefore the phase (e.g., onset or offset of activity) and amplitude metrics would not be good summary statistics, even if the fit is statistically “significant”. As an extreme example, an ECG is rhythmic, but fitting a sinusoid with a few harmonics to the data (including the QRS complex) would not yield a good fit and therefore the derived statistics would not be sensitive or specific for the known pattern of ECG signals.

For example, when a cosinor curve is fit to the bayz/Control sleep efficiency data presented above, results are shown in Figure 4 with cosinors with fundamental only (24 hrs), fundamental plus 1 harmonic (24 and 12 hours) and fundamental plus 2 harmonics (24, 12, and 8 hours). Note that (i) for all of these fits, the fit curves fall below zero, which is not physiologically possible and (ii) no parameter of the fit (e.g., minimum, maximum, time of crossing of mean value) captures the physiologically relevant times of onset of the relatively steep rise at ~1600hr and fall at ~0500hr (Marler et al 2006).

Figure 4. Multiple cosinor fit to longitudinal data.

Raw and cosinor fit to 5 days of temperature data from Control and Bayz animals. The cosinor fits to the data are with 24 hr (fundamental), 24+12 hr (fundamental plus first harmonic), and 24+12+8 hr (fundamental plus first two harmonics).

Even if there is a “good” sinusoidal fit, there is debate about whether the parameters of the fundamental (i.e. ~ 24 hr) component of the composite curve or the parameters of the composite (~24 hr +harmonics) fit should be used. Standard goodness-of-fit tests to compare models (e.g., Akaike Information Criteria or Bayesian Information Criteria) would not address the question of whether to use the composite fit or the parameters of the composite fit in such situations, since the choice of what metric to use depends on the physiological issue being tested.

When fitting the model parameters to data from multiple individual organisms, mixed-models or GEE methods should be used instead of combining the results of separate fitting for each individual data series. However, these results may be misleading if the individual organisms are not homogeneous. For example, the group rhythm data may appear to have a very low amplitude if individual rhythms have high amplitudes but different phases. Once again, we stress the importance of reviewing and plotting the individual data before further analyses are performed.

Some methods of determining amplitude do not require defining the waveform with a function. These include raw maximum, raw maximum-raw minimum, and area under the curve. Once the raw data are fitted with an appropriate descriptive model function, then the amplitude can be calculated from that function. Methods of computing amplitude depend on the waveform and may include fit maximum, difference between fit maximum and minimum, or the coefficient of a parameter (e.g., coefficient of the 24-hr component of a sinusoid but not of the 12-hr component).

Key points.

• The choice of function(s) used to “fit” rhythmic data will affect the resulting summary measures. Pre-specification of methodology is suggested to avoid the temptation of repeated post-hoc fitting attempts.

• Cosinor analyses may not be appropriate if the data do not have a cosinor shape. Multiple harmonics of the cosinor may be necessary to describe some waveforms.

Determining circadian phase

The phase of a biological rhythm can be defined by the investigator using any marker since the phase of an oscillator is a continuous variable in time. Common examples of reference phase markers are timing of onset of wheel running or minimum of core body temperature. If the oscillator is free-running, phase can be interpolated at a specific time point between reference phases before and after that time point. Different reference phase markers may correspond to different points in the cycle of the underlying pacemaker, or may be sensitive to different types of masking, so phase assessments via different phase markers may not always maintain stable time relationships.

A physiologically relevant metric may enable physiologic interpretation of the data. For example, the onset and offset of melatonin secretion are times of physiological change, while the midpoint of melatonin secretion may not be as relevant, since physiologically the midpoint would require the system to “know” the future when offset would occur. Many metrics depend on using a point after data are fit with a curve (e.g., the minimum value of a cosine fit) or crossing a threshold value. When phase markers are based on a measurement crossing a threshold value, several issues commonly arise: (i) how was the threshold chosen, (ii) what is reported if the threshold is crossed more than once in a single cycle, and (iii) what is reported if the value never crosses the threshold in a particular cycle? Low thresholds may be within the statistical “noise” of the signal, but thresholds that are too high may be insensitive to biological variation. If a threshold-based metric is used to define circadian phase, then the effect of changing the threshold on the results should be determined. Does the timing of crossing the threshold shift uniformly later in all data trains or is the difference in crossing of the threshold variable or based on some other parameter (e.g., on the maximum concentration observed)? Ideally, changing the threshold should cause only a minor change in the trend of the results. The experimenter’s knowledge of the physiology underlying the data and the experimental protocol should define such threshold before any analyses are performed.

An alternative approach to defining the phase of a biological rhythm is to use a Hilbert-Huang transform. Unlike the Fourier Transform, which assumes linearity and stationarity, this transform can be applied to nonlinear and nonstationary signals (i.e., signals that are changing their shape or frequency over time, which is common for biological rhythms), providing an instantaneous estimate of phase, amplitude, and period over time across the experiment.

Rayleigh tests can be used to determine whether there is directionality in a distribution such that the points occur at a consistent time in the day that an event occurs. The Rayleigh test can include both amplitude and phase information. For example, we can test if there a consistent phase and amplitude of the wheel running rhythm in bayz vs. Control animals. The test statistics follow a Chi-squared distribution.

A special consideration for analyzing phase is that the temporal data are circular – 359º is close to 0º in the same way that a clock-time of 23:59 is close to 00:00. This consideration is important when computing statistics such as the average circadian phase or average clock-time at which a particular event occurs. If observations of an event occur at 359º, 2º, 358º, 0º, and 1º, a simple arithmetic average would give a misleading average time of 180º. In practice, this issue is often resolved in our field by “recentering” the data. For example, if we add 360º to any observations <180º, the arithmetic average is now 360º. This method of recentering is, however, not robust. If observations are spread across a wide range of phases or clock-times, the computed average will depend on the (arbitrary) cut-off used for recentering. A better method for averaging of circular data is a vector average. For this method, each observation is treated as a vector of length 1 and angle corresponding to the phase of clock-time. Average timing can now be found by computing the average angle of the sum of all vectors. In cases where observations are tightly clustered, the vector average agrees closely with a recentered arithmetic average, but works in all cases without any arbitrary choices needed.

Key points.

• Phase can be defined using markers to define a specific point in the cycle, or using a continuous definition of phase (e.g., using a Hilbert-Huang transform).

• When phase markers are defined based on a measurement crossing a threshold value, challenges can arise due to difficulty in choosing an appropriate threshold and the possibility of multiple threshold crossings.

• Pre-specifying the threshold value is suggested if possible. If not possible, one should explain how the results differ as the threshold changes.

• When computing average phase, care must be taken due to the circular nature of the data. The use of vector averages can circumvent this problem.

Conclusion

In analyzing biological rhythm data, investigators are faced with both generic statistical considerations and considerations specific to the types of time series typically recorded. In Part 1 of this series, we provided a guide to the former, using examples from circadian rhythms and sleep data, and emphasizing the importance of proper data visualization. In Part 2 of this series, we considered the issues that arise in analyzing biological rhythms time series. Because the study of biological rhythms necessarily involves taking measurements at multiple time points from the same organisms, methods that assume independence of measurements are typically inappropriate. In addition, there are statistical considerations germane to the biological structure of the system being studied, as these may alter the interpretation of statistical findings. Whether the potential pitfalls are statistical (e.g., stationarity) or biological (e.g., demasking techniques), recognition is key to avoidance.