Chronobiometry
The single cosinor method involves the least-squares fit to the data of a model consisting of one or several cosine curves with one or several periods anticipated to characterize the data, with or without the inclusion of polynomial trends. To obtain estimates of the MESOR (a rhythm-adjusted mean), the amplitude and acrophase (measures of extent and timing of change within a cycle) of each cosine term entering the equation, as a first approximation, the error term is assumed to have an independent normal distribution with mean 0 and unknown constant variance [sigma]2. When the periods can be anticipated on the basis of prior biologic information, the equation can be linearized in its parameters, so that linear least-squares techniques can be used (Figure 4/I); otherwise the adjustment needs to be done nonlinearly, in which case point-and-interval estimates are also obtained for the period(s) in addition to the MESOR, amplitude(s) and acrophase(s). An F-statistic is used to perform a zero-amplitude test by comparing the sum of squares accounted for by the model either with the residual sum of squares (original or pooled cosinor), or with the sum of squares due to the pure error, uncontaminated by lack of fit (unpooled cosinor). For each cosine term entering the equation, a polar representation can be used to illustrate its characteristics. The amplitude-acrophase pair is represented by a directed line (vector); the length of the vector represents the amplitude and its orientation along the circular scale in relation to the selected reference time (0deg.), the acrophase.
For circadian rhythms, the reference time is usually midnight, mid-sleep or the time of the acrophase of another physiologic variable such as body core temperature; circaseptan rhythms are usually referred to midnight between Saturday and Sunday, and circannual rhythms to midnight on December 22 preceding the start of data collection. The 360deg. of the circular scale are equated to one full cycle (i.e., to the period length). By using a cosine (rather than a sine) function, 0deg. can be placed on top of the circle; by expressing the acrophase in negative degrees, angles vary clockwise. The ellipse shown around the tip of the vector represents the 95% confidence region for the joint estimation of the amplitude and acrophase. From this error ellipse, conservative confidence intervals can be derived for the amplitude and acrophase separately by taking concentric circles tangent to the error ellipse and by drawing the tangents from the center of the graph (pole) to the error ellipse, respectively.
The advantages of rhythmometry (Figure 4/II) are: 1) a more accurate and more precise estimate of the overall mean value provided by the MESOR. When data are non-equidistant, the estimate of the arithmetic mean but not of the MESOR may be biased, for instance, when most of the data are collected near the acrophase of a rhythm and few data are collected near its bathyphase; when data are equidistant, the estimate of the MESOR usually has a smaller standard error (not shown); 2) the double amplitude represents the extent of predictable change within a cycle whereas the range may include outlying values that may but need not be biologically meaningful (in the case of technical blunders); 3) the acrophase represents a more robust measure of timing of overall high values than the time of a single value-based maximum.
Once estimates of the MESOR, amplitude and acrophase of a given periodic component are obtained for several series on a given individual, or on several different individuals for a given variable, the extent of clustering or similarity of these amplitude-acrophase pairs can be estimated by the population-mean cosinor method, which can be applied to multiple series from a given individual as well as from a population. Two abstract examples are shown (Figure 4/III). On the left, the case of 4 series with widely different acrophases and amplitudes is shown, whereas on the right, the case of 4 series with similar acrophases and amplitudes is illustrated. The fitted cosine functions are shown on top, their vectorial representation in the middle, and the cosinor representation at the bottom. Even though the curves are perfectly sinusoidal, when the amplitudes and acrophases are widely different, the error ellipse covers the center (pole) of the plot so that the zero-amplitude (no-rhythm) assumption cannot be rejected. When the amplitudes and acrophases are similar and they cluster in a given region of the plane, the error ellipse does not cover the pole. Thus, a rhythm can be documented by rejection of the zero-amplitude hypothesis; inferences can then be drawn for the "population" as a whole, i.e., for a set of time series from one or several individuals.
The merit of using cosinor rhythmometry in designing experiments (Figure 4/IV) is illustrated theoretically by power considerations (IVA) and practically by actual data on the effect of daily low doses of aspirin on circulating lipoperoxides based on a sample of only 6 women (IVB and C). By assigning subjects more or less evenly along a full cycle of an anticipated periodicity (in this case between awakening and bedtime), it is shown that relatively large power (~80%) can be reached to resolve the rhythmic structure of the data with chronobiologic designs based on a relatively small sample (N=8) for moderate signal-to-noise ratios (~2) when the data are analyzed by cosinor and, to a lesser extent, when the more conventional analysis of variance is used (see also Figure 5).
In practice, a chronobiologic design allowed, for instance, the demonstration of a circadian stage-dependent effect of aspirin on lipoperoxides on the basis of only 6 subjects, each assigned randomly to a given treatment time. Each took 100 mg of aspirin each day for one week and provided blood samples at 4-hour intervals for the determination of lipoperoxides in platelet-rich plasma for two days prior to the start of treatment and during the last two days of the aspirin test span. Whereas the effect is quite pronounced in the morning, it is barely demonstrable 12 hours later.
For the case of an ACTH analogue as well, a chronobiologic pilot design based on only 5 subjects serves to demonstrate that an effect clearly apparent at one circadian stage can be missed at another circadian stage, Figure 6.
