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Menstrual-cycle variability and measurement: further cause for doubt
Psychoneuroendocrinology 25 (2000) 837–847 www.elsevier.com/locate/psyneuen
Menstrual-cycle variability and measurement: further cause for doubt Jeffrey C. Schank
*
Department of Psychology, University of California, One Shields Ave., Davis, CA 47405, USA Received 25 January 2000; received in revised form 26 April 2000; accepted 8 June 2000
Abstract This paper critically examines Weller and Weller’s preferred last month only method for measuring synchrony. Within-woman and between-women menstrual-cycle variability are distinguished. If there is within-woman cycle variability, synchrony requires a process of entrainment. Between-women cycle variability precludes synchrony between rhythms that are not integer multiples of each other. The assumptions of Weller and Weller’s last month only measurement model are tested by computer simulation under conditions of cycle variability. It is demonstrated that these assumptions are biased towards finding synchrony when it does not exist and that the degree of error is an increasing function of cycle variability. Indeed, the error uncovered quantitatively predicts the peculiar skew in their data distributions. Synchrony is almost impossible when there is cycle variability and finding synchrony may be an indicator of a methodological artifact rather than a phenomenon. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Menstrual synchrony; Cycle irregularity; Ovarian-cycle synchrony; Estrous Synchrony; Computer simulation; Synchrony; Coupled-oscillators
1. Introduction Menstrual-cycle synchrony among women is a widely held belief. A cursory examination of the menstrual-cycle synchrony literature suggests more studies finding synchrony (McClintock, 1971; Skandhan et al., 1979; Graham and McGrew, 1980;
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Quadagno et al., 1981; Matteo, 1987; Goldman and Schneider, 1987; Little et al., 1989; Weller and Weller, 1992, 1993a,b, 1995a, 1997a, 1998, 1999a; Weller et al 1995, 1997a) than failing to find it (Jarett, 1984; Trevathan et al., 1993; Wilson et al., 1991; Weller and Weller 1995b, 1998; Weller et al., 1995; Cepicky et al., 1996; Strassmann, 1997). However, the fact that there are a number of studies failing to find synchrony suggests two alternative explanations for these discrepancies. One possibility is that menstrual cycle synchrony is a highly context dependent phenomenon that is far from fully understood (Weller and Weller, 1993c, 1995c, 1997b; McClintock 1998, 2000). Another possibility is that the reports of synchrony are artifacts due to methodological errors interacting with random effects (Wilson, 1992; Strassmann 1997, 1999; Arden and Dye, 1998). Strassmann (1997, 1999) has presented several compelling reasons for suspecting that reports of menstrual-cycle synchrony may be methodological artifacts. First, synchrony reported in the positive studies is very weak; no study has ever shown that women closely match their menses onsets. Second, there is no evidence that synchrony occurs in natural fertility populations where presumably it may have some biological function. Third, and most importantly, cycle variability should be an obstacle to synchrony and cycle variability both within and between women is well documented (Chiazze et al., 1968; Treloar et al., 1967; Vollman, 1977). When cycle variability occurs both within woman and between women, cycle onsets typically do not stably coincide over time (Fig. 1). Fig. 1a illustrates two sequences of cycles with the same mean cycle length, but due to within-woman cycle variability, cycle onsets only randomly match between the first and last cycle onsets even though mean cycle length is exactly the same for both sequences of cycles (i.e. approximately 27 days). Thus, starting out synchronized does not guarantee that two sequences of cycles will remain synchronized even if the mean cycle length is exactly the same for both women over a given period of time. The situation is even worse for between-women cycle variability. The mathematical problem of between-women cycle variability is illustrated in Fig. 1b and c. In Fig. 1b, a 32-day rhythm is plotted next to a 24-day rhythm. They both begin in exact onset synchrony and then because cycles are not integer multiples of each other, they repeatedly diverge and then converge within the 96-day periods of exact onset synchrony (Fig. 1b). This is not synchrony (Winfree, 1980). Fig. 1c illustrates this point even more clearly. In this case, a 31-day rhythm is plotted next to a 27day rhythm. Because the 31-day rhythm is a prime number, cycle onsets only coincide every 27×31=837 days (less than half this time period is illustrated in Fig. 1c). Within this period cycles onsets repeatedly converge and diverge. Thus, stable onset matching cannot be maintained over time unless cycles are at least on average the same length. If cycles are not on average integer multiple lengths of each other (i.e., there is between-women cycle variability), synchrony over time is a mathematical impossibility (Fig. 1b and c). If there is a process of entrainment that brings cycle onsets closer together than expected by chance and maintains this closeness despite cycle variability, then there is still the problem of statistically measuring and detecting synchrony. Measuring synchrony is not straightforward (Schank, 1997). There is always the possibility of
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Fig. 1. Cycle variability and synchrony: within-woman cycle variability, cycle onsets are randomly related (vertical lines indicate when onsets exactly match); (a) between-women cycle variability, cycle onsets repeatedly converge and diverge every 96 days, when rhythms consist of 32- and 24-day cycles; (b) between-women cycle variability, cycle converge and diverge within 847 day periods, when rhythms consist of 31- and 27-day cycles (less than half of this period is depicted); (c) In the latter two cases, synchrony is a mathematical impossibility, since cycle onsets do not persistently correspond to each other over time (Winfree, 1980).
introducing errors into measurement models. Thus, it is critical to analyze measurement models for potential errors that may either increase or decrease the likelihood of detecting synchrony. Weller and Weller (1997b) recently discussed the problem of menstrual-cycle variability in the measurement of synchrony. Upon reviewing five approaches to
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measuring synchrony, they concluded that their last month only measurement model “In terms of dealing with the measurement problems posed by cycle variability and irregular cycles…is preferable [to the other four discussed in their paper], although it addresses synchrony as a state without examining the process which leads to it” (Weller and Weller, 1997b, p. 120). Their last month only measurement model makes several assumptions:
1. The synchrony state of two women can be accurately measured by sampling two (perhaps more) consecutive cycle onsets and women can be assumed to have the same cycle length M for the purpose of measurement. Typically, Weller and Weller (1993a,b, 1995a, 1997b) have assumed that M=28 days. This assumption implies that if a 19-day difference is observed between cycle onsets of women A and B, the minimal onset difference “actually” is 28⫺19=9 days. 2. Weller and Weller (1997b) adopted Wilson’s (1992) method for finding the initial onset difference between two cycles. On this method, two cycle onsets must be observed from individuals A and B. Three comparisons are then made between onsets: (i) the first onset of A and B, |A1⫺B1|, (ii) the first onset of A and the second of B, |A1⫺B2|, and (iii) the second onset of A and the first of B, |A2⫺B1|. For example, if A reports onset dates of February 20 and March 20 and B reports onsets of February 1 and March 1 (and February is not a leap year), then the least of the three possible comparisons is February 20 and March 1, nine days. However, the difference between the observed difference and the assumed cycle length also must be considered (assumption 1). For example, if 19 days is the observed difference between two cycle onsets, 28⫺19=9 is the “real” onset difference. 3. Weller and Weller (1993a) did not explicitly state that they used Wilson’s (1992) method in determining the minimum onset difference between two women. They calculated “…a random variable representing the onset date difference, call it R is the smallest difference between A’s and B’s onset dates over any two consecutive cycles” (Weller and Weller, 1993a, p. 945). The phrasing of this sentence suggests that in addition to the comparisons (2i–iii), they may have made a fourth comparison (iv) the second onset of A and the second of B, |A2⫺B2|, and again, as with (2), observed onset differences are compared with the assumed cycle length according to assumption (1). 4. Consequently, calculating cycle onset differences (with assumptions 1 and 2 or 3) yields a uniform random distribution, where all absolute onset differences between 0 and M/2 occur with probability 2/M, and the onset differences 0 and M/2 occur with probability 1/M (Weller and Weller, 1993a). The Kolomogorov– Smirnov nonparametric test is then used to determine whether the observed frequency distribution of the data differ from the distribution expected by chance. The critical question is whether assumptions (1) and (2 or 3) always imply a uniform random distribution or whether there is an inherent bias towards finding close cycle onsets due to cycle variability.
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2. Computer simulation
The purpose of this simulation was to discover how cycle variability interacts with the assumptions of Weller and Weller’s (1993a,b, 1995a, 1997b) preferred measurement model of synchrony. A program was written in ANSI C and compiled in CodeWarrior. The program generated random sequences of cycles for pairs of hypothetical women, A and B using the pseudo-random number generator (Ran2; ¯ , there was both within-woman and Press et al., 1992). Cycles had a mean length M between-women cycle variation, and cycle variation was normally distributed about the mean for each individual (using the gasdev algorithm, where ran2 was substituted for ran1; Press et al., 1992). Normal distributions are mathematically elegant for independently altering within and between cycle sources of variability. However, a Poisson distribution may be more appropriate because there are often more very long cycles than very short cycles (Strassmann, 1997). The use of a normal distribution in this case, however, is justified for two reasons: First, preliminary computer simulations with different underlying distributions (e.g., uniform random and Poisson) indicated that it is cycle variability per se and not the exact form of the underlying distribution that produces the results reported below. Second, because there was the possibility of randomly generating cycles that were biologically too short with a normal distribution (e.g., 0 days in length), a constraint was placed on the minimum length of cycles: cycles less than 16 days in length were disallowed — allowing cycle lengths less than 16, only increase absolute-onset difference biases. When cycles less than 16 days in length occurred, the simulation looped back through the random cycle generation process until a cycle greater than or equal to 16 days was produced. A consequence of this constraint was that the resulting cycle-length distribution was more Poisson like. Sequences of cycles were generated for A and B and the between-women and within-woman cycle variability was measured as the standard deviation (SD) for all cycles generated in a set of simulations. Rhythms were randomly related to each other using a Monte Carlo “spinning” algorithm: sequences of cycles are connected in a loop, randomly spun, “cut”, then laid out in their same relative positions in time. This permitted the generation of expected probability distributions for cycle onset differences as a function of cycle variability. When standard deviations for menstrual cycle lengths were reported by Weller and Weller (1993a,b, 1995a, 1997b), they were in the range of 3 to 5 (or slightly greater). To systematically explore the effect of cycle variation on the distribution of absolute onset differences expected by chance with assumptions (1) and (2 or 3), cycle distributions ranging from 0 to 5 (i.e. SD⬇0, 1, 2, 3, 4, 5) were generated. ¯ on the distribution of absolute onset To explore the effect of mean cycle length M differences, means of 27, 28, 29, and 31 were simulated. Weller and Weller (1993c) have typically found a mean menstrual cycle length in their studies of 28 days. The initial onsets of A and B were then compared in the two ways prescribed by assumptions (1) and (2 or 3). A range of cycle variability was generated for each method (i.e., 2 or 3) and 1,000,000 simulations run for each degree of variability.
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3. Results Cycle variability introduced a systematic bias towards synchrony relative to the expected frequency of absolute onset differences assumed by Weller and Weller (1993a,b, 1995a, 1997b). The greater the standard deviation in the simulated cycle ¯ differed from the distribution, the greater the bias. If the mean cycle length M assumed cycle length (i.e. M=28 or 29 days), then the bias became even larger. That ¯ =27 or 31 days, the bias observed is, if M=28 and the mean cycle length was either M was larger than if M=28. Assumption (3) produces a greater bias than assumption (2) in conjunction with assumption (1). Assuming that M=29 days does not qualitatively alter the bias inherent in these methods. Systematic results for an assumed ¯ =28 are illustrated in Fig. 2, since cycle length of M=28 with an actual mean of M this condition produced the least bias. Fig. 2 illustrates how variability in cycle length altered the expected distribution of cycle onset differences using the methods assumed in (1; M=28) and (2 or 3) for different standard deviations in cycle length (i.e., SD⬇0, 1, 2, 3, 4, 5). For comparative purposes, the data from Weller and Weller (1993a) are combined into a single frequency distribution and plotted as columns (Fig. 2). The heavy line is Weller and Weller’s (1993a,b, 1995a, 1997b)expected relationship among cycle onsets when the ¯ =28 (SD=0), and the heavy dashed line is the maximum mean cycle length is M degree of variability tested (SD⬇5). Weller and Weller’s expected uniform random distribution (except at the ends) only occurs when all cycles are exactly 28 days in length (SD=0). As cycle variability increased, the expected distributions became increasingly skewed towards shorter onset differences (Fig. 2). As illustrated in Fig. 2, under either method for calculating onset differences (3 produced slightly more skew than did 2), the skew in the expected distributions of cycle onset differences remarkably resemble the actual data for standard deviations ranging from approximately 3 to 5. This implies that even if cycle onsets are completely randomly related, Weller and Weller’s (1997b) method for measuring synchrony will yield data distributions that are skewed towards synchrony, and skewed in a way that is qualitatively and quantitatively like the actual data distributions they report. The degree of skew is a function of the cycle variability of the women measured (Fig. 2). Significant differences from the expected distribution are indicated in the Kolomogorov–Smirnov test by the maximum cumulative difference between the theoretically expected distribution and the data derived distribution. If, however, the Kolomogorov–Smirnov test were applied to their data with an expected distribution corrected for the synchrony introduced by assumptions (1) and (2 or 3), it is almost certainly the case — with the levels of cycle variation reported (SD=2.2 to 5.4) — that there would be no significant effect of synchrony in these studies. 4. Discussion The last month only method used by Weller and Weller (1993a,b, 1995a, 1997b) creates a non-uniform distribution from randomly related cycles that is biased
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Fig. 2. Expected distributions of cycle onset differences for standard deviations (SD) of 0 (solid heavy line) and approximately 1, 2, 3, 4, and 5 (SD⬇5 is indicated by the heavy dashed line). The total data from Weller and Weller (1993a) are plotted as bars. As variability increases, so does the skew in the expected distribution for either method (2), a, or (3), b. Only when all cycles are 28 days in length is Weller and Weller’s (1993a,b, 1995a, 1997b) expected distribution correct (heavy solid line).
towards synchrony as an increasing function of cycle variability. The greater the variability, the more likely it is that their measurement model together with the Kolomogorov–Smirnov test will find synchrony when it does not exist — when the null distribution is inappropriately assumed to be uniform random (assumption 4).
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Indeed, these simulations demonstrate that assumptions (1) and (2 or 3) predict the unusual skew and quantitative form of their data (see Fig. 2). For mean cycle lengths of 28 days, the best fit occurred when the standard deviations for cycles ranged from approximately 3 to 5, which is consistent with the reported range for their mothers/daughters and roommates study (SD=2.2 to 5.4; Weller and Weller, 1993a). Thus, these simulations demonstrated (Fig. 2) that the assumption of a uniform random distribution (4) is not implied by assumptions (1) and (2 or 3), and that results using the last month only method may be explained as methodological artifacts (Fig. 2). The error inherent in the last month only method is not surprising. The assumption that all cycles are the same length is the main culprit. For cycles longer than the mean, assumption (1) truncates the onset differences, thereby increasing the frequency of small onset differences. For cycles shorter than the mean, there is also an increased likelihood of cycle onsets occurring closer than expected. Thus, it is ¯ deviates from the assumed also no surprise that if the actual mean cycle length M cycle length (i.e., M=28 or 29 days), this only further increases the error. It should also be pointed out that Weller and Weller (1997a) and Weller et al., (1999a,b) changed their methodological approach for detecting and statistically analyzing menstrual-cycle synchrony. In these most recent papers, they have adopted a parametric approach in which they calculate synchrony in terms of the expected mean absolute difference in menstrual-cycle onsets over three consecutive cycles using one-sample t–tests. Weller and Weller (1997a) do not explain their change in methodology (Weller and Weller, 1997b), but Arden and Dye (1998) have recently pointed out a serious problem of chance convergent synchrony with this new approach. More generally, if there is within-woman cycle variability (Fig. 1a), synchrony is possible, but only under the very special conditions of a process of entrainment that tends to keep cycle onsets closer together than expected (Winfree, 1980) — but it should also be kept in mind that menstrual-cycle entrainment has never been demonstrated and only shown to be theoretically possible in Norway rats (Schank and McClintock, 1992). The simulation results presented in Schank and McClintock (1992) showed that estrous synchrony among female rats was a very weak phenomenon at best. A subsequent test of the coupled-oscillator hypothesis in rats failed to verify its assumptions (Schank and McClintock, 1997). A recent computer-simulation reanalysis of McClintock’s (1978) report of synchrony among female rats showed that the level of synchrony reported could not be distinguished from chance (Schank, 2000b). Nor does synchrony appear to occur (Schank, 2000a) in the only other rodent species (golden hamsters, Mesocricetus auratus) in which it has been reported (Handelmann et al., 1980). Thus, while a coupled-oscillator mechanism could synchronize ovarian cycles, there is no evidence for this mechanism in mammals. If there is between-women cycle variability (Fig. 1b), synchrony is a mathematical impossibility (Winfree, 1980). Since there is between cycle variability among women (Chiazze et al., 1968; Strassmann, 1997; Treloar et al., 1967; Vollman, 1977), cycle variability is a very serious problem; rendering suspect all reports of synchrony. Indeed, it must be concluded that synchrony is almost impossible when there is cycle
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variability and finding synchrony may be an indicator of a methodological artifact rather than a phenomenon. This general point strongly suggests that all methods, which consistently find synchrony when there is persistent cycle variability (especially between women cycle variability), may be plagued with methodological errors (Wilson, 1992; Strassmann 1997, 1999; Arden and Dye, 1998). These methods should be carefully reanalyzed using simulation approaches like that presented here to assess their potential for error. The simulation results presented here further strengthen the argument that for almost 30 years the search for synchrony may have been the wrong direction to take and the wrong question to ask regarding the relationship of ovarian cycles among women and females of other mammalian species. Cycle variability is well-documented in women (Chiazze et al., 1968; Treloar et al., 1967; Vollman, 1977) and while some irregularity is due to factors such as illness or accidents (Golub, 1992), cycle variability may be inherent to the physiological processes generating ovarian cycles. It is possible that cycle variability may be beneficial in some evolutionary and ecological contexts. For example, cycle irregularity can prevent even accidental synchrony. Avoiding synchrony may enhance female choice among males with differing phenotypic quality by reducing inter-female competition (Pereira, 1991). Cycle variability may also reduce the predictability of the next ovulation by males. With uncertainty regarding the next ovulation, males may be less able to control the matings of females. Nevertheless, even if cycle variability is beneficial to females in some mammalian mating systems, this should not be interpreted as implying that cycle variability is adaptive. Instead, cycle variability may be a generic feature — borrowing this concept from Kauffman (1993), (also see Schank, 1994) — of the physiological processes generating cycles, which may coincidentally have fitness benefits for females in some mating systems.
Acknowledgements This work was supported by NIH (through a subcontract with Indiana University). I also want to thank three anonymous reviewers of this paper for their comments, which greatly improved its logical structure.
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Menstrual-cycle variability and measurement: further cause for doubt Jeffrey C. Schank
*
Department of Psychology, University of California, One Shields Ave., Davis, CA 47405, USA Received 25 January 2000; received in revised form 26 April 2000; accepted 8 June 2000
Abstract This paper critically examines Weller and Weller’s preferred last month only method for measuring synchrony. Within-woman and between-women menstrual-cycle variability are distinguished. If there is within-woman cycle variability, synchrony requires a process of entrainment. Between-women cycle variability precludes synchrony between rhythms that are not integer multiples of each other. The assumptions of Weller and Weller’s last month only measurement model are tested by computer simulation under conditions of cycle variability. It is demonstrated that these assumptions are biased towards finding synchrony when it does not exist and that the degree of error is an increasing function of cycle variability. Indeed, the error uncovered quantitatively predicts the peculiar skew in their data distributions. Synchrony is almost impossible when there is cycle variability and finding synchrony may be an indicator of a methodological artifact rather than a phenomenon. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Menstrual synchrony; Cycle irregularity; Ovarian-cycle synchrony; Estrous Synchrony; Computer simulation; Synchrony; Coupled-oscillators
1. Introduction Menstrual-cycle synchrony among women is a widely held belief. A cursory examination of the menstrual-cycle synchrony literature suggests more studies finding synchrony (McClintock, 1971; Skandhan et al., 1979; Graham and McGrew, 1980;
* Tel.: +1-530-752-6332; fax: +1-530-752-2087. E-mail address: [email protected] (J.C. Schank). 0306-4530/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 6 - 4 5 3 0 ( 0 0 ) 0 0 0 2 9 - 9
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Quadagno et al., 1981; Matteo, 1987; Goldman and Schneider, 1987; Little et al., 1989; Weller and Weller, 1992, 1993a,b, 1995a, 1997a, 1998, 1999a; Weller et al 1995, 1997a) than failing to find it (Jarett, 1984; Trevathan et al., 1993; Wilson et al., 1991; Weller and Weller 1995b, 1998; Weller et al., 1995; Cepicky et al., 1996; Strassmann, 1997). However, the fact that there are a number of studies failing to find synchrony suggests two alternative explanations for these discrepancies. One possibility is that menstrual cycle synchrony is a highly context dependent phenomenon that is far from fully understood (Weller and Weller, 1993c, 1995c, 1997b; McClintock 1998, 2000). Another possibility is that the reports of synchrony are artifacts due to methodological errors interacting with random effects (Wilson, 1992; Strassmann 1997, 1999; Arden and Dye, 1998). Strassmann (1997, 1999) has presented several compelling reasons for suspecting that reports of menstrual-cycle synchrony may be methodological artifacts. First, synchrony reported in the positive studies is very weak; no study has ever shown that women closely match their menses onsets. Second, there is no evidence that synchrony occurs in natural fertility populations where presumably it may have some biological function. Third, and most importantly, cycle variability should be an obstacle to synchrony and cycle variability both within and between women is well documented (Chiazze et al., 1968; Treloar et al., 1967; Vollman, 1977). When cycle variability occurs both within woman and between women, cycle onsets typically do not stably coincide over time (Fig. 1). Fig. 1a illustrates two sequences of cycles with the same mean cycle length, but due to within-woman cycle variability, cycle onsets only randomly match between the first and last cycle onsets even though mean cycle length is exactly the same for both sequences of cycles (i.e. approximately 27 days). Thus, starting out synchronized does not guarantee that two sequences of cycles will remain synchronized even if the mean cycle length is exactly the same for both women over a given period of time. The situation is even worse for between-women cycle variability. The mathematical problem of between-women cycle variability is illustrated in Fig. 1b and c. In Fig. 1b, a 32-day rhythm is plotted next to a 24-day rhythm. They both begin in exact onset synchrony and then because cycles are not integer multiples of each other, they repeatedly diverge and then converge within the 96-day periods of exact onset synchrony (Fig. 1b). This is not synchrony (Winfree, 1980). Fig. 1c illustrates this point even more clearly. In this case, a 31-day rhythm is plotted next to a 27day rhythm. Because the 31-day rhythm is a prime number, cycle onsets only coincide every 27×31=837 days (less than half this time period is illustrated in Fig. 1c). Within this period cycles onsets repeatedly converge and diverge. Thus, stable onset matching cannot be maintained over time unless cycles are at least on average the same length. If cycles are not on average integer multiple lengths of each other (i.e., there is between-women cycle variability), synchrony over time is a mathematical impossibility (Fig. 1b and c). If there is a process of entrainment that brings cycle onsets closer together than expected by chance and maintains this closeness despite cycle variability, then there is still the problem of statistically measuring and detecting synchrony. Measuring synchrony is not straightforward (Schank, 1997). There is always the possibility of
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Fig. 1. Cycle variability and synchrony: within-woman cycle variability, cycle onsets are randomly related (vertical lines indicate when onsets exactly match); (a) between-women cycle variability, cycle onsets repeatedly converge and diverge every 96 days, when rhythms consist of 32- and 24-day cycles; (b) between-women cycle variability, cycle converge and diverge within 847 day periods, when rhythms consist of 31- and 27-day cycles (less than half of this period is depicted); (c) In the latter two cases, synchrony is a mathematical impossibility, since cycle onsets do not persistently correspond to each other over time (Winfree, 1980).
introducing errors into measurement models. Thus, it is critical to analyze measurement models for potential errors that may either increase or decrease the likelihood of detecting synchrony. Weller and Weller (1997b) recently discussed the problem of menstrual-cycle variability in the measurement of synchrony. Upon reviewing five approaches to
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measuring synchrony, they concluded that their last month only measurement model “In terms of dealing with the measurement problems posed by cycle variability and irregular cycles…is preferable [to the other four discussed in their paper], although it addresses synchrony as a state without examining the process which leads to it” (Weller and Weller, 1997b, p. 120). Their last month only measurement model makes several assumptions:
1. The synchrony state of two women can be accurately measured by sampling two (perhaps more) consecutive cycle onsets and women can be assumed to have the same cycle length M for the purpose of measurement. Typically, Weller and Weller (1993a,b, 1995a, 1997b) have assumed that M=28 days. This assumption implies that if a 19-day difference is observed between cycle onsets of women A and B, the minimal onset difference “actually” is 28⫺19=9 days. 2. Weller and Weller (1997b) adopted Wilson’s (1992) method for finding the initial onset difference between two cycles. On this method, two cycle onsets must be observed from individuals A and B. Three comparisons are then made between onsets: (i) the first onset of A and B, |A1⫺B1|, (ii) the first onset of A and the second of B, |A1⫺B2|, and (iii) the second onset of A and the first of B, |A2⫺B1|. For example, if A reports onset dates of February 20 and March 20 and B reports onsets of February 1 and March 1 (and February is not a leap year), then the least of the three possible comparisons is February 20 and March 1, nine days. However, the difference between the observed difference and the assumed cycle length also must be considered (assumption 1). For example, if 19 days is the observed difference between two cycle onsets, 28⫺19=9 is the “real” onset difference. 3. Weller and Weller (1993a) did not explicitly state that they used Wilson’s (1992) method in determining the minimum onset difference between two women. They calculated “…a random variable representing the onset date difference, call it R is the smallest difference between A’s and B’s onset dates over any two consecutive cycles” (Weller and Weller, 1993a, p. 945). The phrasing of this sentence suggests that in addition to the comparisons (2i–iii), they may have made a fourth comparison (iv) the second onset of A and the second of B, |A2⫺B2|, and again, as with (2), observed onset differences are compared with the assumed cycle length according to assumption (1). 4. Consequently, calculating cycle onset differences (with assumptions 1 and 2 or 3) yields a uniform random distribution, where all absolute onset differences between 0 and M/2 occur with probability 2/M, and the onset differences 0 and M/2 occur with probability 1/M (Weller and Weller, 1993a). The Kolomogorov– Smirnov nonparametric test is then used to determine whether the observed frequency distribution of the data differ from the distribution expected by chance. The critical question is whether assumptions (1) and (2 or 3) always imply a uniform random distribution or whether there is an inherent bias towards finding close cycle onsets due to cycle variability.
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2. Computer simulation
The purpose of this simulation was to discover how cycle variability interacts with the assumptions of Weller and Weller’s (1993a,b, 1995a, 1997b) preferred measurement model of synchrony. A program was written in ANSI C and compiled in CodeWarrior. The program generated random sequences of cycles for pairs of hypothetical women, A and B using the pseudo-random number generator (Ran2; ¯ , there was both within-woman and Press et al., 1992). Cycles had a mean length M between-women cycle variation, and cycle variation was normally distributed about the mean for each individual (using the gasdev algorithm, where ran2 was substituted for ran1; Press et al., 1992). Normal distributions are mathematically elegant for independently altering within and between cycle sources of variability. However, a Poisson distribution may be more appropriate because there are often more very long cycles than very short cycles (Strassmann, 1997). The use of a normal distribution in this case, however, is justified for two reasons: First, preliminary computer simulations with different underlying distributions (e.g., uniform random and Poisson) indicated that it is cycle variability per se and not the exact form of the underlying distribution that produces the results reported below. Second, because there was the possibility of randomly generating cycles that were biologically too short with a normal distribution (e.g., 0 days in length), a constraint was placed on the minimum length of cycles: cycles less than 16 days in length were disallowed — allowing cycle lengths less than 16, only increase absolute-onset difference biases. When cycles less than 16 days in length occurred, the simulation looped back through the random cycle generation process until a cycle greater than or equal to 16 days was produced. A consequence of this constraint was that the resulting cycle-length distribution was more Poisson like. Sequences of cycles were generated for A and B and the between-women and within-woman cycle variability was measured as the standard deviation (SD) for all cycles generated in a set of simulations. Rhythms were randomly related to each other using a Monte Carlo “spinning” algorithm: sequences of cycles are connected in a loop, randomly spun, “cut”, then laid out in their same relative positions in time. This permitted the generation of expected probability distributions for cycle onset differences as a function of cycle variability. When standard deviations for menstrual cycle lengths were reported by Weller and Weller (1993a,b, 1995a, 1997b), they were in the range of 3 to 5 (or slightly greater). To systematically explore the effect of cycle variation on the distribution of absolute onset differences expected by chance with assumptions (1) and (2 or 3), cycle distributions ranging from 0 to 5 (i.e. SD⬇0, 1, 2, 3, 4, 5) were generated. ¯ on the distribution of absolute onset To explore the effect of mean cycle length M differences, means of 27, 28, 29, and 31 were simulated. Weller and Weller (1993c) have typically found a mean menstrual cycle length in their studies of 28 days. The initial onsets of A and B were then compared in the two ways prescribed by assumptions (1) and (2 or 3). A range of cycle variability was generated for each method (i.e., 2 or 3) and 1,000,000 simulations run for each degree of variability.
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3. Results Cycle variability introduced a systematic bias towards synchrony relative to the expected frequency of absolute onset differences assumed by Weller and Weller (1993a,b, 1995a, 1997b). The greater the standard deviation in the simulated cycle ¯ differed from the distribution, the greater the bias. If the mean cycle length M assumed cycle length (i.e. M=28 or 29 days), then the bias became even larger. That ¯ =27 or 31 days, the bias observed is, if M=28 and the mean cycle length was either M was larger than if M=28. Assumption (3) produces a greater bias than assumption (2) in conjunction with assumption (1). Assuming that M=29 days does not qualitatively alter the bias inherent in these methods. Systematic results for an assumed ¯ =28 are illustrated in Fig. 2, since cycle length of M=28 with an actual mean of M this condition produced the least bias. Fig. 2 illustrates how variability in cycle length altered the expected distribution of cycle onset differences using the methods assumed in (1; M=28) and (2 or 3) for different standard deviations in cycle length (i.e., SD⬇0, 1, 2, 3, 4, 5). For comparative purposes, the data from Weller and Weller (1993a) are combined into a single frequency distribution and plotted as columns (Fig. 2). The heavy line is Weller and Weller’s (1993a,b, 1995a, 1997b)expected relationship among cycle onsets when the ¯ =28 (SD=0), and the heavy dashed line is the maximum mean cycle length is M degree of variability tested (SD⬇5). Weller and Weller’s expected uniform random distribution (except at the ends) only occurs when all cycles are exactly 28 days in length (SD=0). As cycle variability increased, the expected distributions became increasingly skewed towards shorter onset differences (Fig. 2). As illustrated in Fig. 2, under either method for calculating onset differences (3 produced slightly more skew than did 2), the skew in the expected distributions of cycle onset differences remarkably resemble the actual data for standard deviations ranging from approximately 3 to 5. This implies that even if cycle onsets are completely randomly related, Weller and Weller’s (1997b) method for measuring synchrony will yield data distributions that are skewed towards synchrony, and skewed in a way that is qualitatively and quantitatively like the actual data distributions they report. The degree of skew is a function of the cycle variability of the women measured (Fig. 2). Significant differences from the expected distribution are indicated in the Kolomogorov–Smirnov test by the maximum cumulative difference between the theoretically expected distribution and the data derived distribution. If, however, the Kolomogorov–Smirnov test were applied to their data with an expected distribution corrected for the synchrony introduced by assumptions (1) and (2 or 3), it is almost certainly the case — with the levels of cycle variation reported (SD=2.2 to 5.4) — that there would be no significant effect of synchrony in these studies. 4. Discussion The last month only method used by Weller and Weller (1993a,b, 1995a, 1997b) creates a non-uniform distribution from randomly related cycles that is biased
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Fig. 2. Expected distributions of cycle onset differences for standard deviations (SD) of 0 (solid heavy line) and approximately 1, 2, 3, 4, and 5 (SD⬇5 is indicated by the heavy dashed line). The total data from Weller and Weller (1993a) are plotted as bars. As variability increases, so does the skew in the expected distribution for either method (2), a, or (3), b. Only when all cycles are 28 days in length is Weller and Weller’s (1993a,b, 1995a, 1997b) expected distribution correct (heavy solid line).
towards synchrony as an increasing function of cycle variability. The greater the variability, the more likely it is that their measurement model together with the Kolomogorov–Smirnov test will find synchrony when it does not exist — when the null distribution is inappropriately assumed to be uniform random (assumption 4).
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Indeed, these simulations demonstrate that assumptions (1) and (2 or 3) predict the unusual skew and quantitative form of their data (see Fig. 2). For mean cycle lengths of 28 days, the best fit occurred when the standard deviations for cycles ranged from approximately 3 to 5, which is consistent with the reported range for their mothers/daughters and roommates study (SD=2.2 to 5.4; Weller and Weller, 1993a). Thus, these simulations demonstrated (Fig. 2) that the assumption of a uniform random distribution (4) is not implied by assumptions (1) and (2 or 3), and that results using the last month only method may be explained as methodological artifacts (Fig. 2). The error inherent in the last month only method is not surprising. The assumption that all cycles are the same length is the main culprit. For cycles longer than the mean, assumption (1) truncates the onset differences, thereby increasing the frequency of small onset differences. For cycles shorter than the mean, there is also an increased likelihood of cycle onsets occurring closer than expected. Thus, it is ¯ deviates from the assumed also no surprise that if the actual mean cycle length M cycle length (i.e., M=28 or 29 days), this only further increases the error. It should also be pointed out that Weller and Weller (1997a) and Weller et al., (1999a,b) changed their methodological approach for detecting and statistically analyzing menstrual-cycle synchrony. In these most recent papers, they have adopted a parametric approach in which they calculate synchrony in terms of the expected mean absolute difference in menstrual-cycle onsets over three consecutive cycles using one-sample t–tests. Weller and Weller (1997a) do not explain their change in methodology (Weller and Weller, 1997b), but Arden and Dye (1998) have recently pointed out a serious problem of chance convergent synchrony with this new approach. More generally, if there is within-woman cycle variability (Fig. 1a), synchrony is possible, but only under the very special conditions of a process of entrainment that tends to keep cycle onsets closer together than expected (Winfree, 1980) — but it should also be kept in mind that menstrual-cycle entrainment has never been demonstrated and only shown to be theoretically possible in Norway rats (Schank and McClintock, 1992). The simulation results presented in Schank and McClintock (1992) showed that estrous synchrony among female rats was a very weak phenomenon at best. A subsequent test of the coupled-oscillator hypothesis in rats failed to verify its assumptions (Schank and McClintock, 1997). A recent computer-simulation reanalysis of McClintock’s (1978) report of synchrony among female rats showed that the level of synchrony reported could not be distinguished from chance (Schank, 2000b). Nor does synchrony appear to occur (Schank, 2000a) in the only other rodent species (golden hamsters, Mesocricetus auratus) in which it has been reported (Handelmann et al., 1980). Thus, while a coupled-oscillator mechanism could synchronize ovarian cycles, there is no evidence for this mechanism in mammals. If there is between-women cycle variability (Fig. 1b), synchrony is a mathematical impossibility (Winfree, 1980). Since there is between cycle variability among women (Chiazze et al., 1968; Strassmann, 1997; Treloar et al., 1967; Vollman, 1977), cycle variability is a very serious problem; rendering suspect all reports of synchrony. Indeed, it must be concluded that synchrony is almost impossible when there is cycle
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variability and finding synchrony may be an indicator of a methodological artifact rather than a phenomenon. This general point strongly suggests that all methods, which consistently find synchrony when there is persistent cycle variability (especially between women cycle variability), may be plagued with methodological errors (Wilson, 1992; Strassmann 1997, 1999; Arden and Dye, 1998). These methods should be carefully reanalyzed using simulation approaches like that presented here to assess their potential for error. The simulation results presented here further strengthen the argument that for almost 30 years the search for synchrony may have been the wrong direction to take and the wrong question to ask regarding the relationship of ovarian cycles among women and females of other mammalian species. Cycle variability is well-documented in women (Chiazze et al., 1968; Treloar et al., 1967; Vollman, 1977) and while some irregularity is due to factors such as illness or accidents (Golub, 1992), cycle variability may be inherent to the physiological processes generating ovarian cycles. It is possible that cycle variability may be beneficial in some evolutionary and ecological contexts. For example, cycle irregularity can prevent even accidental synchrony. Avoiding synchrony may enhance female choice among males with differing phenotypic quality by reducing inter-female competition (Pereira, 1991). Cycle variability may also reduce the predictability of the next ovulation by males. With uncertainty regarding the next ovulation, males may be less able to control the matings of females. Nevertheless, even if cycle variability is beneficial to females in some mammalian mating systems, this should not be interpreted as implying that cycle variability is adaptive. Instead, cycle variability may be a generic feature — borrowing this concept from Kauffman (1993), (also see Schank, 1994) — of the physiological processes generating cycles, which may coincidentally have fitness benefits for females in some mating systems.
Acknowledgements This work was supported by NIH (through a subcontract with Indiana University). I also want to thank three anonymous reviewers of this paper for their comments, which greatly improved its logical structure.
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