## Synchronisation and the McClintock effect

I’m not an expert on menstruation. I’ve never done it, or had any reason to take a significant interest in it. But I had heard that women living together often “synchronise”, i.e. start to have their periods at the same time. Synchronisation is something that interests me, so I decided to look it up.

In short, it’s called the McClintock effect and it probably doesn’t really happen. A quick google reveals this Scientific American article about the history of it and where the research is now. It comes from a 1971 study by Martha McClintock who observed it in women living in college dormitories. Since then a lot has been published both for and against the existence of the effect. There are a lot of complications, and not being an expert etc etc, I couldn’t say for certain whether it does or doesn’t exist. Still, I think the question of how you go about looking for synchronisation between two (or more) cyclic systems that you have in front of you is quite interesting.

McClintock’s method was essentially this: she asked women living together for the first time in college dormitories: (a) which female they spent the most time with and (b) when their periods started. “Close friends” were paired up, and the differences in their onset dates in October were compared to the differences in their onset dates the following March – if the latter difference was smaller than the former, this was taken as evidence for synchronisation. Some statistics were done, and it turned out that indeed the March dates did tend to be closer to each other. So far so good.

However, Wilson points out a problem (link via Wikipedia, subscription required): rather than comparing simply the two closest onset dates in March, they made sure to calculate the difference between onsets after the same number of cycles for each member of the pair, as McClintock puts it:

For example, if onset 6 occurred on March 10 for the first member of the pair, and onsets 5 and 6 for the other member occurred on March 1 and March 29 respectively, then the March 10 and March 29 dates were used to calculate the difference in onset. This procedure was used to minimize chance coincidences that did not result from a trend towards synchrony.

Ironically, insisting on matching cycle numbers probably both increased the likelihood of finding synchrony by chance and of failing to spot pairs that were actually synchronising.

To see the problem, imagine the case when the cycles are not synchronising: assume each pair is two perfectly regular cycles, with random lengths of nearly 28 days, but not necessarily exactly. I wrote a program that generates random pairs where the cycle lengths can be 26, 27, 28 or 29 days, with a one in four probability of each. There is also a random difference between the onset dates in cycle zero (the first month), where we assume the second cycle can start up to 13 days after the first.

There are three possibilities for any given pair in this scenario: (a) the two cycle lengths are the same, in which case the onsets will never diverge or synchronise; (b) the longer cycle starts after the shorter cycle, in which case the onsets will diverge; or (c) the longer cycle starts before the shorter cycle, in which case the cycles appear to synchronise to start with, but eventually the longer cycle overtakes the shorter cycle, and they start diverging. On this graph I’ve plotted the difference in onset dates for 40 randomly generated pairs using this model, with case (a) coloured red, case (b) coloured green, and case (c) coloured blue.

(click to zoom)

Notice that after enough cycles, all the pairs are “diverging”. In reality, they are all, always, “diverging” (except the red ones), and having some of them appearing to synchronise or diverge at different points in the study is very bad news for most statistical approaches.

The right way to do it

Ok, I shouldn’t claim that there is only one right way to do this, but there are certainly better ways than the above. One easy way to look at two cycling systems is with a stroboscopic visualisation: simply choose one cycle as the reference, and at the same point in each reference cycle (e.g. the onset date), check the state of the other cycle (e.g. the number of days until the next onset date). In a sense it’s like flashing a strobe light on the second system whenever we reach a fixed point in the first system, hence the name. If the two systems are synchronised, the second system should always be in the same state when the strobe occurs. Because in this case we can’t “see” the state of the second cycle except on onset days, we have to make up some measure based on the distance to the nearest onsets.

I’ve done this here for paired cycles with the same length (left) and different lengths (right).

To extend the “cycle” analogy a bit, I’ve made the state of the second system an angle in degrees. So if at the point where cycle A reaches its onset, cycle B is also reaching its onset, the angle between them is 0°, if cycle B is a quarter of the way from its last onset to its next one, the angle is 90°, half way is 180° and so on. Notice that 180° and -180° are “the same” (half way from the previous onset is the same as half way to the next one), and hence the stroboscopic view on the right returns to 0° every time the faster cycle “laps” the slower one.

Of course this means that we are no longer comparing cycles at the same number of onsets from the original pair at the start. Why is this a good thing? Because the stroboscopic view always looks the same for any diverging (non-synchronised) cycles. In the above graph, the second cycle is slower than the first, so at each strobe it is slightly further behind. If the second cycle was faster than the first, the points would slope upwards, as the second cycle would advance at each strobe. Either way there is a diagonal pattern that clearly shows diverging behaviour.

There is a lot more to the menstrual synchrony research than this, both practical problems and issues with the statistics. Menstruation doesn’t always happen regularly, and in some cases it may regularised by an outside force (such as taking a contraceptive pill). You also want to look at groups of more than two (McClintock did and I haven’t discussed that), and too many other issues to mention here. Anyway, the stroboscopic method is used a lot in studies of synchronisation, and I think it has a nice intuitiveness about it that makes it a good way of looking at cycling systems.

Notes:

1. When I look at it, the red lines in the first graph (for the identical cycles) look like they slant downwards slightly. They don’t, they are definitely horizontal, its just a slightly cool optical illusion.
2. For a nice example of stroboscopic analysis, a paper by Schäfer, Rosenblum and Kurths (free pdf download) uses this method on human heartbeats and breath rate.
3. For more about synchronisation generally, Steven Strogatz has a nice video on the interwebs and book in the library.