Something that I would have thought to be dead simple nearly drove me crazy!

Let's say we make a small study of women who have similar factors for becoming pregnant. Let's say the study runs for two months ($m=2$) and involves two women ($n=2$). Obviously any real study would be much longer and larger, but 2x2 is the simplest case for the problem I'm having, and demonstrates it most clearly.

Suppose woman "A" becomes pregnant in the first month, and woman "B" does not become pregnant in either of the two months. We can express this in a really simple chart:

study participant pregnancy grid

We want to use this data to reason about the probability that some other similar woman will get pregnant within two months. Obviously in practice the error will be huge on such a small sample, but let's ignore that for now because the same problem can be extended to any number of women or months.

There are three different methods, which to me seem equally valid, by which we could reason about the same desired information.

In Method I, we reason simply that half of the observed women got pregnant within two months and so $p=0.5$. Plain and simple.

Method II is a known technique using a "life table", and in that case we would reason that the probability of "surviving" month 1 is $\frac{1}{2}$ and the probability of "surviving" month 2 is $\frac{1}{1}$, so the chances of not "surviving" both months without becoming pregnant are:

$$p = 1 - \left( \frac{1}{2} \cdot \frac{1}{1} \right) = 0.5.$$

In Method III, we reason that three months with the potential of initiating pregnancy (shown in yellow) were observed, and out of them, only one of them actually did, so the per month probability of becoming pregnant is $\frac{1}{3}$. We can then compute the chances for becoming pregnant within two months as:

$$p = 1 - \left(1 - \frac{1}{3}\right)^2 \approx 0.556.$$

Why does Method III get a different answer?

After proofreading the question a few times, I think I more or less have the answer, but I'll post it anyway since I already went to the trouble of making all the pretty formulae and graphics.  :)  And if there are any practical or theoretical nuances I'm missing, please do let me know!


I think this is a matter of dependence.

Method III assumes that the probability of getting pregnant in any given month is independent of other months. In practical terms this would be a mostly safe assumption, although maybe not totally.

(In particular, while for simplicity I'm assuming all women to have identical fecundity, if there is actually any difference, then in Method III, the less fertile women could pull the number down further the longer the study is run.)

I think Method II does not regard probabilities independently, because it basically expresses the 2nd month survival as:

$$P( s_2 | s_1 ),$$

where $s_i$ indicates "surviving" month $i$.

Method I is essentially just an implicit generalization of Method II, as:

$$p = 1 - \left( P(s_1) \prod_{i=2}^m P(s_i | s_{i-1}) \right).$$

And although plus-or-minus months have a biological significance in this context, in principle you could do this on any time interval and it would be equivalent if the intervals add up to the same total duration - the choice of what constitutes a survival "event" is essentially arbitrary. You could also do likewise in Method III.

It's worth noting that Method III is basically just a degenerate case of Method I/II, applicable where the per-interval probability is truly independent. In that case,

$$P( s_i | s_{i-1} ) = P( s_i ) = P( s_{i-1} ),$$

In which case, simplifying the general Method I/II formula results in the general Method III formula.

So, if there really is no time-dependent effect, then both methods will converge to the same answer with a large enough sample. (It isn't possible to see this in the 2x2 case unless there are no pregnancies, but it is possible to construct exact examples, such as for $n = 4, m = 2, p = \frac{3}{4}$.)

From that I would conclude that Method I/II is safer unless you are certain there is no time-dependent effect.


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