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Contraceptive failure [Oct. 25th, 2007|10:41 am]
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This point came up in a conversation over on steerpikelet's LJ, and I thought it bore repeating here. Partly because it's important, but mostly because it allows me to talk about one of my favourite bits of maths, namely probability theory.

The failure rates of contraceptives are invariably quoted per year, rather than per use. So when people say that condoms have a breakage rate of 3%, this does not mean that for every 100 condoms you use, you can expect 3 to split: it means that for every 100 couples who have been using condoms for a year, 3 couples will have had at least one condom split on them. The difference is dramatic; let's see what would have happened if 3% of condoms split. In that case, every time you use a condom, it has a 97% chance of not splitting. So if you use, say, 100 condoms, the chance you won't experience any breaks at all is the product of the probabilities of each condom not breaking, i.e. 0.97 x 0.97 x ... x 0.97 = 0.97^100 = 0.047... . Which is to say that if 3% of condoms split, and you had sex 100 times in a year, you'd have only a 5% chance of not experiencing at least one broken "condom". And you'd have more than an 80% chance of experiencing two or more condom breakages in that year: the probability of exactly one condom breaking is 0.03 (for the broken condom) x 0.97^99 (for the 99 unbroken ones) x 100 (choices for which condom is the broken one) = 0.147, so the probability of two or more breaking is 1 - P(none break) - P(exactly one breaks) = 1 - 0.047 - 0.147 =~ 0.806. Such a "contraceptive" would be effectively useless.

"Blimey, guv'nor," I hear you say, "100 times a year? Are you avin' a laugh? It's all right for you layabout students with your peace marches and your LSD, but some of us have got to work for a livin'." Well, OK, suppose you have sex 20 times a year and use special fundie condoms that break in 3% of uses. Your chance of not experiencing any breaks would be 1 - 0.97^20 = 0.456..., which is to say that you've only got a bit better than a 45% chance of not seeing a broken "condom". Like I said, effectively useless.

Let's return to reality, and try to work out the actual per-use failure rate of real condoms. Suppose, again, that the couples in the studies had sex on average 100 times a year (adjust up or down as you prefer). Then the probability of a given condom not breaking is the 100th root of 0.97, which is 0.999695. So the chance of it breaking is actually 0.03%. If the couples had sex on average 200 times a year, we'd get 1 - (0.97^(1/200)) = 0.015%.

This is important to know about, because the less-scrupulous opponents of contraception will frequently quote per-year failure rates as if they were per-use rates (when they don't just lie outright, that is). And they're right: if condoms did fail on 3% of occasions, there wouldn't be much point in bothering with them. But they're actually much, much better than that. So if anyone ever says "condoms are useless, they fail 3% of the time", now you know what to say to them.

[Beware also the difference between breakage rate (interesting for STD protection) and conception rate (interesting for, well, contraception). Also, according to the site linked above, about half of breakages occur when you're putting the condom on, and are therefore harmless. Assuming you don't do something stupid like say "oh bugger, that was the last one, we'll just have to carry on without one." Mathematics is powerless against such stupidity.]
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Comments:
[User Picture]From: benparker
2007-10-25 02:03 pm (UTC)

Hmmmm

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Whilst I agree with the point you're trying to make, measuring the statistic "% of condom splitting" seems a bit pointless. The statistic used in comparing birth control methods is normally pregnancies per thousand couples per year (which I guess is also misleading, but at least easier to measure accurately).

How would you measure condom split rate? You could ask people randomly on the street (but who actually keeps records?), but I can't imagine you getting 1000 couples together and running an experiment (I would love to see the ethics board approve that one). Or you could do a simulated machine pumping test. Also, sometimes condoms do split but nobody notices- which is the most dangerous part!

Presumably independence doesn't quite work in the way you describe. There are some people who are good with condoms. There are some that are bad. (There are some that remember to take their pills. Some that don't.) More 15 year olds will probably have problems with condoms than 25 year olds. So because of that, the condom failure rate over a year is almost certain for some, and almost zero for others, certainly if you discard pre-use splittage. Also, if one does split, you're more careful next time so...

Anyway, yes, I think the main point is that it's a silly thing to measure, as you can't get good data, and to look at it on a case by case basis is misleading, as people in general have nookie more than once a year, and that's what you're actually concerned with.

Ben xxx
[User Picture]From: half_of_monty
2007-10-25 02:32 pm (UTC)

Re: Hmmmm

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Actually, now you come to mention it, the statistic I've heard before is that `condoms have a 3% failure rate' means `3% of couples using condoms for a year end up with a pregnancy'. Er, yes, that does appear to be the actual statistic
[User Picture]From: pozorvlak
2007-10-25 03:02 pm (UTC)

Re: Hmmmm

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OK: the site linked referred to it as a breakage rate rather than a pregnancy rate. I suppose pregnancy rate is easier to measure, even though breakage rate is also interesting from an STD point of view.
[User Picture]From: andustar
2007-11-10 09:30 am (UTC)

Re: Hmmmm

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It is morning now, I am more coherent. Geek hat on:

With STIs as well as pregnancy, measuring the actual tranmission rates is more helpful than measuring condom breaks. This has been done: HIV, HPV. It would seem that condoms are actually more effective at preventing HIV transmission than they are at preventing pregnancy. Condom break figures are just estimates based on one thing or another.

Also, on lying outright (warning: may inspire more tears).
[User Picture]From: pozorvlak
2007-11-11 12:39 pm (UTC)

Re: Hmmmm

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How interesting!

TBH, I've always thought of condoms as an STD prevention measure rather than a pregnancy-prevention measure: I feel rather happier if there's something else providing a backup on the latter.
[User Picture]From: pozorvlak
2007-10-25 03:08 pm (UTC)

Re: Hmmmm

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Well, OK, and you're quite right about the idependence issue. I wasn't trying to produce a 100% accurate mathematical model of condom use, more to point out the huge difference between the two similar-sounding statistics, both as a lesson in applied probability and as a defence against evil lying bastards who want to stop condom distribution programmes. The site I linked to seemed fairly clear that it was a rate of breakage rather than a rate of conception (I'd imagined they'd recruited a thousand couples, and asked them to record if they had a condom break in a given year, thus neatly solving the "what is the average amount of sex to have in a year" conundrum), but I'm prepared to be corrected on this point.
[User Picture]From: necaris
2007-10-25 03:05 pm (UTC)

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Tags: maths, sex
I don't know why but that strikes me as hilarious...
[User Picture]From: pozorvlak
2007-10-25 03:11 pm (UTC)

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This is actually the third time I've used that pair of tags. And most of my other posts with a "sex" tag have one or more science/computing tags too :-)
[User Picture]From: azrelle
2007-10-25 05:30 pm (UTC)

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When we are taught about contraception methods we normally get told the theoretical failure rates and then the actual failure rates. Condoms are only as reliable as the people using them, for all kinds of reasons. You can put them on wrong, buy crap ones, not leave enough of a pouch on the end, use some kind of lubricant that weakens it, not use lubricant at all.... Hence measuring their effectiveness very much depends on what endpoint you use and whether you study them in controlled circumstances or in actual use. I can't find the exact article just now, but it has been suggested that condoms are something like 70% effective against conception in actual use when used as the only method of contraception.

Similarly while The Pill is reported to be 98% effective in reality the end-users aren't anywhere near strict enough in administering it to achieve this figure. I can't remember what the actual use value is but it was somewhere around 85%.

The most effective contraceptive is the Mirena coil at 99.5%, a form of IUD, as you can do very little to interfere with its functioning short of removing it. The reported failures were also in very specific circumstances, one of which I believe was a woman who had 2 wombs and it hadn't been diagnosed before insertion. Unfortunately it doesnt offer anything in the way of protection against STDs and pelvic infections which still remains the only benefit of using condoms. It has been suggested that its use should be promoted more than The Pill due to its effectiveness and low side effect profile, but funnily enough guess which is cheaper to give out on a large scale?

[User Picture]From: half_of_monty
2007-10-25 07:37 pm (UTC)

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Thought the coil also left open the (very nasty) possibility of eptopic pregnancies - but there may be various sorts of coil; is this the one with hormones in it too?

At this point in the conversation I usually plug the injection of depo-provera, which has nasty side effects for some but is perfectly safe and healthy for many. And it stops your periods! I also think it's a similar price to the pill. However it isn't a good idea to promote it to teens because it interferes with laying down bone density (which is more or less done by the time you're 22ish).
[User Picture]From: pozorvlak
2007-10-25 07:50 pm (UTC)

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Indeed. Or the implant.
[User Picture]From: azrelle
2007-10-25 08:38 pm (UTC)

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After doing my stint in Obs & Gyn I was thoroughly put off Implanon. They go in great but getting them out is a fucker!! We had lots of young women in their 20s and 30s who ended up with scars on their arm after having to have a minor op to go in and retreive the implant. The side effects can be really nasty and unlike all the other methods it's a bit harder to stop using the implant, especially if yours is one of the ones that's got stuck.

Sure, the majority of women get on fine with Implanon. I didn't see scars from any other contraception method though.
[User Picture]From: azrelle
2007-10-25 08:21 pm (UTC)

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Mirena has hormones in it too, so in some cases stops ovulation occuring, definately stops the lining of the womb growing and also act as a physical block in the tubes. As it acts locally it's a lower does of hormones and also leads to reduced period bleeds and in some cases none at all. All the normal side effects and risks of hormones are reduced. There is still a risk of ectopic pregnancy if you are unlucky enough to be in the 0.5% of people in whom there is a failure, but that's still lower than the overal risk for ectopic pregnancy in the general population. I believe there was a paper recently that argued that in women who had already had an ectopic pregnancy and lost one fallopian tube that the Mirena would be the best option for them for contraception as it reduced the risk of ectopics below the normal for the general population and for any means of contraception.

When compared to Depoprovera the Mirena coil beats it on effectiveness and side effect profile in independent studies. The Mirena coil can however be painful to insert in women who have never had a child and that is what puts many women off having it done. Once you've had a kid your cervix is a bit wider and more easily accepts the Mirena going through.
[User Picture]From: half_of_monty
2007-10-25 08:27 pm (UTC)

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Yes, I think I was thinking of the purely physical coil with ectopic pregnancies (which I see I have been hideously mis-spelling). Thanks, very interesting - in fact now I come to think of it I think a doctor friend has also told me I should try that. Does the lower local dose of hormones effect also mean it's not associated to osteoperosis (which they don't seem to make their mind up on with depo)?
[User Picture]From: azrelle
2007-10-25 08:32 pm (UTC)

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It's not been on the market long enough to formally study, but in theory as the dose of hormones is much lower it shouldn't have as much of an effect on bone density as COCP or Depo. They can't prove that in actuality yet. A few studies have started to come out on Depo but have had very different conclusions - some say it's better, some say it's worse.
From: dariusarringto
2008-10-17 10:12 am (UTC)

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Barbie86   Posted: Anon: I'd need to ask a doctor, but one possibility could be that, the Mirena releases hormones over a period of 5 years.
[User Picture]From: pozorvlak
2007-10-25 07:54 pm (UTC)

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Interesting stuff! My point about the difference between per-year and per-use failure rates still stands, but now the numbers are rather different :-(
[User Picture]From: andustar
2007-11-09 08:58 pm (UTC)

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Thost statistics are definitely pregnancy rates. Not all condom breaks cause pregnancies, and not all pregnancies are caused by condom breaks (semen leaking out of the bottom being another issue). I can dig out the references to the original studies if you like.

And can I thank you for injecting some sanity into that thread? I would normally, but I didn't have the stamina to get involved in it around then.
[User Picture]From: pozorvlak
2007-11-09 09:44 pm (UTC)

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Good knowledge, thanks :-) And no problem about the sanity - to be honest, I was extremely surprised that atreic, a maths graduate, was coming out with such obvious nonsense.
[User Picture]From: andustar
2007-11-09 09:55 pm (UTC)

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There was so much crap logic on that page I didn't really know where to begin. Although this and this together made me weep more than the rest combined. I suspect I've never met you, and yet I still think you will share my pain.
[User Picture]From: pozorvlak
2007-11-09 10:08 pm (UTC)

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vampire_kitten generally made me weep.
[User Picture]From: andustar
2007-11-09 10:24 pm (UTC)

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I like you. Friends?
[User Picture]From: pozorvlak
2007-11-09 10:56 pm (UTC)

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Yeah, good idea :-)
[User Picture]From: andustar
2007-11-09 10:36 pm (UTC)

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P.S. More statistics than you can shake a stick at. In particular, Condom breakage and slippage occurs in an estimated 1.6-3.6% of coital acts. These events are related to user experience with condoms. However, the most important factor affecting condom failure is nonuse of the method, rather than breakage or slippage.

This last bit however refers to the rather frightening 'typical use' failure rate of 15% (15 pregnancies per year per 100 couples), mostly result from people only putting condoms on halfway through sex or not using them every time. The 2-3% failure rate you quoted is what happens when people are doing everything right, sadly it's not really representative of how people use them in the real world.

So Now You Know.
From: lxvxjxnkie
2008-04-04 01:29 am (UTC)

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I'm having a really fucking pointless argument about this with user foldl on Reddit, and I came out with an even smaller number (including the likelihood of PEP failure) and he's still going. *ARRGH!*
[User Picture]From: pozorvlak
2008-04-04 09:51 am (UTC)

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You have my sympathy - this post came about as a result of exactly such a pointless argument - with a mathematician, no less!
[User Picture]From: pozorvlak
2008-04-04 10:14 am (UTC)

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I've just read the thread, and it's not pretty. It looks like most of the problem is over what exactly you're arguing about - you and I were sticking to the original claim that becoming a prostitute necessarily significantly increases your risk of getting HIV, and foldl's sticking to the narrow interpretation of his words - that having more sex increases your risk of getting HIV. Which, yes, with condoms that are anything less than 100% effective (and that includes usage errors - maybe you're an expert, but it's dark and you're tired or whatever - we're all human) then he's right. But so are we.

When an argument degenerates that far, as they often seem to on Reddit, I find it best to swallow my pride and walk away - not always easy to do, mind. You're both too entrenched for either of you to win, and it's not worth the hassle.
From: lxvxjxnkie
2008-04-06 03:12 pm (UTC)

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Yep I decided to leave Reddit and concentrate on advocacy with ppl who might actually listen. I just want to find out what this jerk studied - I'm guessing it's computer science. It's definitely something where they've covered some history and philosophy of science, enough to make him think prospective studies are the only valid form of research.
[User Picture]From: pozorvlak
2008-04-06 05:42 pm (UTC)

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Well, foldl is what Haskell programmers use instead of for-loops, so CS seems a reasonable bet :-)
From: lxvxjxnkie
2008-04-07 03:37 am (UTC)

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I asked and he claims it's linguistics, but I'm not convinced. :P With sites like Reddit I think the Turing test is getting progressively easier to pass...