# Estimating abortion rates from contraceptive failure rates via risk compensation: a mathematical model

Quirino Sugon Jr.[1,2], Daniel J. McNamara[1,2], Romeo Intengan[3]

1 Department of Physics, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines
2 Manila Observatory, Ateneo de Manila University Campus, Loyola Heights, Quezon City, Philippines
3 Loyola School of Theology, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines

Date Submitted: 19 March 2012, Feast of St. Joseph, Husband of Mary
Date Published: 26 March 2012, Feast of the Annunciation

ABSTRACT

In this paper, we propose a set of hypotheses for deriving the abortion rate as a function of the intercourse interval in weeks, the number of weeks since the start of first intercourse, the number weeks of pregnancy, the number of weeks of breastfeeding, and the contraceptive failure rate. We also propose risk compensation as feedback: the intercourse interval is proportional to the mth power of the contraceptive failure rate. We show that for different values of m, the abortion rate may become smaller, bigger, or remain the same compared to the case when no contraceptives are used. Thus, one way to settle the RH Bill debate is to determine the correct value of m derived from accurate data on the reproductive health parameters of a large sample of the female population. If this data is not available, it is better not to take risk in approving the bill, because there is a possibility of increasing our national abortion rate through the promotion of contraceptives. Instead, it may be better to use alternative methods to manage our population and reduce our abortion rate to zero by promoting chastity before marriage, late marriages, and breastfeeding—and accepting each child conceived as a gift and not as a burden.

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### 18 Responses to Estimating abortion rates from contraceptive failure rates via risk compensation: a mathematical model

1. WillyJ says:

Great job. Here are my questions (in layman’s terms):

1. Do you know of any counter hypotheses from the pro-RH side? A similar mathematical model that makes the pro-RH strongly conclude that increased contraceptives in the Philippines will lead to lower maternal deaths? And of course the figure on maternal deaths is still in question (11 per day according to the RH bill proponents!).

2. I assume there is yet no comprehensive survey to get the statistical value of “m”. Is that right?

3. Can we conclude from this hypothesis that raising the contraceptive prevalence rate will lead to unpredictable results in terms of the number of maternal deaths? Should we pay large sums for something that promises unpredictable results at best?

• WillyJ,

1. There is a prior work by Bongaarts and Westoff, Studies in Family Planning, Vol. 31, No. 3 (Sep., 2000), pp. 193-202:

TAR = Y_R (1-eu)/I_A – TFR I_B/I_A,

where TAR is the total abortion rate, Y_R is the number of reproductive years, e is the contraceptive effectivity, u is the proportion of the reproductive years where contraception is practiced, TFR is the total fertility rate, I_B is the birth interval, and I_A is the reproductive time allotted for induced abortion. This is Eq. (2). Bongaarts has another equation in their Eq. (4).

Their model does not use risk compensation and the intercourse interval. Thus, it is a zero-sum game. In our paper, Eq. (10) on the abortion rate can also be used by pro-RH side because making the contraceptive failure rate 1-ce = 0 makes the abortion rate Na = 0. But note that Eq. (10) does not still include risk compensation.

2. There is no survey that I know of to determine the statistical value of m. This requires women to log the specific dates and times of their fertility cycles, their intercourse intervals, the contraceptives used, their pregnancy period, their breastfeeding weeks, their abortions, etc. It requires a lifetime of data gathering per woman. Maybe an interview of each woman’s reproductive history may suffice as preliminary data–a toy data set for the development of statistical methods for extracting the parameters of interest as they vary in time. Once the statistical methods are perfected, they would then be applied to real observational data with accuracy comparable to the precise measurements of pressure, temperature, humidity, and wind speed and direction in atmospheric research.

3. We don’t know what would happen whether more contraceptives will decrease or increase the abortion rate. So it is better not to promote contraceptives until we are sure of what will happen. The model is only for computing abortion rate; determining maternal deaths is a separate problem and may require a new set of equations. We can propose a new hypothesis for verification: the abortion rate is proportional to the maternal mortality rate.

–Q. Sugon

2. Shimofuri says:

An assumption in the paper is flawed (there might be more as I just have a bit of time to browse). The authors in proposition 3.3 had asserted that the probability of failure for multiple contraceptives is the product of all the individual failure probabilities. That is flawed for it assumes that the failure of one contraceptive is independent from the others.

Individual failure events are mutually exclusive of each other (barring any consideration for synergistic effects) and for pregnancy to occur, all contraceptives must fail. Total failure is thus a conditional probability and not an independent probability between events. For example, no pregnancy can occur even if a placebo pill is taken (failure probability of 1) as long as the condom doesn’t break.

http://en.wikipedia.org/wiki/Conditional_probability

• Shimofuri,

You may like to propose other models for finding the effective contraceptive failure probability for a combination of several contraceptives. For example, you may like to create a matrix whose elements are the contraceptive failure rates for each combination of two contraceptives. But whatever that model is, the only important thing is the effective contraceptive failure rate which is the parameter used in the succeeding equations to derive the abortion rates.

–Q. Sugon

• michael says:

is this study already published in an international peer-reviewed journal/conference?

• Michael,

I did not submit the article to peer-reviewed journals, because in my experience, it takes about 6 months for the journal to send the referees comments and another 6 months of editing and corresponding with the referee before the paper is published. Besides, journals would make the article difficult to download because you have to pay. Since the RH Bill is being debated in the Congress and time is running fast, I asked the permission of my coauthors to publish the paper online instead in the Department of Physics’s blog which I manage. In this way, people can comment on it, point out its flaws and vague areas, and I can immediately respond.

–Dr. Sugon

• borrico1965 says:

It would be interesting to know how this paper will fare if published in a peer-reviewed journal/conference.

Publishing this online to join the current rh-bill controversy is a praise-worthy move. Nonetheless, I hope the authors will still consider submitting this to an appropriate scientific journal to subject it to an honest-to-goodness peer-review. It will have a wider audience and will be historical.

Kudos to the authors!

3. mikel says:

Dear Professor,

It is to my understanding that you equate number of possible pregnancies discounting infertile periods (Equation 10 by combining Equations 8 and 9) to be abortion rate. I can’t follow the logic behind the case, if this indeed is how it should be read. Btw, I got this link from the Inquirer.

• Mikel,

The woman’s maximum number of pregnancies Np in Eq. (8) is also the maximum number of abortions that she may have if she immediately abort each of those pregnancies through ‘morning after’ pills, for example. Since Na’ is the total number of abortions that she averted due to pregnancy and breastfeeding, then the remaining number of abortions that she may have is Np – Na’, which is what we defined as the total number of abortions Na in Eq. (10).

I am glad Inquirer led you here.

–Q. Sugon

• mikel says:

Thanks for clarifying Dr. Sugon. I believe this assumption is not so clear especially towards the end of the paper. The impression I got from reading the paper is that it is assuming that every possible pregnancy will be aborted. I follow the point on risk compensation hypothesis that it may lead to more sexual intercourse thus a greater chance of getting pregnant, but it does not follow that pregnancy will be aborted. Jumping from equation 8 to 10, which is the cornerstone of the rest of the model, requires a great leap of faith that contraceptives users will readily abort their pregnancy if ever gotten pregnant. Is it correct to say that what you have derived is an upper bound of abortion rate, i.e. possible number of times of getting pregnant, and that the endogenous decision to abort is not modeled in the paper?

• Mikel,

Yes, we are computing the upper bound of the abortion rate. One alternative method of analysis is to put an abortion decision probability for each pregnancy, then followed by a probability that the child will stay in the womb for a given number of weeks, and so on. This is useful if we use a decision tree model, such as in game theory. This would lend itself more to computational rather than analytical approach. We went via an analytical route: we counted the number of weeks that a child is within the womb and the number of weeks that a child is breastfed; a woman cannot be pregnant again during these times. So the number of possible pregnancies (which may be aborted) during this time intervals is subtracted from the maximum possible abortion rate (children aborted immediately after conception). The result is the upper bound of the abortion rate as given in Eq. (10).

4. Allister says:

“We know that 1 in 4 weeks, a woman is fertile. Thus,
1 in 4 intercourse of the woman would result to a
pregnancy if done on any 4 random days in the 4-
week fertility cycle.”

Funny how you state this. In most cases, men in general NEVER have sex with women who are menstruating. 😛

5. dear dr. sugon,

i beg to disagree. the case i presented was not covered in your model.

in the case of m = -1 you would get s = k / (1 – c_e), which would be undefined in the case of c_e = 1. in which case contraceptive methods like vasectomy would not be allowed by your model since it is 100% effective (the vas deferens is permanently cut).

in the alternative model i presented we have s = k / (1 + c_e), which allows for c_e = 1.

• xsaltire,

I deleted my previous comment because it did not update properly, but I think you have seen it. Yes, you are right. Your proposal is not covered by the case m = -1. Thus, in your model, when c_e = 0 (no contraceptives), then s = s_0 = k. And when c_e = 1 (vasectomy), s = s_0/2, so that the intercourse interval would be halved. In this case, you don’t anymore have a free parameter to play with in case experimental data would show that s = s_0/3, for example. Perhaps, you may need a power law such as s = k/(1 + c_e)^m.

There is still another boundary condition that a model must explain: the experimental result that despite the increasing safety of seat belts (which we compare to contraceptives), the number of accidents (which we compare to abortable pregnancies) remain the same. This is the classic case of risk compensation which we were able to accommodate in our paper through the case m = 1. It seems that there is no way in your proposed model that the number of abortions can remain constant despite the increasing contraceptive effectivity, even with a power law modification I proposed.

• Curious says:

Sir, I cannot seem to follow the logic.

You claim that the analogous case is that ” despite the increasing number of safety seat belts, the number of accidents remain the same.” I do not think that the reduction of accidents per se is the entire point of safety seat belts. In fact, the point of seat belts is the reduction of INJURY and not accident. I do not see how seat belts can actually reduce accidents, in the first place.

• Curious,

The claim is not mine but by Adams. He said that though seat belts have reduced the injury of the driver, as you have said, the risk of car accidents is transferred from the driver to pedestrians and cyclists. In summary, this is what he wrote:

“In summary there were two major road safety measures introduced by
the British Government in 1983: the seat belt law and the campaign against
drinking and driving. Figure 5 suggests that in 1983 there was a very small,
temporary, drop in road accident fatalities below the established trend. The
evidence with respect to seat belts suggests that the law had no effect on total
fatalities but was associated with a redistribution of danger from car occupants
to pedestrians and cyclists. The evidence with respect to alcohol suggests that
the decrease in fatalities in 1983 during the drink-drive hours is accounted for
partly by the still-unexplained rise above the trend in 1982, and partly by the
drink-drive campaign in 1983. The evidence from Britain, which has been
singled out as the only jurisdiction in the world in which it is possible to
measure fatality changes directly attributable a seat belt law, suggests that the
law produced no net saving of lives, but redistributed the burden of risk from
those who were already the best protected inside vehicles to those who were
the most vulnerable outside vehicles.”