Tuesday, September 11, 2012

On Sugon, McNamara and Intengan (2012): A Case for the RH Bill

On 26 March 2012, Sugon, McNamara and Intengan of the Ateneo de Manila University web-published a paper on Estimating abortion rates from contraceptive failure rates via risk compensation: a mathematical model, which puts forward a theoretical model linking the abortion rate and contraception (more on this later). More technically inclined readers can download the pdf here. It would probably have gone unnoticed, but Mr. Antonio Montalvan III of the Inquirer featured it in his column and sung it high praises. Bottom line, it presents a mathematical argument for not passing the controversial RH bill, at least at this time, because there is a chance that contraceptives can actually increase the abortion rate.

The paper presents a model for the number of abortions as a function of six determinants: how frequently a woman has intercourse, when she begins sexual activity, cumulative time she is pregnant (infertile time on top of the monthly infertile period), cumulative time she is breastfeeding (they assume a woman is infertile while she is breastfeeding), the contraceptive failure rate, and the risk compensation factor. Notwithstanding the other problems with the paper, this post will just take a look at their model and the conclusions they gather from that.

The Model in Brief (no pun intended)

In a nutshell, their contraception-to-abortion argument hinges on the risk compensation mechanism they set forth in their paper. They argue, using maths, that women (curiously, they make no reference to men in their paper) will be more sexually active if contraception is available (economists will easily recognise this as the moral hazard problem). Sugon, McNamara and Intengan begin by building a model of a woman's likely number of abortions (they couch the discussion in individual language, although I would presume they are referring to population averages) in equation (10):

Na = (1/s)[10(ns - np - nb)(1 - ce)]

where Na = number of abortions with contraception (this is actually the number of pregnancies, which the authors say may be aborted; later on this is just the number of abortions); s = interval between sexual intercourse in weeks (the lower this number, the more frequently she is having sex); ns = number of "womb years" (a proxy for fertile period depending on time being sexually active, so the earlier a woman starts having sex the higher this number); np = number of womb years a woman is pregnant; nb  = number of womb years a woman is breastfeeding (she is assumed to be infertile during this time, even though LAM doesn't apply to all breastfeeding mothers); and ce = the success probability of contraception, with 0 < ce < 1, so (1 - ce) is the contraceptive failure rate.

The authors then lay out their model of risk compensation in section 5 of their paper (page 7). The crucial link between contraceptive use and frequency of sexual intercourse can be seen in equation (12) under hypothesis 3:

s = k(1 - ce)^m

where k is a constant and the other variables are as earlier defined. In describing Hypothesis 3 and equation (12), the authors say that "we propose that a woman's intercourse interval s is proportional to the mth power of the contraceptive failure rate 1 - ce" (page 7). The boundary of equation (12) for s is then analysed when ce approaches zero, or (1 - ce) approaches one. This is done to examine the frequency of sexual intercourse when contraceptives fail miserably such that it is as if one is not using contraceptives at all. They find in equation (14) that

lim (ce -> 0) s = k = s0

where s0 = interval between sexual intercourse in the absence of contraception, or when ce = 0.  Thus, equation (12) becomes equation (15):

s =  s0(1 - ce)^m

Plugging in equation (15) into (10) and doing some algebra, they derive an equation linking the number of abortions with contraceptive use in equation (18):

Na = Na0 / (1 - ce)^(m - 1)

where Na = number of abortions with contraception; Na0 = number of abortions without contraception; and ce and m are as before. One can see that

Na'(ce) = Na0(m - 1) / (1 - ce)^m

so that the sign of  Na'(ce) depends on the sign of (m - 1): any value of m > 1 and we get Na'(ce) > 0, so contraceptive use will increase abortions. In other words, the way contraceptives affect the abortion rate depends on the magnitude of m.

The s(ce) Function and the RH Bill

The value of m is crucial for the paper because this is used to argue that the country should hold off on passing the RH bill.  Since contraceptives could reduce, increase, or have no impact on abortions depending on the value of m, they argue that the government should hold off on the RH bill pending an estimation of m. After all, if m happens to be empirically greater than unity then the RH bill will increase abortions, rather than decrease them as the bill's supporters argue. Other than the obvious line of critique against this argument (risk compensation also applies to seat belts and health insurance), the problem is that there is no explanation for the functional relationship between sexual frequency and contraceptive effectiveness.

Their argument against the RH bill hinges on their arbitrary definition of the risk compensation mechanism as seen in equation (12), which introduces m and leads to equation (18). It is arbitrary because it is neither derived from other equations, the outcome of an optimisation exercise, nor taken from previous literature. While the other parts of the model were derived from other equations, they just assumed a functional form for s(ce) and went from there. There is also no reason, theoretical or empirical, to say that the risk compensation mechanism should follow the power law.

That said, their incorporation of the risk compensation mechanism is a valid exercise: it is reasonable to say that intervals between sexual intercourse could be inversely proportional to the effectiveness of contraception, especially those who do not wish to have children at the moment. Looking at equation (12), one can see that the risk compensation mechanism, through the s(ce) function, needs to satisfy two conditions:

Condition 1: s'(ce) < 0

Condition 2: lim (ce -> 0) s(ce) = k = s0

Taken together, the two conditions state that (1) contraceptive effectiveness is inversely related to the intercourse interval and (2) if contraceptives are totally ineffective s(ce) should revert to the no-contraception case of s0. This would be a more general formulation of their Hypothesis 3.

Using their exact same model, but being just as arbitrary as the authors in defining s(ce), one can define the risk compensation mechanism as

s*(ce) = k / (1 + ce)

which one can easily verify satisfies the two conditions above. This then becomes

s*(ce) =  s/ (1 + ce)

after the limits operation. The equation implies that, on average, women (men are not part of the authors' model) will be twice as sexually active with 100% effective contraception as they are without them. Plugging this into equation (10) gives us the following function for the number of abortions:

Na* = (1/s0)[10(ns - np - nb)(1 - ce^2)] = Na0(1 - ce^2)

So how does increasing contraceptive effectiveness affect the number of abortions relative to the no-contraceptives situation? One can easily see that

Na*'(ce) = - 2Na0c< 0

so promoting contraceptive use will unambiguously reduce the number of abortions. In other words, using their same model while slightly changing the risk compensation function, but keeping the same properties, one can arrive at a totally opposite conclusion: approve the RH bill now because it will reduce abortions. In fact, if contraceptives were 100% effective there would be zero abortions!

So What's the Point?

The Sugon, McNamara and Intengan paper essentially sets out a thought experiment with the aim of bolstering the argument that contraceptives can increase, rather than decrease, the number of abortions. However, what we show here is that the driver of their thesis-- their formulation of the risk compensation mechanism-- is an arbitrary assertion meant to arrive at their desired conclusion. There was no attempt to buttress this formulation through a risk minimisation (or sexual utility maximisation) operation, or to generalise the functional form of the mechanism. What we show here is that, using essentially their same model and being as arbitrary as the authors, one can do a similar thought experiment and arrive at a totally different conclusion and policy prescription. Upon closer examination, the authors' conclusions are really just assertions dressed in the language of mathematics. Take out those assertions and the paper falls apart.

+++

[Postscript: As a critique, it would have been easy to just point out their unfounded assumptions, especially in equation (12), and do away with the paper, but it would be unfair to the authors. I really admire their effort to further their side through the use of hard logic-- this is really much better than the fear mongering and name-calling we have been hearing from other anti-RH campaigners. A bit more of this, and a lot less of Sotto's unoriginal dramatics, would be much appreciated.]

Thursday, July 19, 2012

How did the Bible come about? Really?

Ok, ok, you'll probably say "God" and that's the end of it. I'm not here to challenge that. I'm here to tell you how it happened through history. It shouldn't be a problem, really, since one of the main tenets of Christianity is that God reveals himself to man through history. So there's nothing to fear from a little history. I'll dispense with citations for brevity-- if you don't believe me you can confirm everything on Wikipedia or, better yet, actual historical sources and the writings of the Early Church Fathers. If something ticks your curiosity or if you're lost in a term, just google it.

The Christian Bible, unlike Islam's Koran or Taoism's Tao Te Ching, did not come along in a singular moment in history. Tradition states that the Koran was dictated to Muhammad by the angel Gabriel, while the Tao Te Ching was written by the wise Laozi. Not so the Bible. The individual books of the Bible (which essentially means a collection of books) were written over the span of a few thousand years by scores of authors. These books, among other books with glowing pedigrees and spiritual value, were eventually selected to be part of the Bible. But how did that happen? It would be great if God just sent us a few stone tablets ala-10 Commandments (which aren't really Commandments if you ask the Jews, but that's another matter) saying, "The books of the Bible are etc. etc." But, no, it didn't happen that way.

As is often the case with Christianity, let's begin with the Jews. The Jewish Bible-- properly known as the Tanach-- is basically a collection of scriptures that can be read in synagogues. The Tanach is composed of the Torah or "teaching" (Genesis-Deuteronomy), Neviim or "prophets" (Joshua-Malachi), and the Ketuvim or "writings" (Psalms-Chronicles). The canon (i.e., list) of books in the Jewish Bible was determined during the Great Assembly (or Knesset; yes, the same name for the Israeli parliament) of scribes and sages some time during the 400s BC-- the Talmud (a collection of rabbinic literature) doesn't really give a precise date.

But the story of the Jewish Bible doesn't end there. At around the same time the Tanach was being finalised, Judaism was expanding and experiencing a split in language use. As Judaism spread from the Levant (today's Israel, Jordan, Lebanon, Palestine, and Syria) to what is today's Greece, Turkey, and Northern Africa, its adherents adopted the lingua franca of the time, which was Greek (there are also the Ethiopian Jews who supposedly rescued the Ark of the Covenant when Solomon's Temple was razed, but that's another story). The Talmud records that Ptolemy II, Egypt's king during the 300s BC, had the Jewish scriptures translated into Greek to make them more accessible to Greek Jews. He gathered 72 elders and put them in 72 separate chambers to translate the scriptures independently-- this was done so their translations could be cross-checked to ensure accuracy, Lo and behold, all 72 elders translated the scriptures identically, proof, says tradition, that God himself guided the translations. And thus the Septuagint (from the Greek word for 70) became the Bible for the Greek Jews. However, the Septuagint also contained books not included in the original Tanach defined by the Great Assembly about 100 years before. Among other books, it also included books by Tobit and Judith, and on the exploits of the Maccabaeus brothers (thanks to whom Jews now celebrate Hanukkah).

Fast-forward a few hundred years when this small but growing sect of Jews in the Levant started believing that Jesus of Nazareth was indeed the Christ. This religion, Christianity, which was new and quite revolutionary at the time, got more traction among Greek Jews and non-Jews (i.e., Gentiles) than among the more traditional Hebrew Jews in the Levant. So Christianity spread quickly out of the Levant and into present day Greece, Italy, Turkey, North Africa, and even farther away into France, Spain, and India. However, despite being open to Gentiles, the early Christians retained their Jewish practices of reading scriptures and partaking in ritual meals, this time in churches rather than synagogues. Early Christians, less than 500 years away from Jesus himself, also wrote extensively about their religion and the teachings of Jesus Christ, and their writings eventually made their way to churches.

The problem was, there was no agreed list of books that will be read in church. For the Old Testament it was quite easy: since most of them, ex-Jews or Gentiles, spoke Greek they just used the Septuagint. But the Christian books were more dicey. At the time many books were floating around and read in church. There are the familiar four Gospels, as well as the Pauline epistles, the Acts of the Apostles, and the epistles of Peter and James, among others. But also in the church-reading playlist were less familiar books such as the Teachings of the Twelve Apostles (Didache), Gospel of Thomas, Paul's Epistle to the Laodiceans, Apocalypse of Paul, Shepherd of Hermas, the Passions of various martyrs, etc. And some books we know today-- such as Paul's Epistle to the Hebrews and the Apocalypse of John-- were not widely accepted in the early church. So in 397 AD, Christian bishops (from the Greek "episkopos" meaning leader) gathered in Carthage to discuss what books can be read in church; most readers would be familiar with one attendee, Augustine of Hippo. This is what they decided could be read in church:

"It was also determined that besides the Canonical Scriptures nothing be read in the Church under the title of divine Scriptures. The Canonical Scriptures are these: Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua the son of Nun, Judges, Ruth, four books of Kings, two books of Chronicles, Job, the Psalter, five books of Solomon [i.e., Proverbs, Ecclesiastes, Song of Songs, Wisdom of Solomon, and Ecclesiasticus], the books of the twelve prophets, Isaiah, Jeremiah, Ezechiel, Daniel, Tobit, Judith, Esther, two books of Esdras [i.e., Ezra and Nehemiah], two books of the Maccabees. Of the New Testament: four books of the Gospels, one book of the Acts of the Apostles, thirteen Epistles of the Apostle Paul, one epistle of the same to the Hebrews, two Epistles of the Apostle Peter, three of John, one of James, one of Jude, one book of the Apocalypse of John."

Although the canon of the New Testament has been discussed and thrown around previously, the Council of Carthage in 397 was when the Christian Bible, particularly the New Testament, was made official (more or less) and didn't change since (more or less). Note that the proceedings of the Council of Carthage-- the only surviving account of which is the one by Dionysius Exiguus written 100 years after the fact-- did not mention why some books were included, while others were excluded, from the canon of the New Testament. It just lists them as fit to read in church, and that's that. They might have had good reasons to include the kaleidoscopic Apocalypse of John while excluding the instructive Didache, but we'll never know them now.

After the Council of Carthage, a more official and formal pronouncement on the books in the Bible (with requisite seals and stamps) would not happen until more than 1,000 years later, when the Catholic Council of Trent (1545-1563) and the Orthodox Synod of Jerusalem (1672) defined their respective Bibles in response to the Reformation. Martin Luther famously wanted to get rid of seven New Testament books-- Hebrews, James, Jude, 2 Peter, 2 and 3 John, and the Apocalypse (i.e., Luther's antilegomena)-- and the additional books in the Septuagint not found in the Tanach because he disputed their authenticity. Eventually most Protestants would accept Luther's argument on keeping only the Tanach books-- they would call the additional books in the Septuagint the Apocrypha, while Catholics and Orthodox Christians would call them the Deuterocanonicals-- but rejected his dispute with the seven New Testament books, accepting the Council of Carthage's canon of 27 books.

And that's how the Christian Bible came about. Not so much a giant book coming down from heaven on a chariot of clouds, but more a series of decisions by committees composed of old men. Committees composed of old men inspired by God, perhaps.

Tuesday, July 17, 2012

Woman on Top: Marissa Ann Mayer

Marissa Ann Mayer
President and Chief Executive Officer
Yahoo! Inc.

• Born in in 1975 in Wisconsin.
• Obtained her BS in Symbolic Systems and MS in Computer Science from Stanford University; specialised in artificial intelligence.
• Awarded the Centennial Teaching Award and Forsythe Award by Stanford for her contribution to undergraduate education.
• Worked at the Union Bank of Switzerland research lab and the Stanford Research Institute prior to joining Google.
• Was Google's 20th employee and first female engineer when she was hired in 1999.
• Was one of the three-person team that developed the Ad Words, which is the cornerstone of Google's business model.
• Was crucial in developing Google's search, email, news, and maps products.
• Oversaw the efficient look-and-feel of the Google experience.
• Awarded an honorary doctorate degree by the Illinois Institute of Technology for her work in the development of search engines.
• Among Fortune magazine's 50 Most Powerful Women in Business in 2008-2011.
• Was Google's vice president for search before being moved to local and location services in 2010.
• Appointed President and CEO of Yahoo! on 16 July 2012.

Monday, July 2, 2012

Why is the Catholic Church so stubborn on RH? Really?

If you know me then you know where I stand on reproductive health issues. This post is not to defend the Catholic Church's stand on artificial contraception, but to explain why she takes such a strong stance against it. I also hope to clarify some misconceptions on the Church's stance. If you wish to engage the Church on this issue you need to first understand where she's coming from and what motivates her. Maybe it will also help you evaluate the (f)utility of debating with the Church on this issue in the first place. This post is written with a lay (even irreligious) audience in mind, so I will dispense with the Bible quotes and Magisterial references.

It is no secret that the Catholic Church believes in the existence of God who created the universe. Duh. But corollary to this belief in a sentient and benevolent God is the assumption that everything He created has a purpose: nothing is random in God's creation therefore everything has to have been made for a purpose. So the Sun isn't just an amalgamation of cosmic particles brought together by gravity and heated by nuclear fusion; it was created to eventually sustain life on Earth. This belief system applies to all of creation, including the human body. Everything in the human body from the heart to the toenail you clip off has a purpose willed and designed by God. So far so good. No problem.

The problem begins when we start talking about the reproductive system. The Catholic Church believes that the reproductive system-- not just the womb and testes but also the pleasure-giving glans penis and clitoris-- have a dual purpose: to express mutual love and to procreate. The reproductive system was created to enable humans to express their mutual love for each other through sexual intercourse and to encourage procreation. Now take note of the "and". The Catholic Church's issue with contraception begins when that "and" becomes an "or". Artificial contraception, by removing any possibility of procreation, turns sexual intercourse into an exclusively love-making pleasurable affair. This, believe the Church, is contrary to God's will and purpose for creating the reproductive system.

But how about natural family planning? Or when one spouse is infertile due to natural causes or a needed medical operation (e.g., hysterectomy due to a tumour)? Won't sexual intercourse in those situations be divorced from the procreation purpose too, and therefore against God's will? Well, no. Natural family planning, by virtue of being natural, is part of God's plan: in God's wisdom He recognised the need for families to plan and space their offspring, but also recognised the need for spouses to make love, so He provided windows of opportunity to make love while vastly minimising the chance of conception. As for infertility due to natural causes or a needed medical operation, well, God had reasons for giving someone that affliction, and it definitely wasn't His intention to prevent spouses from expressing their love for each other. So in these cases any dichotomy between love-making and procreation was not man's will but God's, which is fine for the Church.

So the Church's real problem with artificial contraception is that man is divorcing love-making with procreation. In the case of artificial contraception, man wants the love-making part while eliminating the procreation part. Note that the same problem arises when man wants the procreation part while eliminating the love-making part, thus the Church's similar opposition to in-vitro fertilisation. Man cannot, should not, separate the expression of mutual love from the possibility of procreation. God can do it, but not man.

It is thus easy to see why no amount of medical, social, economic, democratic, etc. arguments or evidence will change the Church's position on artificial contraception-- they all pale in comparison to God's will and purpose for creating the reproductive system. Practical circumstances may mitigate the gravity of going against God's will through the use of artificial contraception (or in-vitro fertilisation, for that matter), but it is a sin nonetheless and bishops will be remiss in their duty if they tolerate it. So changing their stance on artificial contraception will require a change in their understanding of God's purpose for creating the reproductive system. It hasn't changed in 2,000 years, so it is quite unlikely that it will change any time soon.

Wednesday, March 30, 2011

hear that, philippine broadcast media? yeah, remember that next time you're thinking of shoving your cameras into the faces of grieving family members.

Saturday, March 19, 2011

Hello?!?

Anybody there?!?

It's been more than a year since my last post. Haven't really had the time to blog lately. Also haven't been on a plane since my trip to Vientiane. Let's see, maybe we can still resuscitate life into this old blog.

Wednesday, February 10, 2010

So when does it make sense to play Lotto 6/49?

Ok, by "make sense" here I mean you don't lose money in expected value terms; i.e., you're not throwing money in the crapper by spending P20 on that Lotto 6/49 ticket (unless, of course, you think PCSO does great social work and that P20 is your corporal work of mercy for the day).

First let me explain what I mean by "expected value". In games of chance such as lotto, expected value is the amount of winnings (or loss) multiplied by the probability of winning. Suppose the game of chance is tossing a fair coin and you get P100 if heads come up and you get zilch if tails come up, so you have a 50/50 chance of winning and getting zilch. In this case, the expected value from the coin-toss game is

EV = (P100)(0.5) + (P0)(0.5) = P50

So basically the expected value is the amount you can expect to gain from playing a game before you actually play it (e.g., when you're thinking of buying the ticket). Of course, it "makes sense" to play if the cost of the ticket is less than or at least equal to how much you can expect to win. In this case you probably won't (or shouldn't) play the game if you have to gamble more than P50 to have a 50% of winning P100, but it'll make sense to play if your bet is only, say, P25.

Now let's look at Lotto 6/49. The game involves a player picking 6 numbers from 1 to 49 (hence the name). A lotto ticket, on the other hand, costs P20, which the player has to pay if he wants to have a chance of winning. PCSO, which manages the game, then (presumably) randomly selects 6 numbers and publishes it. If the player gets 6 out of 6 numbers he gets the pot, which could run to the tens to hundreds of millions of pesos. Getting 5 out of 6 numbers gets you P20,000; 4 out of 6 gets you P500; and 3 out of 6 numbers results in balik taya, or you get your P20 back.

So, and this is where it gets dicey, what are your chances of winning? If you buy one ticket, then you will win the pot if your exact 6 numbers are selected out of all the possible combinations of 6 numbers out of 49, which are 13,983,816 possible combinations (mathematically, this is 49C6). Therefore, the probability of your 6 numbers winning the pot is 1/13,983,816 = 0.00000007151.

But then winning the pot isn't the only way you can win-- you will also win something if you get 5/6, 4/6, or 3/6 numbers correct. Choosing 6 numbers is like getting 6 chances of getting 5 numbers correct since there are 6 possible combinations of 5 numbers from the 6 numbers you chose. Eh? Suppose you very creatively chose the following 6 numbers:

[1, 2, 3, 4, 5, 6]

Then the possible combinations of 5 numbers from the 6 numbers you chose are:

[1, 2, 3, 4, 5], [1, 2, 3, 4, 6], [1, 2, 3, 5, 6], [1, 2, 4, 5, 6] [ 1, 3, 4, 5, 6], and [2, 3, 4, 5, 6]

Similarly, you get 15 chances of getting 4 numbers correct and 20 chances of getting 3 numbers correct from the set of 6 numbers you chose. Using combinatorial mathematics, the expected value from the lotto game can be computed as:

EV = (1/49C6)(X) + (1/49C5)(6C5)^2(P20,000) + (1/49C4)(6C4)^2(P500) + (1/49C3)(6C3)^2(P20)

where X = the winning pot, which can change. The squared terms are there to take into account that both the player and PCSO choose 6 numbers.

Since the cost of a lotto ticket is P20, how much should the pot be so that the expected value from playing equals the cost of a ticket? Setting EV = P20, we can calculate X as... P261 million.

Yes, the pot will have to be at least P261 million before it actually makes sense to play Lotto 6/49. Any pot lower than that and you're basically throwing P20 into the crapper, just like paying P75 to play the coin-toss game above.

But then if buying that P20 ticket gives you the right to hope and to dream about what you will do in the off-chance you will win the pot, then by all means buy the ticket. After all, that's what PCSO (and all casinos, for that matter) are hoping you will think.