Contraceptive Efficacy and Combined Probability

When sex education sources quote numbers on the efficacy of different forms of contraception, they usually report a number like "98% effective". What they really mean is that, on average, 98 out of 100 women using that method of birth control for a whole year won't get pregnant. However, most people don't understand how a seemingly small annual risk of pregnancy might translate to a significant risk of pregnancy over a period of several years. In this post I show you how the probability of pregnancy grows over time and why it's commonly recommended that couples use two forms of contraception.

The annual risk of unintended pregnancy for several different contraceptive methods are given in the table below, taken from the 20th edition of Contraceptive Technology. The effectiveness typically reported by sex education sources is simply 100% minus the annual risk of pregnancy.

I'm not sure how typical use differs from perfect use when it comes to male sterilization...

It should be noted that "typical use" is perhaps misleading because it is an average for the people reportedly using that contraceptive method. For most contraceptive methods, including the condom and the pill, the most common deviation from "perfect use" is conscious user non-compliance (i.e. knowingly not using the contraception). Not using the contraception shouldn't really count in the typical use, but I guess they just do the best they can with the data they get.

So, suppose you're interested in calculating the risk of pregnancy within a period several years. From the table above, we can get the probability of not conceiving during the first year of use, equal to 100% minus the number from the table. We'll call this number p. The probability of not conceiving during a period of n years of use can be taken as p raised to the nth power. The risk of pregnancy is then calculated as

This simply estimates the risk of getting pregnant at least once during an n-year period while using a certain contraceptive method. The graph below compares a few common contraceptive methods for n = 1 to 10 years.

Comparing the risk of pregnancy using different contraceptive methods.

The graph shows that if you are not diligent in using the contraceptive method properly (i.e. you stray from "perfect use"), there's a pretty good chance of unintended pregnancy somewhere down the road. There's greater than 50% chance of unintended pregnancy within 8 years of "typical use" with the pill, within 6 years with the Standard Days method, 4 years with the male condom, and 3 years with the withdrawal method. Even with perfect use, if condoms are the only contraceptive method used, there's still about 18.3% chance of an unintended pregnancy in 10 years. That's 1 in 5.47. If you look at a 10-year period, comparing all the methods from the previous table, you'll find that if you use only a single method, only sterilization or IUDs reduce your risk of unintended pregnancy to less than 1 in 2 (i.e. less than 50%), assuming "typical use".

We can estimate the risk of pregnancy when contraceptive methods are combined by combining probabilities as shown in the following equation:

When two events, A and B, are independent, the probability of both A and B occurring is equal to the product of the probability of A and the probability of B. Applied to pregnancy risk, it means the risk of getting pregnant while using both methods A and B is equal to the probability of A failing multiplied by the probability of B failing. This approach to combining probability assumes that the individual probabilities are independent of each other. Before combining probabilities you should think about whether they actually are reasonably independent. For example, birth control pills and male condoms are probably independent (or nearly independent) because they are unrelated methods and it is unlikely that one method will influence the efficacy of the other. Therefore, the probability of an unintended pregnancy in the first year of use is probably approximately 0.18*0.09 = 1.62% when both male condoms and birth control pills are used (assuming "typical use"). A counterexample is the combination of the Standard Days Method and the TwoDay Method, which are probably not independent because they are both methods of achieving "fertility awareness". It's unlikely that one method negatively influences the other, but combining the two probably has little benefit either. The combination of Standard Days Method with the TwoDay Method is probably only slightly better than either one of them their own. That said, I've calculated the risk of pregnancy with some combinations of contraceptive methods which probably satisfy the assumption of independence in order to demonstrate how using two methods simultaneously dramatically reduces your risk of unplanned pregnancy.

Comparing the risk of pregnancy using different combinations of contraceptive methods.

As you can see, combining contraceptive methods reduces the risk of unintended pregnancy by a pretty significant margin. If using a single contraceptive method, assuming "typical use", only IUDs or sterilization were effective enough to reduce the risk of unintended pregnancy to less than 50% in 10 years. Combining 2 methods very effectively reduces the risk of unintended pregnancy without resorting to invasive or irreversible medical procedures.

Criticism of the Body Mass Index

The body mass index (BMI) is a metric proposed by Adolphe Quetelet during the mid-nineteenth century to assess human body shape. It is defined as a person's body mass in kilograms divided by the square of their height in metres. 

Since the 1800s, the BMI has seen continued use by health professionals as a quick assessment of one's health. Today, the BMI is still used to judge if a person is obese (BMI > 30 kg/m²), overweight (25 < BMI < 30), normal weight (18.5 < BMI < 25), or underweight (BMI < 18.5 kg/m²). But is the BMI a reasonable metric? Does it make sense for one's body mass to be proportional to the square of one's height? People of above average height, particularly men, often find that their BMI seems high despite being lean and fit. Shorter people, particularly women, similarly may find that their BMI seems low even with noticeable excess weight in the midsection. The BMI standards are most applicable for people who are close to average human height, but the standards lose their usefulness for everyone else. Considering that the average man and woman are already naturally about 8 cm above and below the average human height, respectively, it would seem that BMI immediately tends toward classifying men as overweight and women as underweight.

The BMI assumes this is true. The evidence says it isn't.

The simplest approach to estimating a power relation between mass and height would be to assume that the density of the human body is independent of size and that our bodies exhibit isometry. In simple terms, isometry means that height, breadth, and thickness all change in equal proportion. If a person gets 10% taller, they also get 10% wider across the shoulders and 10% thicker from front to back. The assumption of density's independence of size means that the average density of a tall person is the same for a short person. If both assumptions are true, then we would expect mass to be proportional to the cube of height. 

What we'd expect if all humans were scale copies of each other.

A lesser known index, known as Rohrer's Index or the Ponderal Index, actually makes the assumptions I've just mentioned above. Rohrer's Index is defined as mass divided by height cubed. For reference, the equivalent standards for underweight, overweight, and obese using Rohrer's Index instead of BMI are < 11.0 kg/m³, 14.9 < RI < 17.8, and > 17.8 kg/m³, respectively.

While people generally get wider and thicker as they grow taller, humans don't exhibit true isometry. There is a tendency for taller people to be narrower relative to their height than their shorter counterparts. Babies, with their comparatively large heads and short legs, are also far from being miniature adults. 

Their big heads and tiny legs are adorable, but they also force us to abandon the isometry hypothesis.

If you look at actual data, you find that neither index is very good, though Rohrer's Index seems to work better, especially in pediatrics. We'd like to have a working power law relation between mass and height because it would make the whole mass-to-height type index applicable to more people and probably a more useful health metric as a result. To find a power law relation, we can simply plot mass as a function of height and use the power curve fit option in Excel (or see if log(m) ÷ log(h) is approximately constant)

We're looking for a value of 'p' that has better correlation with data.

From Vital and Health Statistics (Series 11, No. 252), which contains anthropometric data from American adults and children collected between 2007 and 2010, we find that p = 2.48 with R² = 0.98 for males and p = 2.50 with R² = 0.97 for females. According to the data used to create the CDC Growth Charts (published in 2000), p = 2.52 and 2.54 for males and females, respectively (R² = 0.99 and 0.98). Data from Britain's 2003 Health Survey suggests that p = 2.49 and 2.69 fit best for males and females, respectively (R² = 0.97 for both sexes). 

Mass vs. Height of Males (2000 CDC Growth Chart Data)

Even the data which appears in Quetelet's Treatise on Man and the Development of His Faculties indicates 2 < p < 3. Using the data tables he gives for height and weight at different ages, p = 2.35 and 2.40 for males and females, respectively (R² = 0.97 for both sexes). Quetelet presents a separate table showing average weight for a given height. Based on that table, p = 2.21 for males (R² = 0.98) and p = 2.27 for females (R² = 0.96). It appears that the exponent is usually about the same in males as in females, so if we simply take the average of all the values of p we get 2.45. Therefore, the mass index formula we should be using is: 

With this formula, the standards become:

• Underweight (MI < 14.6)
• Normal weight (14.6 < MI < 19.8)
• Overweight (19.8 < MI < 23.7)
• Obese (MI > 23.7)

With a correct power law, we eliminate the issue of classifying people as overweight or underweight simply because they are significantly taller or shorter than the average human. While the mass index derived from statistical analysis is an improvement over the BMI, it still doesn't overcome the other serious flaws. First, women naturally have a higher body fat percentage than men. Basically, female hormones cause women to grow breasts full of fatty tissue while male hormones cause men to grow larger muscles. The result is, on average, fat accounts for more of a woman's body weight than a man's by about 6 percentage points.

Second, the index doesn't distinguish between lean mass and body fat. Muscle tissue is about 17% denser than fatty tissue, so athletes and gym rats can be lean and fit but still have a total mass that suggests they are overweight according to the BMI standards.

Finally, lean mass accounts for most of a person's mass (except perhaps in a few extreme cases). Any mass index therefore should only correlate well with body fat percentage among the morbidly obese, but for the majority of people mass index and adiposity will correlate poorly.

Health professionals are aware of issues with using a mass index. Romero-Corral et al. (2008) published a study of over 13,000 Americans in the International Journal of Obesity  to assess the accuracy of BMI as a diagnostic tool. Their discussion of the usefulness of BMI evolves around the inability to distinguish between lean mass and fat. They found that BMI > 30 classified 21% of the men and 31% of the women as obese. However, 50% of the men and 62% of the women were actually obese (defined as having greater than 25% or 35% body fat for men or women, respectively).

The accuracy of diagnostic tests is often assessed by positive and negative predictive values. Positive predictive value (PPV) is the probability that a positive test result indicates a correct diagnosis. Negative predictive value (NPV) is the probabilty that a negative result is correct. In Romero-Corral et al, PPV indicates the probability that BMI > 30 correctly identifies a person as obese and NPV indicates the probability that it correctly identifies a person as not obese. They found that the 30 kg/m² benchmark for obesity has a PPV of 87% for men and 99% for women. The NPV was 60% for men and 54% for women. What this all amounts to is a few false positives and a lot of false negatives; 50% of Americans are misidentified by the BMI-defined threshold for obesity.

Obesity diagnoses of the American population using BMI > 30 kg/m².
For every true positive obesity diagnosis using this test, there are 1.31 false negatives.

Despite well-known flaws and such poor accuracy as a diagnostic tool for obesity, the BMI remains commonplace. There are plenty of online BMI calculators. BMI is often included as part of a fitness assessment (I recall that it was calculated during fitness testing in high school gym class). All I can say is that I hope it goes away and that, however BMI classifies you, there's a pretty good chance that it's wrong.

Natural: Not a Synonym for Healthy

More and more I hear and see the word natural used to market products under the premise that natural things are good for you. Why should people equate natural with healthy? There are a plethora of things that occur naturally and are harmful to us. Asbestos; harmful elements like mercury, lead, arsenic, and radon; plus poisons like amatoxin, botulinum toxin, tetrodotoxin, ricin, tetanospasmin, taicatoxin, and PhTx3 can all be found in nature. Furthermore, how much of our food is truly natural? Even the most basic foods we eat are unnatural. We've hybridized nature's wild fruits, vegetables, and grasses to create nearly all modern crops, artificially increasing yields, edibility, and nutritional value. We've taken nature's wild animals and bred them selectively to artificially make them dumber, more docile, more productive, and meatier. When wild animals see the bountiful feasts we've created for ourselves, they are enticed to take it for themselves. After all, what creature wouldn't want to exploit a new food source which is both plentiful and nutritious? So we have to fight back against birds, rodents, insects, and other natural invaders to preserve our artificially enhanced food sources. Without our intervention, all of the crops and domesticated animals of our creation would disappear. 

Given the opportunity, this robin and his pals will gorge themselves on orchard cherries.

It's difficult to make the argument that we were better off before we started artificially enhancing the human environment. We are smarter, larger, healthier, more comfortable, and have greater longevity than ever before. I doubt that very many of the people buying into the "all-natural" trend would agree that we'd all be better off if we stopped adding Vitamins A & D to milk, iodine to salt, and folic acid to flour. 

Flour fortified with folic acid helps prevent Spina Bifida, much to the chagrin of today's all-naturally fed mothers.

Perhaps instead of letting the latest trend decide for us what is and isn't healthy, we should trust scientific evidence and the people qualified to interpret that evidence. Certified organic food isn't healthier than the regular stuff. There's no significant difference between milk produced by dairy cows and milk produced by dairy cows on the growth hormone rBST. High-fructose corn syrup doesn't hinder weight loss or cause weight gain (consuming too many calories does). 

Even worse than the use of the word natural to market overpriced foods is its use to market herbal supplements. Sure, we've been using some herbal remedies for thousands of years. The efficacy of some natural remedies, like willow bark, is well established and led to improvements in modern medicine.

Powdered willow bark might work, but I'm not convinced it's better for me than Aspirin.

Many other remedies, however, haven't been subjected to the same scientific scrutiny. Herbal supplements don't need to be effective and they don't need to be approved by a regulating body before hitting the market. Health Canada basically only requires that the ingredients of an herbal supplement be labelled correctly and that the stuff doesn't pose a significant health risk if taken as directed. The U.S. FDA has essentially the same relaxed rules. While many supplements are probably harmless, there are a few that could be dangerous. St. John's Wort might help with mild depression, but there isn't enough evidence to use it to treat major depression and it isn't more effective or significantly better tolerated than other antidepressants. Plus, it interacts with a number of different drugs, including oral contraceptives. Comfrey helps heal damaged skin and bone, but its toxic effects on the liver resulted in the FDA banning comfrey preparations intended for ingestion. Kava might help with your anxiety, but it often takes several weeks to start working, and some possible adverse effects include liver damage and dangerous interactions with alcohol and other drugs. Those are just three examples of supplements that have received enough attention to identify some of the risks. There are hundreds of untested, barely regulated supplements out there being marketed with unsubstantiated claims of significant health benefits, and new ones are coming out all the time. Without the clinical testing required of real medicine, the harmful effects of herbal supplements can go undetected longer and affect a much larger number of people. Remember, just because something's been around for a long time doesn't mean that it's right. 

In conclusion, the next time you see the word natural associated with a food or supplement, keep in mind that natural does not equal healthy, many artificial enhancements have made our lives better, and that the word natural has probably been placed there just to sucker you out of some money.