The madness of proportions

This was first posted to my blog on 20th March, 2022.

This post is inspired by a case study mentioned in the book “The Art of Statistics: Learning from Data” by the British statistician Sir David John Spiegelhalter.

If a few people sitting in a hotel and engrossed in eating bacon sandwiches were interrupted to show an article titled “Bacon has the same cancer risk as cigarettes, warn experts”, do a majority of them spit the sandwiches out of their mouths? If they are paranoid (or as some might say health-conscious), they would, or else they would continue eating their sandwiches as if they don’t give a damn about this article.

But should they be giving a damn about this article? If we pose this question to Professor Spiegelhalter, he would say, “it depends”.

But it depends on what? Of course, on the truth behind the facts and stats cited as evidence in this article.

In 2015, WHO placed processed meat among a group of carcinogens that includes but is not limited to – cigarettes, asbestos, alcoholic beverages, and solar radiation.

A few studies dealing with this issue stated that consuming 50g of processed meat every day increased the lifetime risk of bowel cancer from 5% to 6%, whereas a few other studies believed that the initial risk was 6% and the consumption of processed meat increased it to 7%. So we can conclude that on average, these studies reported that there is a 1% increase in the lifetime risk of bowel cancer upon the regular consumption of one or two bacon sandwiches per day (which amounts to around 50g of processed meat).

But a few sections of the media that ran reports around this bacon sandwich and bowel cancer story cited the reports that state eating 50g of processed meat every day gives a person an increased risk of bowel cancer of 17 to 20 percent. So how did this 1% reported by the studies shoot up to 17-20 % in the media articles? Are these media sources exaggerating the stats to create false propaganda against the meat industry? Before we answer that question, let us digress and look into a hypothetical scenario.

Imagine that you went to a shopping mall with 1000 dollars in your pocket. You liked a cute red hoodie that was on sale for 60 dollars in that mall. You purchase it in a heartbeat, return home joyously, and try to put it on. But to your horror, it doesn’t fit you. So you immediately return to the mall and exchange it for a blue hoodie that costs you an additional 10 dollars (because its price is 70 dollars).

Now if I am someone with a predilection for shopping and extravagant spending, I try to convince you that this exchange fiasco caused you to overspend by 10 dollars, which is a mere 1% increase if we compare it with the 1000 dollars you had initially. But if I am someone that hates consumerism and holds a grudge against all these malls, I would try to convince you that this caused you to overspend by 10 dollars, which is a mammoth 16.7% increase if we compare it with the original 60 dollars you spent. So the first proportion (10 out of 1000) is called an ‘absolute proportion’, and the second one (10 out of 60) is called a ‘relative proportion’.

Hence, in the case of the bacon story, the original lifetime risk of bowel cancer in the general population was between 5% to 6%. If one or two bacon sandwiches are consumed every day, there is a 1% increase in the absolute risk or a 17 to 20% increase in the relative risk. So, the media articles were not exaggerating this stat, they were just using the term ‘increased risk’, which is as ambiguous as it can be.

This naturally brings us to the question of where each of these proportions should be used. Absolute proportions are more liberal; overspending of 10 dollars is not a serious problem, hence we can represent such general and not-so-significant things with the absolute proportions. But relative proportions are lethal, especially in the small amounts, as an increase from 1 to 2 would mean a 100% relative increase, and an increase from 2 to 3 would mean a 50% relative increase. So if we are dealing with the scenarios where high precision is non-negotiable, like the drug experiments in the laboratory where a slight change in the drug dosage or the design of nuclear reactors where a slight change in the parameters temperature & pressure, could prove fatal and disastrous, then it is a good idea to use relative proportions.

There might be a question still troubling you – if a 1% absolute increase in the lifetime risk is not a big deal, then why did WHO add processed meat to the dangerous group of carcinogens? It is because as per WHO, these proportions are reported for the general population, which includes a vast majority of healthy people who might not get bowel cancer in their lifetime. But there’s a proportion who already have an increased risk, maybe because of their lifestyle choices, or maybe because of inherent conditions like obesity, and the majority of this 1% increase in the general population comes from this small proportion.

Hence, while there is no need to panic about these reports, it is still a good idea to assess our lifestyles and make wise choices. And since we are talking about making wise choices, it is also wise to ask ourselves the question, “Is this a relative proportion or absolute proportion?”, next time we come across the stats, especially those that talk about various risks.