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Meeting Ghosts In The Chase For Reality

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Sunrise

Sunrise (via faxpilot @ Flickr CC BY-NC-ND license)

Watching the morning sun beaming through the clouds during today’s morning jog, I was struck by an epiphany. What ultimately transpired was a streak of thoughts, that left me in a overwhelming sense of awe and humility for its profound implications.

Perhaps the rejuvenating air, the moist earth from the previous night’s rains and the scent of the fresh Golden Flamboyant trees lining my path made the sun’s splendor much more obvious to see. Like in a photograph coming to life, when objects elsewhere in the scene enhance the main subject’s impact.

As I gazed in its direction wondering about the sunspots that neither I nor anyone else around me could see (but that I knew were really there, from reading the work of astronomers), I began thinking about my own positional coordinates. So this was the East, I found. But how did I know that? Well as you might have guessed, from the age old phrase: “the sun rises in the East and sets in the West”. Known in Urdu as “سورج مشرق میں نکلتا ہے اور مغرب میں ڈوبتا ہے ” or in Hindi, “सूरज पूरव में निकलता है और पश्चिम में डूबता है” and indeed to be found in many other languages, we observe that man has come to form an interesting model to wrap his mind around this majestic phenomenon. Indeed, many religious scriptures and books of wisdom, from ancient history to the very present, find use of this phrase in their deep moral teachings.

But we’ve come to think that we know this model is not really “correct”, is it? We’ve come to develop this thinking with the benefit of hindsight (a relative term, given Einstein’s famous theory, by the way. One man’s hindsight could actually be another man’s foresight!). We’ve ventured beyond our usual abode and looked at our planet from a different vantage point – that of Space. From the Moon and satellites. The sun doesn’t actually rise or set. That experience occurs because of our peculiar vantage point – of relatively slow or immobile creatures grounded here on Earth. One could say that it is an interesting illusion. Indeed, you could sit on a plane and with the appropriate speed, chase that sliver of sunlight as the Sol (as it’s lovingly called by scientists) appears or disappears in the horizon, never letting it vanish from view and do so essentially indefinitely.

Notes In The Margin About Language

Coming back, for a moment, to this amusing English phrase that helped me gauge my position, I thought about how language itself can shape one’s thinking. A subject matter upon which I’ve reflected before. There really comes a point when our models of the world and the universe get locked within the phraseology of a language that can actually reach the limits of its power of expression fairly unexpectedly. Speak in English and your view is different from somebody who can speak in Math. Even within Math, the coming about of algebra expanded the language’s power of expression incredibly from its meager beginnings. New models get incorporated into the lexicon of a language and because we tend to feed off of such phrases to make sense of ourselves and our universe, there is the potential for an inertia to develop, whereby it becomes easy to stay put with our abstractions of reality and not move on to radically new ones – models that are beyond the power of expression of a language and that haven’t yet been captured in its lexicon. In a way we find that models influence languages and languages themselves influence models and ultimately there is this interesting potential for a peculiar steady state to be reached – which may or may not be such a good thing.

So when it comes to this phenomenon, we’ve moved from one model to another. We began with “primitive” maxims. Perhaps during a time when people used to think of the Earth as flat and stars as pin-point objects too. And then progressed to geocentrism and then heliocentrism, both of which were basically formulated by careful and detailed observations of the sky using telescopes, long before the luxury of satellites and space travel came into being. And now that we see the Earth from this improved vantage point – of Space – our model for understanding reality has been refined. And actually, really shifted in profound ways.

So what does this all mean? It looks like reality is one thing, that exists out there. And we as humans make sense of reality through abstractions or models. How accurate we are with our abstractions really depends on how much information we’ve been able to gather. New information (through ever more detailed experiments or observations and indeed as Godel and Poincare showed, sometimes by mere pontification), drives us to alter our existing models. Sometimes in radically different ways (a classic example is our model of matter: one minute particle, one minute wave). There is this continuous flux about how we make sense of the cosmos, and it will likely go on this way until the day mankind has been fully informed – which may never really happen if pondered upon objectively. There have been moments in the past where man has thought that this precipice had been finally reached, that he was at last fully informed, only to realize with utter embarrassment that this was not the case. Can man ever know, by himself, that he has finally reached such a point? Especially, given that this is like a student judging his performance at an exam without the benefit of an independent evaluator? The truth is that we may never know. Whether we think we will ever reach such a precipice really does depend on a leap of faith. And scientists and explorers who would like to make progress, depend on this faith – that either such a precipice will one day be reached or at least that their next observation or experiment will increase them in information on the path to such a glorious point. When at last, a gestalt vision of all of reality can be attained. It’s hard to stay motivated otherwise, you see. And you thought you heard that faith had nothing to do with science or vice versa!

It is indeed quite remarkable the extent to which we get stuck in this or that model and keep fooling ourselves about reality. No sooner do we realize that we’ve been had and move on from our old abstraction to a new one and one what we think is much better, are we struck with another blow. This actually reminds me of a favorite quote by a stalwart of modern Medicine:

And not only are the reactions themselves variable, but we, the doctors, are so fallible, ever beset with the common and fatal facility of reaching conclusions from superficial observations, and constantly misled by the ease with which our minds fall into the rut of one or two experiences.

William Osler in Counsels and Ideals

The World According To Anaximander

The World According To Anaximander (c. 610-546 BCE)

The phenomenon is really quite pervasive. The early cartographers who divided the world into various regions thought funny stuff by today’s standards. But you’ve got to understand that that’s how our forefathers modeled reality! And whether you like it or not someday many generations after our time, we will be looked upon with similar eyes.

Watching two interesting Royal Society lectures by Paul Nurse (The Great Ideas of Biology) and Eric Lander (Beyond The Human Genome Project: Medicine In The 21st Century) the other day, this thought kept coming back to me. Speaking about the advent of Genomic Medicine, Eric Lander (who trained as a mathematician, by the way) talked about the discovery of the EGFR gene and the realization that its mutations strongly increase the risk for a type of lung cancer called Adenocarcinoma. He mentioned how clinical trials of the drug Iressa – a drug whose mechanism of action scientists weren’t sure of yet but was nevertheless proposed as a viable option for lung adenocarcinomas – failed to show statistically significant differences from standard therapy. Well, that was because the trial’s subjects were members of the broad population of all lung adenocarcinoma cases. Many doctors realizing the lack of conclusive evidence of a greater benefit, felt no reason to choose Iressa over standard therapy and drastically shift their practice. Which is what Evidence-Based-Medical practice would have led them to do, really. But soon after the discovery of the EGFR gene, scientists decided to do a subgroup analysis using patients with EGFR mutations, and it was rapidly learned that Iressa did have a statistically significant effect in decreasing tumor progression and improving survival in this particular subgroup. A significant section of patients could now have hope for cure! And doctors suddenly began to prescribe Iressa as the therapy of choice for them.

As I was thinking about what Lander had said, I remembered that Probability Theory as a science, which forms the bedrock of such things as clinical trials and indeed many other scientific studies, had not even developed until the Middle Ages. At least, so far as we know. And modern probability theory really began much later, in the early 1900s.

Front page of "Doctrine of Chance – a method for calculating the probabilities of events in plays" by Abraham de Moivre, London, 1718

Abraham de Moivre's "Doctrine of Chances" published in 1718, was the first textbook on Probability Theory

You begin to realize what a quantum leap this was in our history. We now think of patterns and randomness very differently from ancient times. Which is pretty significant, given that for some reason our minds are drawn to looking for patterns even where there might not be any. Over the years, we’ve developed the understanding that clusters (patterns) of events or cases could occur in a random system just as in a non-random one. Indeed, such clusters (patterns) would be a fundamental defining characteristic of a random process. Absence of clusters would indicate that a process wasn’t truly random. Whether such clusters (patterns) would fit with a random process as opposed to a non-random one would depend on whether or not we find an even greater pattern of how these clusters are distributed. A cluster of cases (such as an epidemic of cholera) would be considered non-random if by hypothesis testing we found that the probability of such a cluster coming about by random chance was so small as to be negligible. And even when thinking about randomness, we’ve learned to ask ourselves if a random process could be pseudo-random as opposed to truly random – which can sometimes be a difficult thing to establish. So unlike our forefathers, we don’t immediately jump to conclusions about what look to our eyes as patterns. It’s all quite marvelous to think about, really. What’s even more fascinating, is that Probability Theory is in a state of flux and continues to evolve to this day, as mathematicians gather new information. So what does this mean for the validity of our models that depend on Probability Theory? If a model could be thought of as a chain, it is obvious that such a model would be as strong as the links with which it is made! So we find that statisticians keep finding errors in how old epidemiologic studies were conducted and interpreted. And the science of Epidemiology itself improves as Probability Theory is continuously polished. This goes to show the fact that the validity of our abstractions keeps shifting as the foundations upon which they are based themselves continue to transform. A truly intriguing idea when one thinks about it.

Some other examples of the shifting of abstractions with the gathering of new information come to mind.

Image from Andreas Vesalius's De humani corporis fabrica (1543), page 190.

An image from Vesalius's "De Humani Corporis Fabrica" (1543)

Like early cartographers, anatomists never really understood human anatomy very well back in the days of cutting open animals and extrapolating their findings to humans. There were these weird ideas that diseases were caused by a disturbance in the four humors. And then Vesalius came along and by stressing on the importance of dissecting cadavers, revolutionized how anatomy came to be understood and taught. But even then, our models for the human body were until recently plagued by ideas such as the concept that the seat of the soul lay in the pineal gland and some of the other stuff now popularly characterized as folk-medicine. In our models for disease causation, we’ve progressed over the years from looking at pure environmental factors to pure DNA factors and now to a multifactorial model that stresses on the idea that many diseases are caused by a mix of the two.

The Monty Hall paradox, about which I’ve written before is another good example. You’re presented with new information midway in the game and you use this new information to re-adjust the old model of reality that you had in your mind. The use of decision trees in genetic counseling, is yet another example. Given new information about a patient’s relatives and their genotype, your model for what is real and its accuracy improves. You become better at diagnosis with each bit of new information.

The phenomenon can often be found in how people understand Scripture too. Mathematician, Gary Miller has an interesting article that describes how some scholars examining the word Iram have gradually transformed their thinking based on new information gathered by archeological excavations.

So we see how abstractions play a fundamental role in our perceptions of reality.

One other peculiar thing to note is that sometimes, as we try to re-shape our abstractions to better congrue with any new information we get, there is the tendency to stick with the old as much as possible. A nick here or a nudge there is acceptable but at its heart we are usually loath to discard our old model entirely. There is a potential danger in this. Because it could be that we inherit flaws from our old model without even realizing it, thus constraining the new one in ways yet to be understood. Especially when we are unaware of what these flaws could be. A good example of abstractions feeding off of each other are the space-time fabric of relativity theory and the jitteriness of quantum mechanics. In our quest for a new model – a unified theory or abstraction – we are trying to mash these two abstractions together in curious ways, such that a serene space-time fabric exists when zoomed out, but when zoomed in we should expect to see it behave erratically with jitters all over the place. Our manner of dealing with such inertia when it comes to building new abstractions is basically to see if these mash-ups agree with experiments or observations much better than our old models. Which is an interesting way to go about doing things and could be something to think about.

Making Sense Of Reality Through The Looking Glass

Making Sense Of Reality Through The Looking Glass (via Jose @ Flickr, CC BY-SA-NC license)

Listening to Paul Nurse’s lecture I also learned how Mendel chose Pea plants for his studies on inheritance rather than other complicated vegetation because of the simplicity and clarity with which one could distinguish their phenotypes, making the experiment much easier to carry out. Depending on how one crossed them, one could trace the inheritance of traits – of color of fruit, height of plant, etc. very quickly and very accurately. It actually reminded me of something I learned a long time ago about the various kinds of data in statistics. That these data could be categorized into various types based on the amount of information they contain. The highest amount of information is seen in Ratio data. The lowest is seen in Nominal data. The implication of this is that the more your experiment or scientific study uses Ratio data rather than Nominal data, the more accurate will your inferences about reality be. The more information you throw out, the weaker will your model be. So we see that there is quite an important caveat when we build abstractions based on keeping it simple and stripping away intricacy. When we are stuck with having to use an ape thumb with a fine instrument. It’s primitive, but it often gets us ahead in understanding reality much faster. The cost we pay though, is that our abstraction congrues better with a simpler and more artificial version of the reality that we seek to understand. And reality usually is quite complex. So when we limit ourselves to examining a bunch of variables in say for example the clinical trial of a drug, and find that it has a treatment benefit, we can be a lot more certain that this would be the case in the real world too provided that we prescribe the drug to as similar a patient pool as in our experiment. Which rarely happens as you might have guessed! And that’s why you find so many cases of treatment failure and unpredictable disease outcomes. How the validity of an abstraction is influenced by the KISS principle is something to think about. Epidemiologists get sleepless nights when pondering over it sometimes. And a lot of time is spent in trying to eliminate selection bias (i.e. when errors of inference creep in because the pool of patients in the study doesn’t match to an acceptable degree, the kinds of patients doctors would interact with out in the real world). The goal is to make an abstraction agree with as much of reality as possible, but in doing so not to make it so far removed from the KISS principle that carrying out the experiment would be impractical or impossible. It’s such a delicate and fuzzy balance!

So again and again we find that abstractions define our experiences. Some people get so immersed and attached with their models of reality that they make them their lifeblood, refusing to move on. And some people actually wonder if life as we know it, is itself an abstraction :-D! I was struck by this when I came upon the idea of the Holographic principle in physics – that in reality we and our universe are bound by an enveloping surface and that our real existence is on this plane. That what we see, touch or smell in our common experience is simply a projection of what is actually happening on that surface. That these everyday experiences are essentially holograms :-D! Talk about getting wild, eh :-D?!

The thought that I ultimately came with at the end of my jog was that of maintaining humility in knowledge. For those of us in science, we find that it is very common for arrogance to creep in. When the fact is that there is so much about reality that we don’t know anything about and that our abstractions may never agree with it to full accuracy, ever! When pondered upon deeply this is a very profound and humbling thing to realize.

Even the arrogance in Newton melted away for a moment when he proclaimed:

If I have seen a little further it is by standing on the shoulders of Giants.

Isaac Newton in a letter to rival Robert Hooke

Here’s to Isaac Newton for that spark of humility, even if it was rather fleeting :-). I’m guessing there must have been times when he might have had stray thoughts of cursing at himself for having said that :-)! Oh well, that’s how they all are …


Copyright Firas MR. All Rights Reserved.

“A mote of dust, suspended in a sunbeam.”



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Written by Firas MR

November 16, 2010 at 12:18 am

Let’s Face It, We Are Numskulls At Math!

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Noted mathematician, Timothy Gowers, talks about the importance of math

I’ve often written about Mathematics before Footnotes. As much as math helps us better understand our world (Modern Medicine’s recent strides have a lot to do with applied math for example), it also tells us how severely limited man’s common thinking is.

Humans and yes some animals too, are born with or soon develop an innate ability for understanding numbers. Yet, just like animals, our proficiency with numbers seems to stop short of the stuff that goes beyond our immediate activities of daily living (ADL) and survival. Because we are a higher form of being (or allegedly so, depending on your point of view), our ADLs are a lot more sophisticated than say those of, canaries or hamsters. And consequently you can expect to see a little more refined arithmetic being used by us. But fundamentally, we share this important trait – of being able to work with numbers from an early stage. A man who has a family with kids knows almost by instinct that if he has two kids to look after, that would mean breakfast, lunch and dinner times 2 in terms of putting food on the table. He would have to buy two sets of clothes for his kids. A kid soon learns that he has two parents. And so on. It’s almost natural. And when someone can’t figure out their way doing simple counting or arithmetic, we know that something might be wrong. In Medicine, we have a term for this. It’s called acalculia and often indicates the presence of a neuropsychiatric disorder.

It’s easy for ‘normal’ people to do 2 + 2 in their heads. Two oranges AND two oranges make a TOTAL of four oranges. This basic stuff helps us get by day-to-day. But how many people can wrap their heads around 1 divided by 0? If you went to school, yea sure your teachers must have hammered the answer into you: infinity. But how do you visualize it? Yes, I know it’s possible. But it takes unusual work. I think you can see my point, even with this simple example. We haven’t even begun to speak about probability, wave functions, symmetries, infinite kinds of infinities, multiple-space-dimensions, time’s arrow, quantum mechanics, the Higgs field or any of that stuff yet!

As a species, it is so obvious that we aren’t at all good at math. It’s almost as if we construct our views of the universe through this tunneled vision that helps us in our day-to-day tasks, but fails otherwise.

We tend to think of using math as an ability when really it should be thought of as a sensory organ. Something that is as vital to understanding our surroundings as our eyes, ears, noses, tongues and skins. And despite lacking this sense, we tend to go about living as though we somehow understand everything. That we are aware of what it is to be aware of. This can often lead to trouble down the road. I’ve talked about numerous PhDs having failed at the Monty Hall Paradox before. But a recent talk I watched, touched upon something with serious consequences that meant people being wrongfully convicted because of a stunted interpretation of DNA, fingerprint evidence, etc. by none other than “expert” witnesses. In other words, serious life and death issues. So much for our expertise as a species, eh?!

How the human mind struggles with math!

We recently also learned that the hullabaloo over H1N1 pandemic influenza had a lot do with our naive understanding of math, the pitfalls of corporate-driven public-interest research notwithstanding.

Anyhow, one of my main feelings is that honing one’s math not only helps us survive better, but it can also teach us about our place in the universe. Because we can then begin to fully use it as a sensory organ in its own right. Which is why a lot of pure scientists have argued that doing math for math’s own sake can not only be great fun (if done the right way, of course :-P) but should also be considered necessary. Due to the fact that such research has the potential to reveal entirely new vistas that can enchant us and surprise us at the same time (take Cantor’s work on infinity for example). For in the end, discovery, really, is far more enthralling than invention.

UPDATE 1: Check out the Khan Academy for a virtually A-Z education on math — and all of it for free! This is especially a great resource for those of us who can’t even recall principles of addition, subtraction, etc. let alone calculus or any of the more advanced stuff.

Copyright © Firas MR. All rights reserved.


# Footnotes:

  1. ذرا غور فرمائیے اپنے انسان ہونے کی حیثیت پر
  2. Decision Tree Questions In Genetics And The USMLE
  3. The Story Of Sine
  4. On The Impact Of Thinking Visually
  5. A Brief Tour Of The Field Of Bioinformatics
  6. Know Thy Numbers!

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The Doctor’s Apparent Ineptitude

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ineptitude

via Steve Kay@Flickr (by-nc-nd license)

As a fun project, I’ve decided to frame this post as an abstract.

AIMS/OBJECTIVES:

To elucidate factors influencing perceived incompetence on the part of the doctor by the layman/patient/patient’s caregiver.

MATERIALS & METHODS:

Arm-chair pontification and a little gedankenexperiment based on prior experience with patients as a medical trainee.

RESULTS:

Preliminary analyses indicate widespread suspicions among patients on the ineptitude of doctors no matter what the level of training. This is amply demonstrated in the following figure:

As one can see, perceived ineptitude forms a wide spectrum – from most severe (med student) to least severe (attending). The underlying perceptions of incompetence do not seem to abate at any level however, and eyewitness testimonies include phrases such as ‘all doctors are inept; some more so than others’. At the med student level, exhausted patients find their anxious questions being greeted with a variety of responses ranging from the dumb ‘I don’t know’, to the dumber ‘well, I’m not the attending’, to the dumbest ‘uhh…mmmm..hmmm <eyes glazed over, pupils dilated>’. Escape routes will be meticulously planned in advance both by patients and more importantly by med students to avert catastrophe.

As for more senior medics such as attendings, evasion seems to be just a matter of hiding behind statistics. A gedankenexperiment was conducted to demonstrate this. The settings were two patients A and B, undergoing a certain surgical procedure and their respective caregivers, C-A and C-B.

Patient A

Consent & Pre-op

C-A: (anxious), Hey doc, ya think he’s gonna make it?

Doc: It’s difficult to say and I don’t know that at the moment. There are studies indicating that 95% live and 5% die during the procedure though.

C-A: ohhh kay (slightly confused) (murmuring)…’All this stuff about knowing medicine. What does he know? One simple question and he gives me this? What the heck has this guy spent all these years studying for?!’

Post-op & Recovery

C-A: Ah, I just heard! He made it! Thank you doctor!

Doc: You’re welcome (smug, god-complex)! See, I told ya 95% live. There was no reason for you to worry!

C-A: (sarcastic murmur) ‘Yeah, right. Let him go through the pain of not knowing and he’ll see. Look at him, so full of himself – as if he did something special; luck was on our side anyway. Heights of incompetence!’

Patient B

Consent & Pre-op

C-B: (anxious) Hey doc, ya think he’s gonna make it?

Doc: It’s difficult to say and I don’t know that at the moment. There are studies indicating that 95% live and 5% die during the procedure though.

C-B: ohhh kay (slightly confused) (murmuring)…’All this stuff about knowing medicine. What does he know? One simple question and he gives me this? What the heck has this guy spent all these years studying for?!’

Post-op & Recovery

C-B: (angry, shouting numerous explicatives) What?! He died on the table?!

Doc: Well, I did mention that there was a 5% death rate.

C-B: (angry, shouting numerous explicatives).. You (more explicatives) incompetent quack! (murmuring) “How convenient! A lawsuit should fix him for good!”

The Doctor’s Coping Strategy

Although numerous psychology models can be applied to understand physician behavior, the Freudian model reveals some interesting material. Common defense strategies that help doctors include:

Isolation of affect: eg. Resident tells Fellow, “you know that patient with the …well, she had a massive MI and went into VFib..died despite ACLS..poor soul…so hey, I hear they’re serving pizza today at the conference…(the conference about commercializing healthcare and increasing physician pay-grades for ‘a better  and healthier tomorrow’)”

Intellectualization: eg. Attending tells Fellow, “so you understand why that particular patient bled to death? Yeah it was DIC in the setting of septic shock….plus he had a prior MI with an Ejection Fraction of 33% so there was that component as well..but we couldn’t really figure out why the antibiotics didn’t work as expected…ID gave clearance….(ad infinitum)…so let’s present this at our M&M conference this week..”

Displacement: eg. Caregiver yells at Fellow, “<explicatives>”. Fellow yells at intern, “You knew that this was a case that I had a special interest in and yet you didn’t bother to page me? Unacceptable!…” Intern then yells at med student, “Go <explicatives> disimpact Mr. X’s bowels…if I don’t see that done within the next 15 minutes, you’re in for a class! Go go go…clock’s ticking…tck tck tck!”

We believe there are other coping mechanisms that are important too, but in our observations these appear to be the most common. Of the uncommon ones, we think med students as a group in particular, are the most vulnerable to Regression & Dissociation, duly accounting for confounding factors.

All of these form a systematic ego-syntonic pattern of behavior, but for reasons we are still exploring, is not included in the DSM-IV manual’s section on Personality Disorders.

CONCLUSIONS:

Patients and their caregivers seem to think that ALL doctors are fundamentally inept, period. Ineptitude follows a wide spectrum however – ranging from the bizarre to the mundane. Further studies (including but not limited to arm-chair pontification) need to be carried out to corroborate these startling results and the factors that we have reported. Other studies need to elucidate remedial measures that can be employed to save the doctor-patient relationship.

NOTE: I wrote this piece as a reminder of how the doctor-patient relationship is experienced from the patient’s side. In our business-as-usual frenzy, we as medics often don’t think about these things. And these things often DO matter a LOT to our patients!

Copyright © Firas MR. All rights reserved.

USMLE – Designing The Ultimate Questions

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Question

Shot courtesy crystaljingsr @ Flickr (Creative Commons Attribution, Non-Commercial License)

 

There are strategies that examiners can employ to frame questions that are designed to stump you on an exam such as the USMLE. Many of these strategies are listed out in the Kaplan Qbook and I’m sure this stuff will be familiar to many. My favorite techniques are the ‘multi-step’ and the ‘bait-and-switch’.

The Multi-Step

Drawing on principles of probability theory, examiners will often frame questions that require you to know multiple facts and concepts to get the answer right. As a crude example:

“This inherited disease exclusive to females is associated with acquired microcephaly and the medical management includes __________________.”

Such a question would be re-framed as a clinical scenario (an outpatient visit) with other relevant clinical data such as a pedigree chart. To get the answer right, you would need:

  1. Knowledge of how to interpret pedigree charts and identify that the disease manifests exclusively in females.
  2. Knowledge of Mendelian inheritance patterns of genetic diseases.
  3. Knowledge of conditions that might be associated with acquired microcephaly.
  4. Knowledge of medical management options for such patients.

Now taken individually, each of these steps – 1, 2, 3 and 4 – has a probability of 50% that you could get it right purely by random guessing. Combined together however, which is what is necessary to get the answer, the probability would be 50% * 50% * 50% * 50% = 6.25% [combined probability of independent events]. So now you know why they actually prefer multi-step questions over one or two-liners! :) Notice that this doesn’t necessarily have anything to do with testing your intelligence as some might think. It’s just being able to recollect hard facts and then being able to put them together. They aren’t asking you to prove a math theorem or calculate the trajectory of a space satellite :P !

The Bait-and-Switch

Another strategy is to riddle the question with chock-full of irrelevant data. You could have paragraph after paragraph describing demographic characteristics, anthropometric data, and ‘bait’ data that’s planted there to persuade you to think along certain lines and as you grind yourself to ponder over these things you are suddenly presented with an entirely unrelated sentence at the very end, asking a completely unrelated question! Imagine being presented with the multi-step question above with one added fly in the ointment. As you finally finish the half-page length question, it ends with ‘<insert-similar-disease> is associated with the loss of this enzyme and/or body part: _______________’. Very tricky! Questions like these give flashbacks and dejavu of  days from 2nd year med school, when that patient with a neck lump begins by giving you his demographic and occupational history. As an inexperienced med student you immediately begin thinking: ‘hmmm..okay, could the lump be related to his occupation? …hmm…’. But wait! You haven’t even finished the physical exam yet, let alone the investigations. As medics progress along their careers they tend to phase out this kind of analysis in favor of more refined ‘heuristics’ as Harrison’s puts it. A senior medic will often wait to formulate opinions until the investigations are done and will focus on triaging problems and asking if management options are going to change them. The keyword here is ‘triage’. Just as a patient’s clinical information in a real office visit is filled with much irrelevant data, so too are many USMLE questions. That’s not to say that demographic data, etc. are irrelevant under all conditions. Certainly, an occupational history of being employed at an asbestos factory would be relevant in a case that looks like a respiratory disorder. If the case looks like a respiratory disorder, but the question mentions an occupational history of being employed as an office clerk, then this is less likely to be relevant to the case. Similarly if it’s a case that overwhelmingly looks like an acute abdomen, then a stray symptom of foot pain is less likely to be relevant. Get my point? That is why many recommend reading the last sentence or two of a USMLE question before reading the entire thing. It helps you establish what exactly is the main problem that needs to be addressed.

Hope readers have found the above discussion interesting :). Adios for now!

Copyright © Firas MR. All rights reserved.

Decision Tree Questions In Genetics And The USMLE

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Courtesy cayusa@flickr. (creative commons by-nc license)

Courtesy cayusa@flickr. (creative commons by-nc license)

Just a quick thought. It just occurred to me that some of the questions on the USMLE involving pedigree analysis in genetics, are actually typical decision tree questions. The probability that a certain individual, A, has a given disease (eg: Huntington’s disease) purely by random chance is simply the disease’s prevalence in the general population. But what if you considered the following questions:

  • How much genetic code do A and B share if they are third cousins?
  • If you suddenly knew that B has Huntington’s disease, what is the new probability for A?
  • What is the disease probability for A‘s children, given how much genetic code they share with B?

When I’d initially written about decision trees, it did not at all occur to me at the time how this stuff was so familiar to me already!

Apply a little Bayesian strategy to these questions and your mind is suddenly filled with all kinds of probability questions ripe for decision tree analysis:

  • If the genetic test I utilize to detect Huntington’s disease has a false-positive rate x and a false-negative rate y, now what is the probability for A?
  • If the pre-test likelihood is m and the post-test likelihood is n, now what is the probability for A?

I find it truly amazing how so many geneticists and genetic counselors accomplish such complex calculations using decision trees without even realizing it! Don’t you :-) ?

Copyright © Firas MR. All rights reserved.

Elegance In Inelegance

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Courtesy Lydia Elle @ Flickr (by-nc license)

Courtesy Lydia Elle @ Flickr (by-nc license)

I just finished a great lecture series on the history of mathematics by Dr. David Bressoud recently1. Remember how I once spoke about elegance in inelegance? How some people have argued (eg: Lee Smolin) that the universe just might be complex by nature? How mankind might just be wrong about looking for simple and thus elegant solutions to explain physical phenomena?

Well, I was pretty intrigued by some of the stuff I learned about Henri Poincare‘s work in this regard. Poincare is famous for a number of things, his Poincare conjecture being the most obvious of them. A Russian math guru, Grigori Perelman, apparently proved this conjecture some years back and among other peculiar things, not only declined the Fields medal but also a million dollar prize for solving one of the toughest math problems ever known.

But I was particularly piqued by how Poincare was fascinated by this idea of finding elegance and hidden patterns even where one might expect junk. Here are what might be interesting questions as crude examples:

Take a random set of 100 beads. Throw these beads on the floor. They scatter randomly. How many throws would be needed to find at least three beads on the floor that yield an equilateral triangle when they are connected? How many throws would you need to find a cluster of beads that is of a certain shape or size?

That there is some sense of order even in randomness and chaos, is truly an enchanting concept.

Have any thoughts of your own? Do send in your feedback :-)!

1. Queen Of The Sciences (Lectures by David Bressoud)

Copyright © Firas MR. All rights reserved.

Written by Firas MR

August 31, 2009 at 11:49 pm

The Story Of Sine

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zaveqna@flickr (by-nc-sa license)

zaveqna@flickr (by-nc-sa license)

I’ve been studying mathematics lately and really enjoying it. Here’s an interesting story about the history of the trigonometric function, ‘sine‘.

Early in the 1st millenium A.D., a new way of thinking about chords was coming about. The chord is defined as the straight line that joins two points on the circumference of a circle. The ancient Greeks had developed trigonometric functions to calculate the length of arbitrary chords. But several centuries later, by the early 1st millenium A.D., mathematicians in India began to think about calculating and working with half-chord lengths instead. For this, they developed the familiar ‘sine’ and ‘cosine‘ functions that we still use to this day. The earliest accounts of the use of the half-chord in Indian texts, is from the Surya Siddhanta (c. 300 – 400 AD), written in Sanskrit. The sound of the Sanskrit word used for ‘half-chord’ was ardha-jya [ardha = half, jya = chord]. Perhaps they found this word too long and eventually it was shortened to jya or jiva for all practical purposes.

By roughly the end of the 1st millenium A.D., the vanguard of scientific growth was now in the hands of the Arab world. In translating the works from Sanskrit into Arabic, scholars in the Arab world transliterated and pronounced jiva as jiba [جب]. The sound ‘jiba‘ is recorded in Arabic as two consonants j [ج] and b [ب] with no vowels explicitly written between them. The vowel sounds are merely implied.

Several centuries later, after the decline of scientific growth in the Arab world, came the Europeans. When they in turn came upon the Arabic word for jiva and tried to translate it, they of course ended up with a word, ‘jb‘ [pronounced as 'jay bee']. Apparently, they were oblivious of the implied vowel sounds. Things were dandy for the Arab scientists, but the Europeans couldn’t make any sense of the sound ‘jay bee‘ because such a sound doesn’t exist in any of the words in the Arabic language. They found that the closest sound to ‘jay bee‘, was the sound ‘jaib‘ or ‘ja-eeb‘, in the Arabic word for the mammary gland! And so the Europeans assumed that the half-chord was to be referred to with a Latin word that meant mamma, mammary gland or any of its other synonyms. Perhaps out of modesty, it was ultimately instead decided that the word used for the fold of a cloth utilized to cover a mamma would be appropriate to refer to a half-chord. This word was ‘sinus’. And from this Latin word ‘sinus‘, ultimately came the English word ‘sine‘ that is in use today!

Remarkable, isn’t it?

Feel free to send in your feedback, corrections and comments :-) .

References:

  1. Queen Of The Sciences (Lectures by David Bressoud)

Copyright © Firas MR. All rights reserved.

Written by Firas MR

August 27, 2009 at 5:35 pm

Posted in Math, Science

Tagged with , , ,

On The Impact Of Thinking Visually

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Romanesco broccoli - One of many examples of fractals in nature. (Wikipedia)

Romanesco broccoli - One of many examples of fractals in nature. (Wikipedia)

What do Mandelbrot and Einstein have in common?

They were/are both math aficionados. But more importantly, they both laid down the foundations for thinking about abstract natural phenomena in a geometrical way. The impact was reverberating.

Before Einstein came along, people had no real sense of gravity at all. Yea sure, there was Newton’s universal law of gravitation. But no one really could make any sense whatsoever of how exactly gravity might operate. Was it a wave? If so, at what speed could it act? Was there something particulate about it? Gravity was so mystical. And as always, so have been the concepts of time and space. Einstein’s greatest achievement in my view is that not only was he able to lay out the underpinnings of such phenomena in the form of a couple of abstract equations, but perhaps more importantly, that he devised a method to think about them visually. In developing his theories of special and general relativity, Einstein proposed the idea of the space-time fabric. It has a 3-D structure, yet represents four dimensions – 3 in space and 1 in time. Gravity would result from distortions in this fabric. The speed with which gravity could influence an object would depend on how fast these distortions could travel. And this central notion of ‘distortions in a fabric’ would also influence our understanding of the more difficult to grasp concepts of time and space. Time and space could mean different things to different observers depending on how this fabric was warped or sliced.

Mandelbrot achieved the same thing with his theory of fractals. How can complex natural structures and phenomena be represented mathematically? How to mathematically model a plant, the form of a human or a mountain range? In spite of how incredibly difficult it all sounds, these complex shapes could all be simplified into repeating units of tiny yet geometrically simple components – fractals. Mandelbrot went on to write his epic, “The Fractal Geometry Of Nature” and there was no turning back. Suddenly so many of nature’s workings could now be analyzed mathematically. An immensely significant step for mankind indeed. What I find absolutely fascinating about fractals, is the discovery that many intangible natural phenomena also contain a fractal component. Dr. Ary Goldberger and his team of researchers at Harvard Medical School have been working on applying fractal theory to medicine and biology. For those of you who might not be familiar with Dr. Goldberger, the name might ring a bell if you’ve read his books on electrocardiography. For Dr. Goldberger, interest in electrocardiography runs in the family, his father having invented the augmented limb leads back in the day. Among some of the things I learned about his work on electrocardiography, is that his team has shown that there is a fractal nature to ECG waveforms! This isn’t something like representing the heart itself in fractal form. It’s the activities of the heart that we are talking about here. Something really quite abstract. By looking at these fractal patterns, one could potentially detect pathology at a much earlier stage. Fractal patterns and their aberrations could help detect diseases in ways that no one had ever imagined! If you want to dig what’s cool, check out what’s been going on in the world of fractals in medicine – from human vasculature, to the brain and beyond. A quick PubMed query would lead you to a lot of riveting literature on the topic. Don’t forget to also take a look at the excellent documentary on fractal theory from PBS NOVA, “Hunting The Hidden Dimensions“.

Copyright © Firas MR. All rights reserved.

Readability grades for this post:

Flesch reading ease score: 61.1
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Flesch-Kincaid grade level: 8.2
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Gunning fog index: 11.6
SMOG index: 11

Written by Firas MR

August 18, 2009 at 11:48 pm

Why Equivalence Studies Are So Fascinating

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Bronze balance pans and lead weights from the Vapheio tholos tomb, circa 15th century BC. National Museum, Athens. Shot courtesy dandiffendale@Flickr. by-nc-ca license.

Bronze balance pans and lead weights from the Vapheio tholos tomb, circa 15th century BC. National Museum, Athens. Shot courtesy dandiffendale@Flickr. by-nc-sa license.

Objectives and talking points:

  • To recap basic concepts of hypothesis testing in scientific experiments. Readers should read-up on hypothesis testing in reference works.
  • To contrast drug vs. placebo and drug vs. standard drug study designs.
  • To contrast non-equivalence and equivalence studies.
  • To understand implications of these study designs, in terms of interpreting study results.

——————————————————————————————————–

Howdy readers! Today I’m going to share with you some very interesting concepts from a fabulous book that I finished recently – “Designing Clinical Research – An Epidemiologic Approach” by Stephen Hulley et al. The book speaks fairly early on, on what are called “equivalence studies”. Equivalence studies are truly fascinating. Let’s see how.

When a new drug is tested for efficacy, there are multiple ways for us to do so.

A Non-equivalence Study Of Drug vs. Placebo

A drug can be compared to something that doesn’t have any treatment effect whatsoever – a ‘placebo’. Examples of placebos include sugar tablets, distilled water, inert substances, etc. Because pharmaceutical companies try hard to make drugs that have a treatment effect and that are thus different from placebos, the objective of such a comparison is to answer the following question:

Is the new drug any different from the placebo?

Note the emphasis on ‘any different’. As is usually the case, a study of this kind is designed to test for differences between drug and placebo effects in both directions1. That is:

Is the new drug better than the placebo?

OR

Is the new drug worse than the placebo?

The boolean operator ‘OR’, is key here.

Since we can not conduct such an experiment on all people in the target ‘population’ (eg. all people with diabetes from the whole country), we conduct it on a random and representative ‘sample’ of this population (eg. randomly selected diabetes patients from the whole country). Because of this, we can not directly extrapolate our findings to the target population without doing some fancy roundabout thinking and a lot of voodoo first – a.k.a. ‘hypothesis testing’. Hypothesis testing is crucial to take in to account random chance (error) effects that might have crept in to the experiment.

In this experiment:

  • The null hypothesis is that the drug and the placebo DO NOT differ in the real world2.
  • The alternative hypothesis is that the drug and the placebo DO differ in the real world.

So off we go, with our experiment with an understanding that our results might be influenced by random chance (error) effects. Say that, before we start, we take the following error rates to be acceptable:

  1. Even if the null hypothesis is true in the real world, we would find that the drug and the placebo DO NOT differ only 95% of the time, purely by random chance. [Although this rate doesn't have a name, it is equal to (1 - Type 1 error)].
  2. Even if the null hypothesis is true in the real world, we would find that the drug and the placebo DO differ 5% of the time, purely by random chance. [This rate is also called our Type 1 error, or critical level of significance, or critical α level, or critical 'p' value].
  3. Even if the alternative hypothesis is true in the real world, we would find that the drug and the placebo DO differ only 80% of the time, purely by random chance. [This rate is also called the 'Power' of the experiment. It is equal to (1 - Type 2 error)].
  4. Even if the alternative hypothesis is true in the real world, we would find that the drug and the placebo DO NOT differ 20% of the time, purely by random chance. [This rate is also called our Type 2 error].

The strategy of the experiment is this:

If we are able to accept these error rates and show in our experiment that the null hypothesis is false (that is ‘reject‘ it), the only other hypothesis left on the table is the alternative hypothesis. This has then, GOT to be true and we thus ‘accept’ the alternative hypothesis.

Q: With what degree of uncertainty?

A: With the uncertainty that we might arrive at such a conclusion 5% of the time, even if the null hypothesis is true in the real world.

Q: In English please!

A: With the uncertainty that we might arrive at a conclusion that the drug DOES differ from the placebo 5% of the time, even if the drug DOES NOT differ from the placebo in the real world.

Our next question would be:

Q: How do we reject the null hypothesis?

A: We proceed by initially assuming that the null hypothesis is true in the real world (i.e. Drug effect DOES NOT differ from Placebo effect in the real world). We then use a ‘test of statistical significance‘ to calculate the probability of observing a difference in treatment effect in the real world, as large or larger than that actually observed in the experiment.  If this probability is <5%, we reject the null hypothesis. We do this with the belief that such a conclusion is within our pre-selected margin of error. Our pre-selected margin of error, as mentioned previously, is that we would be wrong about rejecting the null hypothesis 5% of the time (our Type 1 error rate)3.

If we fail to show that this calculated probability is <5%, we ‘fail to reject‘ the null hypothesis and conclude that a difference in effect has not been proven4.

A lot of scientific literature out there is riddled with drug vs. placebo studies. This kind of thing is good if we do not already have an effective drug for our needs. Usually though, we already have a standard drug that we know works well. It is of more interest to see how a new drug compares to our standard drug.

A Non-equivalence Study Of Drug vs. Standard Drug

These studies are conceptually the same as drug vs. placebo studies and the same reasoning for inference is applied. These studies ask the following question:

Is the new drug any different than the standard drug?

Note the emphasis on ‘any different’. As is often the case, a study of this kind is designed to test the difference between the two drugs in both directions1. That is:

Is the new drug better than the standard drug?

OR

Is the new drug worse than the standard drug??

Again, the boolean operator ‘OR’, is key here.

In this kind of experiment:

  • The null hypothesis is that the new drug and the standard drug DO NOT differ in the real world2.
  • The alternative hypothesis is that the new drug and the standard drug DO differ in the real world.

Exactly like we discussed before, we initially assume that the null hypothesis is true in the real world (i.e. the new drug’s effect DOES NOT differ from the standard drug’s effect in the real world). We then use a ‘test of statistical significance‘ to calculate the probability of observing a difference in treatment effect in the real world, as large or larger than that actually observed in the experiment.  If this probability is <5%, we reject the null hypothesis – with the belief that such a conclusion is within our pre-selected margin of error. Just to repeat ourselves here, our pre-selected margin of error, is that we would be wrong about rejecting the null hypothesis 5% of the time (our Type 1 error rate)3.

If we fail to show that this calculated probability is <5%, we ‘fail to reject’ the null hypothesis and conclude that a difference in effect has not been proven4.

An Equivalence Study Of Drug vs. Standard Drug

Sometimes all you want is a drug that is as good as the standard drug. This can be for various reasons – the standard drug is just too expensive, just too difficult to manufacture, just too difficult to administer, … and so on. Whereas the new drug might not have these undesirable qualities yet retain the same treatment effect.

In an equivalence study, the incentive is to prove that the two drugs are the same. Like we did before, let’s explicitly formulate our two hypotheses:

  • The null hypothesis is that the new drug and the standard drug DO NOT differ in the real world2.
  • The alternative hypothesis is that the new drug and the standard drug DO differ in the real world.

We are mainly interested in proving the null hypothesis. Since this can’t be done4, we’ll be content with ‘failing to reject’ the null hypothesis. Our strategy is to design a study powerful enough to detect a difference close to 0 and then ‘fail to reject’ the null hypothesis. In doing so, although we can’t ‘prove’ for sure that the null hypothesis is true, we can nevertheless be more comfortable saying that it in fact is true.

In order to detect a difference close to 0, we have to increase the Power of the study from the usual 80% to something like 95% or higher. We wan’t to maximize power to detect the smallest difference possible. Usually though, it’s enough if we are able to detect the the largest difference that doesn’t have clinical meaning (eg: a difference of 4mm on a BP measurement). This way we can compromise a little on Power and choose a less extreme figure, say 88% or something.

And then just as in our previous examples, we proceed with the assumption that the null hypothesis is true in the real world. We then use a ‘test of statistical significance‘ to calculate the probability of observing a difference in treatment effect in the real world, as large or larger than that actually observed in the experiment.  If this probability is <5%, we reject the null hypothesis – with the belief that such a conclusion is within our pre-selected margin of error. And to repeat ourselves yet again (boy, do we like doing this :-P ), our pre-selected margin of error is that we would be wrong about rejecting the null hypothesis 5% of the time (our Type 1 error rate)3.

If we fail to show that this calculated probability is <5%, we ‘fail to reject‘ the null hypothesis and conclude that although a difference in effect has not been proven, we can be reasonably comfortable saying that there is in fact no difference in effect.

So Where Are The Gotchas?

If your study isn’t designed or conducted properly (eg: without enough power, inadequate  sample size, improper randomization, loss of subjects to followup, inaccurate measurements, etc.)  you might end up ‘failing to reject’ the null hypothesis whereas if you had taken the necessary precautions, this might not have happened and you would have come to the opposite conclusion. Purely because of random chance (error) effects. Such improper study designs usually dampen any obvious differences in treatment effect in the experiment.

In a non-equivalence study, researchers, whose incentive it is to reject the null hypothesis, are thus forced to make sure that their designs are rigorous.

In an equivalence study, this isn’t the case. Since researchers are motivated to ‘fail to reject’ the null hypothesis from the get go, it becomes an easy trap to conduct a study with all kinds of design flaws and very conveniently come to the conclusion that one has ‘failed to reject’ the null hypothesis!

Hence, it is extremely important, more so in equivalence studies than in non-equivalence studies, to have a critical and alert mind during all phases of the experiment. Interpreting an equivalence study published in a journal is hard, because one needs to know the very guts of everything the research team did!

Even though we have discussed these concepts with drugs as an example, you could apply the same reasoning to many other forms of treatment interventions.

Hope you’ve found this post interesting :-) . Do send in your suggestions, corrections and comments!

Adios for now!

Copyright © Firas MR. All rights reserved.

Readability grades for this post:

Flesch reading ease score: 71.4
Automated readability index: 8.1
Flesch-Kincaid grade level: 7.4
Coleman-Liau index: 9
Gunning fog index: 11.8
SMOG index: 11

1. An alternative hypothesis for such a study is called a ‘two-tailed alternative hypothesis‘. A study that tests for differences in only one direction has an alternative hypothesis that is called a ‘one-tailed alternative hypothesis‘.
2. This situation is a good example of a ‘null’ hypothesis also being a ‘nil’ hypothesis. A null hypothesis is usually a nil hypothesis, but it’s important to realize that this isn’t always the case.
4. Note that we never use the term, ‘accept the null hypothesis’.

Does Changing Your Anwer In The Exam Help?

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monty hall paradox

The Monty Hall Paradox

One of the 3 doors hides a car. The other two hide a goat each. In search of a new car, the player picks a door, say 1. The game host then opens one of the other doors, say 3, to reveal a goat and offers to let the player pick door 2 instead of door 1. Is there an advantage if the the player decides to switch? (Courtesy: Wikipedia)

Hola amigos! Yes, I’m back! It’s been eons and I’m sure many of you may have been wondering why I was MIA. Let’s just say it was academia as usual.

This post is unique as it’s probably the first where I’ve actually learned something from contributors and feedback. A very critical audience and pure awesome discussion. The main thrust was going to be an analysis of the question, “If you had to pick an answer in an MCQ randomly, does changing your answer alter the probabilities to success?” and it was my hope to use decision trees to attack the question. I first learned about decision trees and decision analysis in Dr. Harvey Motulsky’s great book, “Intuitive Biostatistics“. I do highly recommend his book. As I pondered over the question, I drew a decision tree that I extrapolated from his book. Thanks to initial feedback from BrownSandokan (my venerable computer scientist friend from yore :P) and Dr. Motulsky himself, who was so kind as to write back to just a random reader, it turned out that my diagram was wrong and so was the original analysis. The problem with the original tree (that I’m going to maintain for other readers to see and reflect on here) was that the tree in the book is specifically for a math (or rather logic) problem called the Monty Hall Paradox. You can read more about it here. As you can see, the Monty Hall Paradox is a special kind of unequal conditional probability problem, in which knowing something for sure, influences the probabilities of your guesstimates. It’s a very interesting problem, and has bewildered thousands of people, me included. When it was originally circulated in a popular magazine,  “nearly 1000 PhDs” (cf. Wikipedia) wrote back to say that the solution put forth was wrong, prompting numerous psychoanalytical studies to understand human behavior. A decision tree for such a problem is conceptually different from a decision tree for our question and so my original analysis was incorrect.

So what the heck are decision trees anyway? They are basically conceptual tools that help you make the right decisions given a couple of known probabilities. You draw a line to represent a decision, and explicitly label it with a corresponding probability. To find the final probability for a number of decisions (or lines) in sequence, you multiply or add their individual probabilities. It takes skill and a critical mind to build a correct tree, as I learned. But once you have a tree in front of you, its easier to see the whole picture.

Let’s just ignore decision trees completely for the moment and think in the usual sense. How good an idea is it to change an answer on an MCQ exam such as the USMLE? The Kaplan lecture notes will tell you that your chances of being correct are better off if you don’t. Let’s analyze this. If every question has 1 correct option and 4 incorrect options (the total number of options being 5), then any single try on a random choice gives you a probability of 20% for the correct choice and 80% for the incorrect choice. The odds are higher that on any given attempt, you’ll get the answer wrong. If your choice was correct the first time, it still doesn’t change these basic odds. You are still likely to pick the incorrect choice 80% of the time. Borrowing from the concept of “regression towards the mean” (repeated measurements of something, yield values closer to said thing’s mean), we can apply the same reasoning to this problem. Since the outcomes in question are categorical (binomial to be exact), the measure of central tendency used is the Mode (defined as the most commonly or frequently occurring thing in a series). In a categorical series – cat, dog, dog, dog, cat – the mode is ‘dog’. Since the Mode in this case happens to be the category “incorrect”, if you pick a random answer and repeat this multiple times, you are more likely to pick an incorrect answer! See, it all make sense :) ! It’s not voodoo after all :D !

Coming back to decision analysis, just as there’s a way to prove the solution to the Monty Hall Paradox using decision trees, there’s also a way to prove our point on the MCQ problem using decision trees. While I study to polish my understanding of decision trees, building them for either of these problems will be a work in progress. And when I’ve figured it all out, I’ll put them up here. A decision tree for the Monty Hall Paradox can be accessed here.

To end this post, I’m going to complicate our main question a little bit and leave it out in the void. What if on your initial attempt you have no idea which of the answers is correct or incorrect but on your second attempt, your mind suddenly focuses on a structure flaw in one or more of the options? Assuming that an option with a structure flaw can’t be correct, wouldn’t this be akin to Monty showing the goat? One possible structure flaw, could be an option that doesn’t make grammatical sense when combined with the stem of the question. Does that mean you should switch? Leave your comments below!

Hope you’ve found this post interesting. Adios for now!

Copyright © Firas MR. All rights reserved.

Readability grades for this post:

Flesch reading ease score:  72.4
Automated readability index: 7.8
Flesch-Kincaid grade level: 7.3
Coleman-Liau index: 8.5
Gunning fog index: 11.4
SMOG index: 10.7

Readings:

Intuitive Biostatistics, by Harvey Motulsky

The Monty Hall Problem: The Remarkable Story Of Math’s Most Contentious Brain Teaser, by Jason Rosenhouse

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