Effect Measure

[A series of posts explaining a paper on the mathematical modeling of the spread of antiviral resistance. Links to other posts in the series by clicking tags, "Math model series" or "Antiviral model series" under Categories, left sidebar. Preliminary post here. Table of contents at end of this post.]

The Modeling Series (click Math Modeling Series under Categories in the left sidebar) is a moving target for me. Even though the first draft of the post you read each day was written at least three weeks earlier, each has also been freshly worked on as I go back and tinker and adjust and try to improve my explanations or find holes in the exposition. By this time I already know my audience is shrinking as we go deeper into the details of a specialized scientific paper. But it has become a personal challenge to see if I can explain even the details to a readership I know is interested in the subject, motivated and intelligent (as well as a little crazy). If I can’t do that, I’m not much of a teacher. So I feel compelled to pause for a moment to discuss the challenges for non-specialists in understanding a mathematical modeling paper. This pertains not just to non-scientists, but to many talented and brilliant scientists who shy away from the mathematics and resignedly “take it on faith.”

“Mathematics” isn’t hard. We all use it constantly when we file our taxes or count our change at the store or read public opinion polls. But some mathematics is hard. Very hard. This isn’t surprising. Making “music” isn’t hard, either. Many of us do it all the time, even if it’s only to enjoy it or sing along in the car. But making music at the level of a professional musician is hard. Very hard. I’ve played a number of musical instruments, strictly amateur and very badly. A couple of them were fretted string instruments (5-string banjo, guitar). I love violin music and in my middle age decided I’d take it up. How hard could it be? After all, I already know how to play some string instruments. For an adult beginner who never touched a fiddle in his life, the answer to that question turned out to be that it was unbelievably hard. It felt like an exercise in “how many things can I think about doing all at once.” Although I eventually was able to play full classical repertoire in the second violin section of our community symphony orchestra, I was still so bad I couldn’t stand to hear myself play and finally gave it up. Meanwhile, ten year olds were playing violin after two years ten times better than I did after five. Of course their mothers made them practice. That helps.

What’s this got to do with reading a mathematical modeling paper? In the next post of the series (part IV) we will start the process of “thinking mathematically.” There will be no formulas or equations. Still, it will be very hard for some of you, and you will be tempted to give up. I hope you won’t, so I want to try to guess at the source of some of your difficulties. It isn’t because you are mathematically inept or not “mathematically talented” or smart. On the contrary, I would suspect for some of you it is because you are very smart and insist on asking certain questions that get in your way. Like, “Why?” You want to know how it “really works” or you see all sorts of problems with what’s been said. You will read the mathematical reasoning and feel you “don’t get it.” You must be missing something. You probably are, but it isn’t what you think.

The best explanation I know has been attributed to one of the greatest mathematical physicists of the 20th century, John von Neumann:

“In mathematics you don’t understand things. You just get used to them.” (von Neumann, attributed)

Like a ten year old playing a violin. After a while, she stops thinking about why she used fourth position instead of third to play that note. She just did it. She got used to the fact that fourth position generally works out better in that passage because of the notes that came before and afterward, or because she or her teacher preferred it, or her fingers were long or short or fat. Those are “reasons” but they aren’t forced by a mysterious logic you don’t understand. A lot of the applied mathematics is like that (in pure mathematics you are forced by logic, but there is no claim it relates to the real world question of “why?”). After a while, you get so used to it you think you understand it. But if von Neumann can admit he never really understood it but just got used to it, then the rest of us can also feel somewhat relieved that maybe our discomfort at not understanding Hilbert space the way he did is because we haven’t spent as much time living there as von Neumann did.

Don’t get me wrong. I’m not saying we are all potential von Neumann’s any more than the average ten year old is a potential Heifitz on the violin. But almost all of us are capable of appreciating Heifitz or Doc Watson or Earl Scruggs or any other supremely gifted musician we couldn’t hope to become. We can even make a claim to “understanding” them. In the same way, you can appreciate what mathematical modelers are doing in the Lipsitch et al. paper we are discussing. What will help is to suspend your discomfort at not “understanding” everything long enough to start the process of getting used to it.

On the other hand, there are reasons applied mathematicians do things. For example, I will try to explain the reasons that led them to guess that the Law of Mass Action might be a useful tool in epidemiology (part IV). It was speculation at first but experience has shown that despite many reasons you might think it wouldn’t work, for many purposes it works extremely well. Once they saw that, they got used to it. Fast.

The bottom line here is that I hope you give yourself a chance to get used to it, too. Keep reading. Modeling research has been maturing for a century with fits and starts and blind pathways, as in any science, but it has made great progress, with the help of wonderful new mathematical research and powerful new computational tools. Molecular biology is a younger science and in ten years will probably look very different than it does now. Yet we wouldn’t ignore today’s advances because we know tomorrow we will be smarter.

Enough pep talk. In part IV we start uncovering the mathematical machinery in Lipsitch et al.

Table of contents for posts in the series:

What is a model?

A modeling paper

The Introduction. What’s the paper about?

The essential assumption.

Sidebar: thinking mathematically

The model variables

The rule book

More on the rule book

Finishing the rule book

The rule book in equation form

Ready to run the model

Effects of treatment and prophylaxis on resistance

Effects of Tamiflu use and non drug interventions

Effects of fitness costs of resistance

Discussion

A few words about model assumptions

Conclusion and take home messages

Comments

  1. #1 Melanie
    March 24, 2007

    That is a brilliant analogy. FWIW, the cognitive psychologists I know say that math ability is related to music and languages in the brain. I must simply be somewhat deficient in the area in which I made my living for so many decades.

    In cognitive psych, this area is called “symbolic systems,” and the possessors of skills here are usually good at cards and chess, as well, along with logic and rhetoric. Well, I’m good at the ponies and puppies, but DemFromCT regulary critiques my logic and rhet skills. Being “good” at the ponies and the puppies “may” have something to do with being “good” at philosophy and theology, but there aren’t any studies, other than some work being done by a neurophysiologist here in town who won’t publish. Damn.

    You just started too late for a stringed instrument. If you don’t pick one up before the age of 12, you can never warp your skeletal system around one to become a successful performer. This has nothing to do with your brain and everything to do with when bones finally fuse in the adult skeleton. Stringed instruments require you to become a slave to them. Sorry.

  2. #2 sharpstick
    March 24, 2007

    I’m one that asks “why” or how it “really works”. At my age though I don’t really feel there is any hope for me when it comes to advanced math, physics and chemistry. I just accept that it is beyond my capability and trust that individuals like you will do that heavy lifting. I take it on faith.

    The music analogy fascinates me. My wife is a pianist trained in the classics. I can’t play a note. In discussions over the years I contend that some music stirs something deeply instinctual and its ability to do so is related to the skilled musician’s style and interpretation. She places the higher value on technical ability. I argue that we sometimes hear piano pieces played with a very high level of skill that otherwise leave one somehow feeling shortchanged. There is, I believe, an explanation for this that is probably mathematical. The Golden Ratio maybe?

  3. #3 Tim
    March 24, 2007

    A slight disagreement here. Calculus was developed because differential equations described the way many things worked to Newton. Ultimately the why is that this stuff is based on actual reality.

  4. #4 Ethan Romero
    March 24, 2007

    I would even take your claim that “‘mathematics’ isn’t hard” further by claiming that mathematics is the *only* thing that isn’t hard. Of course, by hard I mean something other than difficult. I mean that mathematics and its similarly founded siblings (logic, mathematical statistics, formal systems, etc…) are the only area of academics that we can assertively claim an objective (i.e. not probabilistic in the Bayesian sense) reality. We may disagree on a pure mathematical result, and we may even be currently unable to resolve our disagreement, but the problem has a definite and knowable answer (even if that answer is that there exists no answer). The difficulty in modeling is not in the mathematics, it is in the metaphor. We take models on faith not because the mathematics with which they are expressed are hard; but, rather, because the internal calculus required to evaluate the model as being representative of reality is hard if not fully impossible. Well, at least, that’s my opinion.

    The relationship between math and metaphor in modeling is further complicated by the fact that in the simplest models�that is, the ones where we have abstracted the farthest from reality�have the easiest mathematical structures. As the math becomes easier, the models become more abstract, less like reality, and more difficult to think of as a valid model of reality. Mathematically complex modes require more investment to understand, but once there, they are much easier to ‘accept on faith’.

  5. #5 revere
    March 24, 2007

    Tim: I am not sure with whom you are disagreeing, me or von Neumann and about what. Newton’s notion of mathematics was quite different than ours in many ways, but the more modern notion is probably best described by Weyl’s description of the unreasonable effectiveness of mathematics. There is o reason why it should work, but it does.

    Ethan: You have towed us out into some pretty deep water, philosophically. You sound like a thoroghgoing Platonist. The question of when a model simple and when it is complex is also not quite so clear to me. Many very simple systems produce complex and difficult to understand consequences (e.g., even the simple problem of mapping the unit interval to itself). On the other hand, you can “reporduce” all sorts of real phenomena with a mathematical model if you have enough adjustable parameters. The complex becomes simple. Oh, well. We won’t face those problems with this model. It behaves fairly nicely.

  6. #6 Melanie
    March 24, 2007

    Since I rarely run into problems that “behave nicely,” this should be fun.

  7. #7 Greg
    March 25, 2007

    This is unfortunate, Revere. I am inundated. I intended to save up these posts and study them leisurely, and almost certainly without comment, over the weeks ahead.

    However, I realize, from what you wrote above and in ‘post zero’, that this is precisely when you need comments to keep up your spirits and to inform you that we are indeed finding your labours useful. And precisely, also, when we, finding them useful, are busy studying them, and refraining from wasting your time with irrelevant comments.

    I bet those who find themselves puzzled by any point are rereading, and nagging innocent bystanders, rather than intrude and magnify your labours.

    .

    In this post you conflate two, maybe three, separable problems.

    One is pedagogical : We tell people that mathematics is too difficult for them. And people obey us and fail,,, by refusing to do it, or by doing it wrong.

    One is logical : We don’t distinguish mathematics as an activity (I was taught, a game) from mathematics as a language. A game which sometimes we can use to predict reality. A very elegant, spare yet comprehensive, descriptive language. I suppose, using the same marks for symbols of our language and for tokens of our game, we are guilty as much of forgetting ourselves as of misleading the laity.

    The third is pathological : We obfuscate. We present scientific theories, whether expressed in English language or in Mathematical language, as Laws to be obeyed by reality, instead of as tentative descriptions of reality, which serve our purposes for so long as they describe accurately enough the bits of reality which happen to interest us. Sure, sure, in practice, we sometimes present our theories as contingent; but separately we train people that science is immutable law.

    Only when confronted, do we agree with Popper, that our theories are useful only until contrary evidence shows them inaccurate or false.

    Computer models bring several of our misconceptions together. Then we add another, that computer arithmetic is the same as mathematics.

    A computer model is no more than a theory expressed in peculiar form. Our perceptions are dazzled by the burning bushes, mathematics, computers, natural law, model, and so forth.

    Then, the modellers misrepresent their theories. They downplay the tinkering and adjustments they sweated over, until their theories could successfully ‘predict’ past observations. They emphasize the predictions of their theories,, throwing sheets over the spit and string,, energetically diverting our attention from the fact that every useful theory is a model which makes predictions. That we do not much value theories which make no testable predictions. And that we use those predictions to test the theory, not to test the reality.

    No (successful) modeller dumps a random collection of empirical measurements and of abstract equations into a machine (or a room full of women with pencils and paper, as we did six/seven decades ago). The empiricals and abstractions are chosen carefully and tested repeatedly, severally and jointly.

    If we do not understand the model, maybe the modeller has not explained it properly (as Revere’s embarquing on this series of posts suggests) (for us, as Revere took care to explain), maybe we are bedazzled and forget that it is merely a scientific theory like any other, maybe we are obeying the order to flunk math.

  8. #8 revere
    March 25, 2007

    Greg: First, thanks for the encouragement. You are right that I feel the dread that by the time I reach the end, I’ll be the only one reading this. That’s why I started out by saying I thought it was a failed experiment, but I am comitted to running it out to the end. I’m learning by doing it and I have to teach modeling again in the fall, so I get a lot out of it, anyway. The other questions you raise are more complicated and raise may issues. I am not a Popperian (I am more a spontaneous Bayesian), so a complete reply would take us in a different direction, but I enjoyed reading your reaction to the posts. That was substantial reward in itself. Thanks. You are right that most modelers reflect deeply on what they are doing, even if they pull the ladder up after themselves. This is a common pedagogical problem with textbooks, too. A text on physics makes it all sound logical and cut and dried, so students worry they are stupid because they are having a hard time wrapping their young brains around questions that took geniuses hundreds of years to figure out. But I don’t blame the textbook writers. There is room for all sorts of books and presentations and modelers usually write for other modelers. When they don’t, they are forced to give bottom line implications and inferences or else write a text on modeling, which the scientific literature doesn’t allow. Hence these posts.

  9. #9 Greg
    March 25, 2007

    My other questions don’t require answers, unless egregiously wrong. I hope that nobody thinks I am correct (except Kruger); rather, that they think and inform their own corrects.

    My paragraph “the modellers misrepresent their theories” is more harsh than I intended. You are right, predicaments of the trade cause unfortunate facts and appearances. Although, they interact strongly with our general injunctions to obey experts and not to examine them closely. Perhaps open publishing will help. And your posts.

    Textbooks, and lectures too. I chanced to read the other day, that Foucault didn’t like his large public lectures (although he read more than his contract required); he preferred small seminars where people could question and argue with him. Doubtless most teachers prefer the same.

  10. #10 Ethan Romero
    March 25, 2007

    Revere: Yea, I let my mind wander out into the philosophical depths as a break from working on my thesis, part of which is on relaxing the random mixing assumption in models of HIV transmission. So, I really appreciate the posts and I want to offer you encouragement to ‘[run] it out to the end’. Anything that helps people to think critically about models is “a good thing”.

    Greg: As you noted, I don’t necessarily agree with all of your points. But, I greatly appreciate the skepticism that you bring to the table. I think that part of the responsibility of being a rational thinker is to maintain skepticism towards models, ideas, theories, etc… (especially the shiny new ones!).

    I’ve recently had the pleasure of presenting some of my models to various groups ranging from undergrads, doctoral students, and mixed professor/doctoral student groups; and, I’ve noticed that the proportion of people that appear to either uncritically accept my model or dogmatically reject it is higher that I expected. A self styled empiricist even said to me (correctly) that my model was a pale approximation of reality and then proceeded to reference a paper that modeled the pathology of HIV as a well-mixed homogenous cell culture! It seems that people tend to evaluate the rigor of a theory based in part on the media in which it is expressed. Mathematical symbols somehow become more ‘theoretical’ than natural language symbols, and physical/empirical models are not even theories at all. Amazing.

  11. #11 Tim
    March 25, 2007

    I think I am slightly disagreeing with the statement that there is no reason for mathematics to work, it just does. In the case of calculus, it works because it was developed to represent reality. The velocity is the first derivative of position, the acceleration is the derivative of velocity, etc. If you like calculus works because it was set up that way from the start, although that’s maybe a tad too simplistic. But it’s definitely not a collection of empirical formulae that work which people just happened to stumble on.

  12. #12 revere
    March 25, 2007

    Tim: Yes, it works. That it is effective, Weyl considered unreasonable. There is no particular reason it should work. Astrology was also set up to represent the world and for millenia people thought it worked. Do you believe there are infinitesimals “in the world”? You know, too, that Newton’s Laws aren’t correct. They are a good approximation at the usual velocities and distances of the human world.

    So, yes, I think we are disagreeing, but not in the way you seem to think. I am not a pure formalist (many mathematicians are, but I’m not). Like Weyl I am struck by and awed by the unreasonable effeciveness of mathematics. But mathematics is also a game, played by certain rules and not all of the games relate to the world we live in. So the question becomes which games are related to the world and which aren’t? Certainly not every mathematical construction is a model of the real world (or are you such a Platonist you think every mathematical game has a “real world” counterpart?). If you accept what I’ve said so far, then you are part way to where I am. The other part — where we disagree — is that I don’t yet understand how the relation to the real world comes about, and you seem to. That is a deep philosophical mystery most mathematicians also wonder about.

    Finally, your ideas of velocity, position and limiting processes in general aren’t part of the real world. They are abstractions. There is no “point” or “line” or limit as x goes to zero or anything like it you can find in the real world. There are also different notions of derivative and integral and lots of other things in mathematics. I think you are taking a lot for granted here. When you feel back the covers, things aren’t so tidy as the first chapter of a college physics text.

  13. #13 Tim
    March 25, 2007

    The other part — where we disagree — is that I don’t yet understand how the relation to the real world comes about, and you seem to.

    Well, if I did, I would edit the universal quantum function so that George Bush could speak coherent sentences.

    Seriously, I don’t understand a lot of things. And every mathematical construct obviously doesn’t have a reality counterpart. But if you play the odds, it just might. For instance, there are points in reality, they’re called electrons and they’re fairly common. When I learned that there actually were zero-dimension point masses it went a long way to convince me that math working was not just an accident.

  14. #14 revere
    March 25, 2007

    Tim: Zero dimensional point masses? That sounds interesting. Where can I read about this?

  15. #15 Tim
    March 25, 2007

    From wikipedia:

    The electron is currently described as a fundamental particle or an elementary particle. It has no substructure. Hence, for convenience, it is usually defined or assumed to be NEWBa point-like mathematical point charge, with no spatial extension.

  16. #16 Melanie
    March 25, 2007

    It is fascinating to watch you scientists converge around a point from your various disciplines.

  17. #17 revere
    March 25, 2007

    Tim: Ah. That’s a bit different. “For convenience.” For convenience we say points in the real world are zero dimension, too. OTOH, if you describe an electron by its wave function, then it occupies three dimensions. These are mathematical models of electrons — whatever electrons are. Yes, they exist (at least in my opinion). But that doesn’t mean I know what they are, or more importantly, that my mental model of what they are carries any ontological freight.

  18. #18 Greg
    March 26, 2007

    Here is a perfect, right in front of our noses, example of something I tried to say after this week’s Freethinker Sunday Sermonette.

    Some people might say, if Tim were replying to a survey, he is lying. However, that is clearly NOT the case.

    Tim asserted a proposition. Revere asked him to justify it. Tim replied, (a) immediately, (b) with NO trace whatsoever of guile or obfuscation, (c) identifying his authority, (d) quoting his authority, (e) accurately. Clearly, Revere does not agree that “for convenience, it is usually defined or assumed to be” means “is”.

    I agree with Revere. I bet everybody here agrees with Revere. Including, now that it has been shown to him, Tim.

    The same sort of thing happens to those Christians, about whom Revere was complaining. And to Global Warming Deniers, and to Evolution Deniers, and to all the rest of those nasty Deniers.

    They aren’t Evil. They simply have syllogisms in their heads, which they haven’t examined at all (probably for dire reason), and somesort of trained reflex, which makes them, whenever they brush up against it, think “is”, or “is not”.

    Indeed, Tim’s authority is misleading. Grammatically, and as people normally interpret English language, the sentence “It has no substructure.” is a positive assertion about ‘the electron’. Logically, however, it should be merely a subordinate clause in the previous sentence, a descriptive qualifier of either the “particle”(s) or the (implied) “description”. The word “hence”, in the next sentence, should not appear at all : there is nothing to which it can refer back as justification for the assumption “point-like” etc.

    Tim misunderstood. However, despite the usual warning signs, such as “for convenience” and “assumed”, the text offered him considerable assistance. Look at “It has no substructure.” again. See how it stands out from the surrounding sentences. Note how its meaning dangles loose connections, ready to latch onto and strengthen whatever misperceptions the reader might embrace.

    Any successful politician or preacher trains to speak that way smoothly and automatically. Few simple lies. Truths and warnings lurking in the bloviage. Simple assertions ingenuously detached, ready to plug into whatever concepts comfortably inhabit each listeners head.

  19. #19 Tim
    March 26, 2007

    Asserting my rights as a quantum wave function, I maintain both me and revere are right, at least until the box is opened.

  20. #20 revere
    March 26, 2007

    tim: Open the box? You trying to kill me?

  21. #21 Melanie
    March 26, 2007

    reveres,

    remember what Pandora found at the bottom of the box.

  22. #22 Greg
    March 26, 2007

    Science?

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