Page 94 of Gelman, Carlin, Stern, Dunson, Vehtari, Rubin “Bayesian Data Analysis” 3rd Edition (which we will call BDA3) provides a great example of what happens when common broad frequentist bias criticisms are over-applied to predictions from ordinary linear regression: the predictions appear to fall apart. BDA3 goes on to exhibit what might be considered the kind of automatic/mechanical fix responding to such criticisms would entail (producing a bias corrected predictor), and rightly shows these adjusted predictions are far worse than the original ordinary linear regression predictions. BDA3 makes a number of interesting points and is worth studying closely. We work their example in a bit more detail for emphasis. Continue reading
The goal of Zumel/Mount: Practical Data Science with R is to teach, through guided practice, the skills of a data scientist. We define a data scientist as the person who organizes client input, data, infrastructure, statistics, mathematics and machine learning to deploy useful predictive models into production.
Our plan to teach is to:
- Order the material by what is expected from the data scientist.
- Emphasize the already available bread and butter machine learning algorithms that most often work.
- Provide a large set of worked examples.
- Expose the reader to a number of realistic data sets.
Some of these choices may put-off some potential readers. But it is our goal to try and spend out time on what a data scientist needs to do. Our point: the data scientist is responsible for end to end results, which is not always entirely fun. If you want to specialize in machine learning algorithms or only big data infrastructure, that is a fine goal. However, the job of the data scientist is to understand and orchestrate all of the steps (working with domain experts, curating data, using data tools, and applying machine learning and statistics).
Once you define what a data scientist does, you find fewer people want to work as one.
We expand a few of our points below. Continue reading
Controlled experiments embody the best scientific design for establishing a causal relationship between changes and their influence on user-observable behavior.
A/B tests are one of the simplest ways of running controlled experiments to evaluate the efficacy of a proposed improvement (a new medicine, compared to an old one; a promotional campaign; a change to a website). To run an A/B test, you split your population into a control group (let’s call them “A”) and a treatment group (“B”). The A group gets the “old” protocol, the B group gets the proposed improvement, and you collect data on the outcome that you are trying to achieve: the rate that patients are cured; the amount of money customers spend; the rate at which people who come to your website actually complete a transaction. In the traditional formulation of A/B tests, you measure the outcomes for the A and B groups, determine which is better (if either), and whether or not the difference observed is statistically significant. This leads to questions of test size: how big a population do you need to get reliably detect a difference to the desired statistical significance? And to answer that question, you need to know how big a difference (effect size) matters to you.
The irony is that to detect small differences accurately you need a larger population size, even though in many cases, if the difference is small, picking the wrong answer matters less. It can be easy to lose sight of that observation in the struggle to determine correct experiment sizes.
There is an alternative formulation for A/B tests that is especially suitable for online situations, and that explicitly takes the above observation into account: the so-called multi-armed bandit problem. Imagine that you are in a casino, faced with K slot machines (which used to be called “one-armed bandits” because they had a lever that you pulled to play (the “arm”) — and they pretty much rob you of all your money). Each of the slot machines pays off at a different (unknown) rate. You want to figure out which of the machines pays off at the highest rate, then switch to that one — but you don’t want to lose too much money to the suboptimal slot machines while doing so. What’s the best strategy?
The “pulling one lever at a time” formulation isn’t a bad way of thinking about online transactions (as opposed to drug trials); you can imagine all your customers arriving at your site sequentially, and being sent to bandit A or bandit B according to some strategy. Note also, that if the best bandit and the second-best bandit have very similar payoff rates, then settling on the second best bandit, while not optimal, isn’t necessarily that bad a strategy. You lose winnings — but not much.
Traditionally, bandit games are infinitely long, so analysis of bandit strategies is asymptotic. The idea is that you test less as the game continues — but the testing stage can go on for a very long time (often interleaved with periods of pure exploitation, or playing the best bandit). This infinite-game assumption isn’t always tenable for A/B tests — for one thing, the world changes; for another, testing is not necessarily without cost. We’ll look at finite games below.
The Facebook data science blog shared some fun data explorations this Valentine’s Day in Carlos Greg Diuk’s “The Formation of Love”. They are rightly receiving positive interest in and positive reviews of their work (for example Robinson Meyer’s Atlantic article). The finding is also a great opportunity to discuss the gap between cool data mining results and usable predictive models. Data mining results like this (and the infamous “Beer and Diapers story”) face an expectation that one is immediately ready to implement something like what is claimed in: “Target Figured Out A Teen Girl Was Pregnant Before Her Father Did” once an association is plotted.
Producing a revenue improving predictive model is much harder than mining an interesting association. And this is what we will discuss here. Continue reading
I was watching my cousins play Unspeakable Words over Christmas break and got interested in the end game. The game starts out as a spell a word from cards and then bet some points game, but in the end (when you are down to one marker) it becomes a pure betting game. In this article we analyze an idealized form of the pure betting end game. Continue reading
This article is a break from data-science, and is instead about the kind of problem you can try on the train. It is inspired by the problems in Bollobas’s “The art of mathematics.”
One of the many irritating things about airlines is the fact that the cary-on bag restrictions are often stated as “your maximum combined linear measurement (length + width + height) must not exceed 45 inches” when they really mean your bag must fit into a 14 inch by 9 inch by 22 inch box (so they actually may not accept a 43 inch by one inch by one inch pool spear as your carry-on). The “total linear measure” seems (at first glance) “gameable,” but can (through some hairy math) at least be seen to at least be self-consistent. It turns out you can’t put a box with longer total linear measurements into a box with smaller total linear measurements.
Let’s work out why this could be problem and then why the measure works. Continue reading
It occurred to us recently that we don’t have any articles about Bayesian approaches to statistics here. I’m not going to get into the “Bayesian versus Frequentist” war; in my opinion, which style of approach to use is less about philosophy, and more about figuring out the best way to answer a question. Once you have the right question, then the right approach will naturally suggest itself to you. It could be a frequentist approach, it could be a bayesian one, it could be both — even while solving the same problem.
Let’s take the example that Bayesians love to hate: significance testing, especially in clinical trial style experiments. Clinical trial experiments are designed to answer questions of the form “Does treatment X have a discernible effect on condition Y, on average?” To be specific, let’s use the question “Does drugX reduce hypertension, on average?” Assuming that your experiment does show a positive effect, the statistical significance tests that you run should check for the sorts of problems that John discussed in our previous article, Worry about correctness and repeatability, not p-values: What are the chances that an ineffective drug could produce the results that I saw? How likely is it that another researcher could replicate my results with the same size trial?
We can argue about whether or not the question we are answering is the correct question — but given that it is the question, the procedure to answer it and to verify the statistical validity of the results is perfectly appropriate.
So what is the correct question? From your family doctor’s viewpoint, a clinical trial answers the question “If I prescribe drugX to all my hypertensive patients, will their blood pressure improve, on average?” That isn’t the question (hopefully) that your doctor actually asks, though possibly your insurance company does. Your doctor should be asking “If I prescribe drugX to this patient, the one sitting in my examination room, will the patient’s blood pressure improve?” There is only one patient, so there is no such thing as “on average.”
If your doctor has a masters degree in statistics, the question might be phrased as “If I prescribe drugX to this patient, what is the posterior probability that the patient’s blood pressure will improve?” And that’s a bayesian question. Continue reading
A fair complaint when seeing yet another “data science” article is to say: “this is just medical statistics” or “this is already part of bioinformatics.” We certainly label many articles as “data science” on this blog. Probably the complaint is slightly cleaner if phrased as “this is already known statistics.” But the essence of the complaint is a feeling of claiming novelty in putting old wine in new bottles. Rob Tibshirani nailed this type of distinction in is famous machine learning versus statistics glossary.
I’ve written about statistics v.s. machine learning , but I would like to explain why we (the authors of this blog) often use the term data science. Nina Zumel explained being a data scientist very well, I am going to take a swipe at explaining data science.
We (the authors on this blog) label many of our articles as being about data science because we want to emphasize that the various techniques we write about are only meaningful when considered parts of a larger end to end process. The process we are interested in is the deployment of useful data driven models into production. The important components are learning the true business needs (often by extensive partnership with customers), enabling the collection of data, managing data, applying modeling techniques and applying statistics criticisms. The pre-existing term I have found that is closest to describing this whole project system is data science, so that is the term I use. I tend to use it a lot, because while I love the tools and techniques our true loyalty is to the whole process (and I want to emphasize this to our readers).
The phrase “data science” as in use it today is a fairly new term (made popular by William S. Cleveland, DJ Patil, and Jeff Hammerbacher). I myself worked in a “computational sciences” group in the mid 1990’s (this group emphasized simulation based modeling of small molecules and their biological interactions, the naming was an attempt to emphasize computation over computers). So for me “data science” seems like a good term when your work is driven by data (versus driven from computer simulations). For some people data science is considered a new calling and for others it is a faddish misrepresentation of work that has already been done. I think there are enough substantial differences in approach between traditional statistics, machine learning, data mining, predictive analytics, and data science to justify at least this much nomenclature. In this article I will try to describe (but not fully defend) my opinion. Continue reading
In data science work you often run into cryptic sentences like the following:
Age adjusted death rates per 10,000 person years across incremental thirds of muscular strength were 38.9, 25.9, and 26.6 for all causes; 12.1, 7.6, and 6.6 for cardiovascular disease; and 6.1, 4.9, and 4.2 for cancer (all P < 0.01 for linear trend).
(From “Association between muscular strength and mortality in men: prospective cohort study,” Ruiz et. al. BMJ 2008;337:a439.)
The accepted procedure is to recognize “p” or “p-value” as shorthand for “significance,” keep your mouth shut and hope the paper explains what is actually claimed somewhere later on. We know the writer is claiming significance, but despite the technical terminology they have not actually said which test they actually ran (lm(), glm(), contingency table, normal test, t-test, f-test, g-test, chi-sq, permutation test, exact test and so on). I am going to go out on a limb here and say these type of sentences are gibberish and nobody actually understands them. From experience we know generally what to expect, but it isn’t until we read further we can precisely pin down what is actually being claimed. This isn’t the authors’ fault, they are likely good scientists, good statisticians, and good writers; but this incantation is required by publishing tradition and reviewers.
We argue you should worry about the correctness of your results (how likely a bad result could look like yours, the subject of frequentist significance) and repeatability (how much variance is in your estimation procedure, as measured by procedures like the bootstrap). p-values and significance are important in how they help structure the above questions.
The legitimate purpose of technical jargon is to make conversations quicker and more precise. However, saying “p” is not much shorter than saying “significance” and there are many different procedures that return p-values (so saying “p” does not limit you down to exactly one procedure like a good acronym might). At best the savings in time would be from having to spend 10 minutes thinking which interpretation of significance is most approbate to the actual problem at hand versus needing a mere 30 seconds to read about the “p.” However, if you don’t have 10 minutes to consider if the entire result a paper is likely an observation artifact due to chance or noise (the subject of significance) then you really don’t care much about the paper.
In our opinion “p-values” have degenerated from a useful jargon into a secretive argot. We are going to discuss thinking about significance as “worrying about correctness” (a fundamental concern) instead of as a cut and dried statistical procedure you should automate out of view (uncritically copying reported p’s from fitters). Yes “p”s are significances, but there is no reason to not just say what sort of error you are claiming is unlikely. Continue reading