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# Tag Archives: Statistics

# Data Science, Machine Learning, and Statistics: what is in a name?

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

# Don’t use correlation to track prediction performance

Using correlation to track model performance is “a mistake that nobody would ever make” combined with a vague “what would be wrong if I did do that” feeling. I hope after reading this feel a least a small urge to double check your work and presentations to make sure you have not reported correlation where R-squared, likelihood or root mean square error (RMSE) would have been more appropriate.

It is tempting (but wrong) to use correlation to track the performance of model predictions. The temptation arises because we often (correctly) use correlation to evaluate possible model inputs. And the correlation function is often a convenient built-in function. Continue reading

# Level fit summaries can be tricky in R

Model level fit summaries can be tricky in R. A quick read of model fit summary data for factor levels can be misleading. We describe the issue and demonstrate techniques for dealing with them. Continue reading

# How robust is logistic regression?

Logistic Regression is a popular and effective technique for modeling categorical outcomes as a function of both continuous and categorical variables. The question is: how robust is it? Or: how robust are the common implementations? (note: we are using robust in a more standard English sense of performs well for all inputs, not in the technical statistical sense of immune to deviations from assumptions or outliers.)

Even a detailed reference such as “Categorical Data Analysis” (Alan Agresti, Wiley, 1990) leaves off with an empirical observation: “the convergence … for the Newton-Raphson method is usually fast” (chapter 4, section 4.7.3, page 117). This is a book that if there is a known proof that the estimation step is a contraction (one very strong guarantee of convergence) you would expect to see the proof reproduced. I always suspected there was some kind of Brouwer fixed-point theorem based folk-theorem proving absolute convergence of the Newton-Raphson method in for the special case of logistic regression. This can not be the case as the Newton-Raphson method can diverge even on trivial full-rank well-posed logistic regression problems. Continue reading

# Modeling Trick: Impact Coding of Categorical Variables with Many Levels

One of the shortcomings of regression (both linear and logistic) is that it doesn’t handle categorical variables with a very large number of possible values (for example, postal codes). You can get around this, of course, by going to another modeling technique, such as Naive Bayes; however, you lose some of the advantages of regression — namely, the model’s explicit estimates of variables’ explanatory value, and explicit insight into and control of variable to variable dependence.

Here we discuss one modeling trick that allows us to keep categorical variables with a large number of values, and at the same time retain much of logistic regression’s power.

# Modeling Trick: Masked Variables

A primary problem data scientists face again and again is: how to properly adapt or treat variables so they are best possible components of a regression. Some analysts at this point delegate control to a shape choosing system like neural nets. I feel such a choice gives up far too much statistical rigor, transparency and control without real benefit in exchange. There are other, better, ways to solve the reshaping problem. A good rigorous way to treat variables are to try to find stabilizing transforms, introduce splines (parametric or non-parametric) or use generalized additive models. A practical or pragmatic approach we advise to get some of the piecewise reshaping power of splines or generalized additive models is: a modeling trick we call “masked variables.” This article works a quick example using masked variables. Continue reading

# Correlation and R-Squared

What is R^{2}? In the context of predictive models (usually linear regression), where *y* is the true outcome, and *f* is the model’s prediction, the definition that I see most often is:

In words, R^{2} is a measure of how much of the variance in *y* is explained by the model, *f*.

Under “general conditions”, as Wikipedia says, R^{2} is also the square of the correlation (correlation written as a “p” or “rho”) between the actual and predicted outcomes:

I prefer the “squared correlation” definition, as it gets more directly at what is usually my primary concern: prediction. If R^{2} is close to one, then the model’s predictions mirror true outcome, tightly. If R^{2} is low, then either the model does not mirror true outcome, or it only mirrors it loosely: a “cloud” that — hopefully — is oriented in the right direction. Of course, looking at the graph always helps:

The question we will address here is : how do you get from R^{2} to correlation?

# The equivalence of logistic regression and maximum entropy models

Nina Zumel recently gave a very clear explanation of logistic regression ( The Simpler Derivation of Logistic Regression ). In particular she called out the central role of log-odds ratios and demonstrated how the “deviance” (that mysterious

quantity reported by fitting packages) is both a term in “the pseudo-R^2″ (so directly measures goodness of fit) and is the quantity that is actually optimized during the fitting procedure. One great point of the writeup was how simple everything is once you start thinking in terms of derivatives (and that it isn’t so much the functional form of the sigmoid that is special but its relation to its own derivative that is special).

We adapt these presentation ideas to make explicit the well known equivalence of logistic regression and maximum entropy models. Continue reading

# The Simpler Derivation of Logistic Regression

Logistic regression is one of the most popular ways to fit models for categorical data, especially for binary response data. It is the most important (and probably most used) member of a class of models called generalized linear models. Unlike linear regression, logistic regression can directly predict probabilities (values that are restricted to the (0,1) interval); furthermore, those probabilities are well-calibrated when compared to the probabilities predicted by some other classifiers, such as Naive Bayes. Logistic regression preserves the marginal probabilities of the training data. The coefficients of the model also provide some hint of the relative importance of each input variable.

While you don’t have to know how to derive logistic regression or how to implement it in order to use it, the details of its derivation give important insights into interpreting and troubleshooting the resulting models. Unfortunately, most derivations (like the ones in [Agresti, 1990] or [Hastie, et.al, 2009]) are too terse for easy comprehension. Here, we give a derivation that is less terse (and less general than Agresti’s), and we’ll take the time to point out some details and useful facts that sometimes get lost in the discussion. Continue reading