In our article How robust is logistic regression? we pointed out some basic yet deep limitations of the traditional full-step Newton-Raphson or Iteratively Reweighted Least Squares methods of solving logistic regression problems (such as in R‘s standard glm() implementation). In fact in the comments we exhibit a well posed data fitting problem that can not be fit using the traditional methods starting at the traditional (0,0) start point. And we cited an example where the traditional methods fail to compute the average from a non-zero start. The question remained: can we prove the standard methods always compute the average correctly if started at zero? It turns out they can, and the proof isn’t as messy as I anticipated. Read more…

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. Read more…

Categories: data science, Expository Writing, Mathematics, Statistics, Tutorials Tags: diverge, generalized linear model, GLM, Logistic Regression, Newton-Raphson, R, Statistics
What does a generalized linear model do? R supplies a modeling function called `glm()`

that fits generalized linear models (abbreviated as GLMs). A natural question is what does it do and what problem is it solving for you? We work some examples and place generalized linear models in context with other techniques. Read more…

Categories: data science, Expository Writing, Pragmatic Machine Learning, Statistics, Tutorials Tags: Genearlized Linear Model, GLM, Logistic Regression, logit, R, sigmoid, What does logistic regression do
Dr. Nina Zumel recently published an excellent tutorial on a modeling technique she called impact coding. It is a pragmatic machine learning technique that has helped with more than one client project. Impact coding is a bridge from Naive Bayes (where each variable’s impact is added without regard to the known effects of any other variable) to Logistic Regression (where dependencies between variables and levels is completely accounted). A natural question is can pick up more of the positive features of each model? Read more…