Tag Archives: Regression

Reading the Gauss-Markov theorem

What is the Gauss-Markov theorem?

From “The Cambridge Dictionary of Statistics” B. S. Everitt, 2nd Edition:

A theorem that proves that if the error terms in a multiple regression have the same variance and are uncorrelated, then the estimators of the parameters in the model produced by least squares estimation are better (in the sense of having lower dispersion about the mean) than any other unbiased linear estimator.

This is pretty much considered the “big boy” reason least squares fitting can be considered a good implementation of linear regression.

Suppose you are building a model of the form:

    y(i) = B . x(i) + e(i)

where B is a vector (to be inferred), i is an index that runs over the available data (say 1 through n), x(i) is a per-example vector of features, and y(i) is the scalar quantity to be modeled. Only x(i) and y(i) are observed. The e(i) term is the un-modeled component of y(i) and you typically hope that the e(i) can be thought of unknowable effects, individual variation, ignorable errors, residuals, or noise. How weak/strong assumptions you put on the e(i) (and other quantities) depends on what you know, what you are trying to do, and which theorems you need to meet the pre-conditions of. The Gauss-Markov theorem assures a good estimate of B under weak assumptions.

How to interpret the theorem

The point of the Gauss-Markov theorem is that we can find conditions ensuring a good fit without requiring detailed distributional assumptions about the e(i) and without distributional assumptions about the x(i). However, if you are using Bayesian methods or generative models for predictions you may want to use additional stronger conditions (perhaps even normality of errors and even distributional assumptions on the xs).

We are going to read through the Wikipedia statement of the Gauss-Markov theorem in detail.

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Automatic bias correction doesn’t fix omitted variable bias

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

Skimming statistics papers for the ideas (instead of the complete procedures)

Been reading a lot of Gelman, Carlin, Stern, Dunson, Vehtari, Rubin “Bayesian Data Analysis” 3rd edition lately. Overall in the Bayesian framework some ideas (such as regularization, and imputation) are way easier to justify (though calculating some seemingly basic quantities becomes tedious). A big advantage (and weakness) of this formulation is statistics has a much less “shrink wrapped” feeling than the classic frequentist presentations. You feel like the material is being written to peers instead of written to calculators (of the human or mechanical variety). In the Bayesian formulation you don’t feel like you will be yelled at for using 1 tablespoon of sugar when the recipe calls for 3 teaspoons (at least if you live in the United States).

Some other stuff reads differently after this though. Continue reading

Trimming the Fat from glm() Models in R

One of the attractive aspects of logistic regression models (and linear models in general) is their compactness: the size of the model grows in the number of coefficients, not in the size of the training data. With R, though, glm models are not so concise; we noticed this to our dismay when we tried to automate fitting a moderate number of models (about 500 models, with on the order of 50 coefficients) to data sets of moderate size (several tens of thousands of rows). A workspace save of the models alone was in the tens of gigabytes! How is this possible? We decided to find out.

As many R users know (but often forget), a glm model object carries a copy of its training data by default. You can use the settings y=FALSE and model=FALSE to turn this off.

set.seed(2325235)


# Set up a synthetic classification problem of a given size
# and two variables: one numeric, one categorical
# (two levels).
synthFrame = function(nrows) {
   d = data.frame(xN=rnorm(nrows),
      xC=sample(c('a','b'),size=nrows,replace=TRUE))
   d$y = (d$xN + ifelse(d$xC=='a',0.2,-0.2) + rnorm(nrows))>0.5
   d
}


# first show that model=F and y=F help reduce model size

dTrain = synthFrame(1000)
model1 = glm(y~xN+xC,data=dTrain,family=binomial(link='logit'))
model2 = glm(y~xN+xC,data=dTrain,family=binomial(link='logit'),
             y=FALSE)
model3 = glm(y~xN+xC,data=dTrain,family=binomial(link='logit'),
              y=FALSE, model=FALSE)

#
# Estimate the object's size as the size of its serialization
#
length(serialize(model1, NULL))
# [1] 225251
length(serialize(model2, NULL))
# [1] 206341
length(serialize(model3, NULL))
# [1] 189562

dTest = synthFrame(100)
p1 = predict(model1, newdata=dTest, type='response')
p2 = predict(model2, newdata=dTest, type='response')
p3 = predict(model3, newdata=dTest, type='response')
sum(abs(p1-p2))
# [1] 0
sum(abs(p1-p3))
# [1] 0

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What is meant by regression modeling?

What is meant by regression modeling?

Linear Regression is one of the most common statistical modeling techniques. It is very powerful, important, and (at first glance) easy to teach. However, because it is such a broad topic it can be a minefield for teaching and discussion. It is common for angry experts to accuse writers of carelessness, ignorance, malice and stupidity. If the type of regression the expert reader is expecting doesn’t match the one the writer is discussing then the writer is assumed to be ill-informed. The writer is especially vulnerable to experts when writing for non-experts. In such writing the expert finds nothing new (as they already know the topic) and is free to criticize any accommodation or adaption made for the intended non-expert audience. We argue that many of the corrections are not so much evidence of wrong ideas but more due a lack of empathy for the necessary informality necessary in concise writing. You can only define so much in a given space, and once you write too much you confuse and intimidate a beginning audience. Continue reading

Unprincipled Component Analysis

As a data scientist I have seen variations of principal component analysis and factor analysis so often blindly misapplied and abused that I have come to think of the technique as unprincipled component analysis. PCA is a good technique often used to reduce sensitivity to overfitting. But this stated design intent leads many to (falsely) believe that any claimed use of PCA prevents overfit (which is not always the case). In this note we comment on the intent of PCA like techniques, common abuses and other options.

The idea is to illustrate what can quietly go wrong in an analysis and what tests to perform to make sure you see the issue. The main point is some analysis issues can not be fixed without going out and getting more domain knowledge, more variables or more data. You can’t always be sure that you have insufficient data in your analysis (there is always a worry that some clever technique will make the current data work), but it must be something you are prepared to consider. Continue reading

The cranky guide to trying R packages

This is a tutorial on how to try out a new package in R. The summary is: expect errors, search out errors and don’t start with the built in examples or real data.

Suppose you want to try out a novel statistical technique? A good fraction of the time R is your best bet for a first trial. Take as an example general additive models (“Generalized Additive Models,” Trevor J Hastie, Robert Tibshirani, Statistical Science (1986) vol. 1 (3) pp. 297-318); R has a package named “gam” written by Trevor Hastie himself. But, like most R packages, trying the package from the supplied documentation brings in unfamiliar data and concerns. It is best to start small and quickly test if the package itself is suitable to your needs. We give a quick outline of how to learn such a package and quickly find out if the package is for you.

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