Blog Archives

Robustness and tests for equal variance

May 16, 2018
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The two-sample t-test is a way to test whether two data sets come from distributions with the same mean. I wrote a few days ago about how the test performs under ideal circumstances, as well as less than ideal circumstances. This is an analogous post for testing whether two data sets come from distributions with the same […]

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Two-sample t-test and robustness

May 11, 2018
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Two-sample t-test and robustness

A two-sample t-test is intended to determine whether there’s evidence that two samples have come from distributions with different means. The test assumes that both samples come from normal distributions. Robust to non-normality, not to asymmetry It is fairly well known that the t-test is robust to departures from a normal distribution, as long as the actual […]

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Robust statistics

May 8, 2018
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P. J. Huber gives three desiderata for a statistical method in his book Robust Statistics: It should have a reasonably good (optimal or nearly optimal) efficiency at the assumed model. It should be robust in the sense that small deviations from the model assumptions should impair the performance only slightly. Somewhat larger deviations from the […]

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Probit regression

May 5, 2018
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Probit regression

The previous post looked at how probability predictions from a logistic regression model vary as a function of the fitted parameters. This post goes through the same exercise for probit regression and compares the two kinds of nonlinear regression. Generalized linear models and link functions Logistic and probit regression are minor variations on a theme. […]

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Sensitivity of logistic regression prediction on coefficients

May 4, 2018
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Sensitivity of logistic regression prediction on coefficients

The output of a logistic regression model is a function that predicts the probability of an event as a function of the input parameter. This post will only look at a simple logistic regression model with one predictor, but similar analysis applies to multiple regression with several predictors. Here’s a plot of such a curve […]

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Obesity index: Measuring the fatness of probability distribution tails

April 23, 2018
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A probability distribution is called “fat tailed” if its probability density goes to zero slowly. Slowly relative to what? That is often implicit and left up to context, but generally speaking the exponential distribution is the dividing line. Probability densities that decay faster than the exponential distribution are called “thin” or “light,” and densities that […]

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Categorical Data Analysis

April 14, 2018
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Categorical Data Analysis

Categorical data analysis could mean a couple different things. One is analyzing data that falls into unordered categories (e.g. red, green, and blue) rather than numerical values (e..g. height in centimeters). Another is using category theory to assist with the analysis of data. Here “category” means something more sophisticated than a list of items you […]

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Hypothesis testing vs estimation

April 3, 2018
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I was looking at my daughter’s statistics homework recently, and there were a pair of questions about testing the level of lead in drinking water. One question concerned testing whether the water was safe, and the other concerned testing whether the water was unsafe. There’s something bizarre, even embarrassing, about this. You want to do […]

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Generalized normal distribution and kurtosis

March 30, 2018
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Generalized normal distribution and kurtosis

The generalized normal distribution adds an extra parameter β to the normal (Gaussian) distribution. The probability density function for the generalized normal distribution is Here the location parameter μ is the mean, but the scaling factor σ is not the standard deviation unless β = 2. For small values of the shape parameter β, the […]

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Making sense of a probability problem in the WSJ

January 1, 2018
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Making sense of a probability problem in the WSJ

Someone wrote to me the other day asking if I could explain a probability example from the Wall Street Journal. (“Proving Investment Success Takes Time,” Spencer Jakab, November 25, 2017.) Victor Haghani … and two colleagues told several hundred acquaintances who worked in finance that they would flip two coins, one that was normal and […]

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