Rick Wicklin, PhD, is a distinguished researcher in computational statistics at SAS and is a principal developer of SAS/IML software. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Rick is author of the books Statistical Programming with SAS/IML Software and Simulating Data with SAS.
Modern statistical software provides many options for computing robust statistics. For example, SAS can compute robust univariate statistics by using PROC UNIVARIATE, robust linear regression by using PROC ROBUSTREG, and robust multivariate statistics such as robust principal component analysis. Much of the research on robust regression was conducted in the
The eigenvalues of a matrix are not easy to compute. It is remarkable, therefore, that with relatively simple mental arithmetic, you can obtain bounds for the eigenvalues of a matrix of any size. The bounds are provided by using a marvelous mathematical result known as Gershgorin's Disc Theorem. For certain
Recently I wrote about how to compute the Kolmogorov D statistic, which is used to determine whether a sample has a particular distribution. One of the beautiful facts about modern computational statistics is that if you can compute a statistic, you can use simulation to estimate the sampling distribution of
Have you ever run a statistical test to determine whether data are normally distributed? If so, you have probably used Kolmogorov's D statistic. Kolmogorov's D statistic (also called the Kolmogorov-Smirnov statistic) enables you to test whether the empirical distribution of data is different than a reference distribution. The reference distribution
In SAS/IML programs, a common task is to write values in a matrix to a SAS data set. For some programs, the values you want to write are in a matrix and you use the CREATE FROM/APPEND FROM syntax to create the data set, as follows: proc iml; X =
At SAS Global Forum 2019, Daymond Ling presented an interesting discussion of binary classifiers in the financial industry. The discussion is motivated by a practical question: If you deploy a predictive model, how can you assess whether the model is no longer working well and needs to be replaced? Daymond
Here's a simulation tip: When you simulate a fixed-effect generalized linear regression model, don't add a random normal error to the linear predictor. Only the response variable should be random. This tip applies to models that apply a link function to a linear predictor, including logistic regression, Poisson regression, and
SAS regression procedures support several parameterizations of classification variables. When a categorical variable is used as an explanatory variable in a regression model, the procedure generates dummy variables that are used to construct a design matrix for the model. The process of forming columns in a design matrix is called
Did you know that SAS provides built-in support for working with probability distributions that are finite mixtures of normal distributions? This article shows examples of using the "NormalMix" distribution in SAS and describes a trick that enables you to easily work with distributions that have many components. As with all
The CUSUM test has many incarnations. Different areas of statistics use different assumption and test for different hypotheses. This article presents a brief overview of CUSUM tests and gives an example of using the CUSUM test in PROC AUTOREG for autoregressive models in SAS. A CUSUM test uses the cumulative
Many statistical tests use a CUSUM statistic as part of the test. It can be confusing when a researcher refers to "the CUSUM test" without providing details about exactly which CUSUM test is being used. This article describes a CUSUM test for the randomness of a binary sequence. You start
I think every course in exploratory data analysis should begin by studying Anscombe's quartet. Anscombe's quartet is a set of four data sets (N=11) that have nearly identical descriptive statistics but different graphical properties. They are a great reminder of why you should graph your data. You can read about
A quadratic form is a second-degree polynomial that does not have any linear or constant terms. For multivariate polynomials, you can quickly evaluate a quadratic form by using the matrix expression x` A x This computation is straightforward in a matrix language such as SAS/IML. However, some computations in statistics
In numerical linear algebra, there are often multiple ways to solve a problem, and each way is useful in various contexts. In fact, one of the challenges in matrix computations is choosing from among different algorithms, which often vary in their use of memory, data access, and speed. This article
Suppose you need to assign 100 patients equally among 3 treatment groups in a clinical study. Obviously, an equal allocation is impossible because the second number does not evenly divide the first, but you can get close by assigning 34 patients to one group and 33 to the others. Mathematically,
I've previously written about how to deal with nonconvergence when fitting generalized linear regression models. Most generalized linear and mixed models use an iterative optimization process, such as maximum likelihood estimation, to fit parameters. The optimization might not converge, either because the initial guess is poor or because the model
Many SAS procedures support the BY statement, which enables you to perform an analysis for subgroups of the data set. Although the SAS/IML language does not have a built-in "BY statement," there are various techniques that enable you to perform a BY-group analysis. The two I use most often are
In simulation studies, sometimes you need to simulate outliers. For example, in a simulation study of regression techniques, you might want to generate outliers in the explanatory variables to see how the technique handles high-leverage points. This article shows how to generate outliers in multivariate normal data that are a
An important concept in multivariate statistical analysis is the Mahalanobis distance. The Mahalanobis distance provides a way to measure how far away an observation is from the center of a sample while accounting for correlations in the data. The Mahalanobis distance is a good way to detect outliers in multivariate
An analyst was using SAS to analyze some data from an experiment. He noticed that the response variable is always positive (such as volume, size, or weight), but his statistical model predicts some negative responses. He posted the data and asked if it is possible to modify the graph so
Statisticians often emphasize the dangers of extrapolating from a univariate regression model. A common exercise in introductory statistics is to ask students to compute a model of population growth and predict the population far in the future. The students learn that extrapolating from a model can result in a nonsensical
It's time to celebrate Pi Day! Every year on March 14th (written 3/14 in the US), math-loving folks celebrate "all things pi-related" because 3.14 is the three-decimal approximation to the mathematical constant, π. Although children learn that pi is approximately 3.14159..., the actual definition of π is the ratio of
A SAS programmer posted an interesting question on a SAS discussion forum. The programmer wanted to iterate over hundreds of SAS data sets, read in all the character variables, and then do some analysis. However, not every data set contains character variables, and SAS complains when you ask it to
A previous article shows how to use a scatter plot to visualize the average SAT scores for all high schools in North Carolina. The schools are grouped by school districts and ranked according to the median value of the schools in the district. For the school districts that have many
Standardized tests like the SAT and ACT can cause stress for both high school students and their parents, but according to a Wall Street Journal article, the SAT and ACT "provide an invaluable measure of how students are likely to perform in college and beyond." Naturally, students wonder how their
Box plots are a great way to compare the distributions of several subpopulations of your data. For example, box plots are often used in clinical studies to visualize the response of patients in various cohorts. This article describes three techniques to visualize responses when the cohorts have a nested or
When I run a bootstrap analysis, I create graphs to visualize the distribution of the bootstrap statistics. For example, in my article about how to bootstrap the difference of means in a two-sample t test, I included a histogram of the bootstrap distribution and added reference lines to indicate a
Last year I published a series of blogs posts about how to create a calibration plot in SAS. A calibration plot is a way to assess the goodness of fit for a logistic model. It is a diagnostic graph that enables you to qualitatively compare a model's predicted probability of
Maybe if we think and wish and hope and pray It might come true. Oh, wouldn't it be nice? The Beach Boys Months ago, I wrote about how to use the EFFECT statement in SAS to perform regression with restricted cubic splines. This is the modern way to use splines
When you use maximum likelihood estimation (MLE) to find the parameter estimates in a generalized linear regression model, the Hessian matrix at the optimal solution is very important. The Hessian matrix indicates the local shape of the log-likelihood surface near the optimal value. You can use the Hessian to estimate
