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.
My article about Fisher's transformation of the Pearson correlation contained a simulation. The simulation uses the RANDNORMAL function in SAS/IML software to simulate multivariate normal data. If you are a SAS programmer who does not have access to SAS/IML software, you can use the SIMNORMAL procedure in SAS/STAT software to
Pearson's correlation measures the linear association between two variables. Because the correlation is bounded between [-1, 1], the sampling distribution for highly correlated variables is highly skewed. Even for bivariate normal data, the skewness makes it challenging to estimate confidence intervals for the correlation, to run one-sample hypothesis tests ("Is
Toe bone connected to the foot bone, Foot bone connected to the leg bone, Leg bone connected to the knee bone,... — American Spiritual, "Dem Bones" Last week I read an interesting article on Robert Kosara's data visualization blog. Kosara connected the geographic centers of the US zip codes in
This article shows how to simulate data from a mixture of multivariate normal distributions, which is also called a Gaussian mixture. You can use this simulation to generate clustered data. The adjacent graph shows three clusters, each simulated from a four-dimensional normal distribution. Each cluster has its own within-cluster covariance,
Did you know that you can get SAS to compute symbolic (analytical) derivatives of simple functions, including applying the product rule, quotient rule, and chain rule? SAS can form the symbolic derivatives of single-variable functions and partial derivatives of multivariable functions. Furthermore, the derivatives are output in a form that
If you use SAS regression procedures, you are probably familiar with the "stars and bars" notation, which enables you to construct interaction effects in regression models. Although you can construct many regression models by using that classical notation, a friend recently reminded me that the EFFECT statement in SAS provides
Correlation is a fundamental statistical concept that measures the linear association between two variables. There are multiple ways to think about correlation: geometrically, algebraically, with matrices, with vectors, with regression, and more. To paraphrase the great songwriter Paul Simon, there must be 50 ways to view your correlation! But don't
A previous article discussed the mathematical properties of the singular value decomposition (SVD) and showed how to use the SVD subroutine in SAS/IML software. This article uses the SVD to construct a low-rank approximation to an image. Applications include image compression and denoising an image. Construct a grayscale image The
The singular value decomposition (SVD) could be called the "billion-dollar algorithm" since it provides the mathematical basis for many modern algorithms in data science, including text mining, recommender systems (think Netflix and Amazon), image processing, and classification problems. Although the SVD was mathematically discovered in the late 1800s, computers have
All statisticians are familiar with the classical arithmetic mean. Some statisticians are also familiar with the geometric mean. Whereas the arithmetic mean of n numbers is the sum divided by n, the geometric mean of n nonnegative numbers is the n_th root of the product of the numbers. The geometric
When you implement a statistical algorithm in a vector-matrix language such as SAS/IML, R, or MATLAB, you should measure the performance of your implementation, which means that you should time how long a program takes to analyze data of varying sizes and characteristics. There are some general tips that can
Visualizing the correlations between variables often provides insight into the relationships between variables. I've previously written about how to use a heat map to visualize a correlation matrix in SAS/IML, and Chris Hemedinger showed how to use Base SAS to visualize correlations between variables. Recently a SAS programmer asked how
When someone refers to the correlation between two variables, they are probably referring to the Pearson correlation, which is the standard statistic that is taught in elementary statistics courses. Elementary courses do not usually mention that there are other measures of correlation. Why would anyone want a different estimate of
Recently, I was asked whether SAS can perform a principal component analysis (PCA) that is robust to the presence of outliers in the data. A PCA requires a data matrix, an estimate for the center of the data, and an estimate for the variance/covariance of the variables. Classically, these estimates
A SAS customer asked, "I computed the eigenvectors of a matrix in SAS and in another software package. I got different answers? How do I know which answer is correct?" I've been asked variations of this question dozens of times. The answer is usually "both answers are correct." The mathematical
Last week I blogged about the broken-stick problem in probability, which reminded me that the broken-stick model is one of the many techniques that have been proposed for choosing the number of principal components to retain during a principal component analysis. Recall that for a principal component analysis (PCA) of
A SAS user needed to convert a program from MATLAB into the SAS/IML matrix language and asked whether there is a SAS/IML equivalent to the fliplr and flipud functions in MATLAB. These functions flip the columns or rows (respectively) of a matrix; "LR" stands for "left-right" and "UD" stands for
A classical problem in elementary probability asks for the expected lengths of line segments that result from randomly selecting k points along a segment of unit length. It is both fun and instructive to simulate such problems. This article uses simulation in the SAS/IML language to estimate solutions to the
For a time series { y1, y2, ..., yN }, the difference operator computes the difference between two observations. The kth-order difference is the series { yk+1 - y1, ..., yN - yN-k }. In SAS, the DIF function in the DATA step computes differences between observations. The DIF function
Skewness is a measure of the asymmetry of a univariate distribution. I have previously shown how to compute the skewness for data distributions in SAS. The previous article computes Pearson's definition of skewness, which is based on the standardized third central moment of the data. Moment-based statistics are sensitive to
An important problem in machine learning is the "classification problem." In this supervised learning problem, you build a statistical model that predicts a set of categorical outcomes (responses) based on a set of input features (explanatory variables). You do this by training the model on data for which the outcomes
I recently showed how to compute a bootstrap percentile confidence interval in SAS. The percentile interval is a simple "first-order" interval that is formed from quantiles of the bootstrap distribution. However, it has two limitations. First, it does not use the estimate for the original data; it is based only
I previously wrote about how to compute a bootstrap confidence interval in Base SAS. As a reminder, the bootstrap method consists of the following steps: Compute the statistic of interest for the original data Resample B times from the data to form B bootstrap samples. B is usually a large
A SAS customer asked how to use SAS to conduct a Z test for the equality of two proportions. He was directed to the SAS Usage Note "Testing the equality of two or more proportions from independent samples." The note says to "specify the CHISQ option in the TABLES statement
Students in introductory statistics courses often use summary statistics (such as sample size, mean, and standard deviation) to test hypotheses and to compute confidence intervals. Did you know that you can provide summary statistics (rather than raw data) to PROC TTEST in SAS and obtain hypothesis tests and confidence intervals?
Suppose you roll six identical six-sided dice. Chance are that you will see at least one repeated number. The probability that you will see six unique numbers is very small: only 6! / 6^6 ≈ 0.015. This example can be generalized. If you draw a random sample with replacement from
My presentation at SAS Global Forum 2017 was "More Than Matrices: SAS/IML Software Supports New Data Structures." The paper was published in the conference proceedings several months ago, but I recently recorded a short video that gives an overview of using the new data structures in SAS/IML 14.2: If your
One way to assess the precision of a statistic (a point estimate) is to compute the standard error, which is the standard deviation of the statistic's sampling distribution. A relatively large standard error indicates that the point estimate should be viewed with skepticism, either because the sample size is small
Most numerical optimization routines require that the user provides an initial guess for the solution. I have previously described a method for choosing an initial guess for an optimization, which works well for low-dimensional optimization problems. Recently a SAS programmer asked how to find an initial guess when there are
In a previous article, I showed two ways to define a log-likelihood function in SAS. This article shows two ways to compute maximum likelihood estimates (MLEs) in SAS: the nonlinear optimization subroutines in SAS/IML and the NLMIXED procedure in SAS/STAT. To illustrate these methods, I will use the same data
