Last week I compared the overhand shuffle to the riffle shuffle. I used random operations to simulate both kinds of shuffles and then compared how well they mix cards. The article caused one my colleague and fellow blogger, Rob Pratt, to ask if I was familiar with a bit of

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Every day I’m shufflin'. Shufflin', shufflin'. -- "Party Rock Anthem," LMFAO The most popular way to mix a deck of cards is the riffle shuffle, which separates the deck into two pieces and interleaves the cards from each piece. Besides being popular with card players, the riffle shuffle is

The SAS/IML language and the MATLAB language are similar. Both provide a natural syntax for performing high-level computations on vectors and matrices, including basic linear algebra subroutines. Sometimes a SAS programmer will convert an algorithm from MATLAB into SAS/IML. Because the languages are not identical, I am sometimes asked, "what

Have you ever tried to type a movie title by using a TV remote control? Both Netflix and Amazon Video provide an interface (a virtual keyboard) that enables you to use the four arrow keys of a standard remote control to type letters. The letters are arranged in a regular

Given a rectangular grid with unit spacing, what is the expected distance between two random vertices, where distance is measured in the L1 metric? (Here "random" means "uniformly at random.") I recently needed this answer for some small grids, such as the one to the right, which is a 7 x 6

Continued fractions show up in surprising places. They are used in the numerical approximations of certain functions, including the evaluation of the normal cumulative distribution function (normal CDF) for large values of x (El-bolkiny, 1995, p. 75-77) and in approximating the Lambert W function, which has applications in the modeling

In the SAS/IML language, you can only concatenate vectors that have conforming dimensions. For example, to horizontally concatenate two vectors X and Y, the symbols X and Y must have the same number of rows. If not, the statement Z = X || Y will produce an error: ERROR: Matrices

A SAS programmer recently asked me how to compute a kernel regression in SAS. He had read my blog posts "What is loess regression" and "Loess regression in SAS/IML" and was trying to implement a kernel regression in SAS/IML as part of a larger analysis. This article explains how to

A frequent topic on SAS discussion forums is how to check the assumptions of an ordinary least squares linear regression model. Some posts indicate misconceptions about the assumptions of linear regression. In particular, I see incorrect statements such as the following: Help! A histogram of my variables shows that they

A SAS programmer recently asked how to interpret the "standardized regression coefficients" as computed by the STB option on the MODEL statement in PROC REG and other SAS regression procedures. The SAS documentation for the STB option states, "a standardized regression coefficient is computed by dividing a parameter estimate by

Video killed the radio star.... We can't rewind, we've gone too far. -- The Buggles (1979) "You kids have it easy," my father used to tell me. "When I was a kid, I didn't have all the conveniences you have today." He's right, and I could say the same

This article shows how to perform an optimization in SAS when the parameters are restricted by nonlinear constraints. In particular, it solves an optimization problem where the parameters are constrained to lie in the annular region between two circles. The end of the article shows the path of partial solutions

My colleague, Robert Allison, recently published an interesting visualization of the relationship between chess ratings and age. His post was inspired by the article "Age vs Elo — Your battle against time," which was published on the chess.com website. ("Elo" is one of the rating systems in chess.) Robert Allison's

The SGPLOT procedure in SAS makes it easy to create graphs that overlay various groups in the data. Many statements support the GROUP= option, which specifies that the graph should overlay group information. For example, you can create side-by-side bar charts and box plots, and you can overlay multiple scatter

This article shows how to score (evaluate) a quantile regression model on new data. SAS supports several procedures for quantile regression, including the QUANTREG, QUANTSELECT, and HPQUANTSELECT procedures. The first two procedures do not support any of the modern methods for scoring regression models, so you must use the "missing

When you use a regression procedure in SAS that supports variable selection (GLMSELECT or QUANTSELECT), did you know that the procedures automatically produce a macro variable that contains the names of the selected variables? This article provides examples and details. A previous article provides an overview of the 'SELECT' procedures

A programmer recently asked a question on a SAS discussion forum about design matrices for categorical variables. He had generated a design matrix by using PROC GLMMOD and wanted to use the design columns in a subsequent procedure. However, the columns were named COL1, COL2, COL3,..., so he couldn't tell

Back in SAS 9.3M2 (SAS/STAT 12.1), PROC FREQ introduced mosaic plots to visualize the joint frequencies in a contingency table. By default, the cells in a mosaic plot are colored according to levels of one of the categorical variables in the analysis. However, in 2013 I showed how you can

Since the late 1990s, SAS has supplied macros for basic bootstrap and jackknife analyses. This article provides an example that shows how to use the %BOOT and %BOOTCI macros. The %BOOT macro generates a bootstrap distribution and computes basic statistics about the bootstrap distribution, including estimates of bias, standard error,

This article shows how to implement balanced bootstrap sampling in SAS. The basic bootstrap samples with replacement from the original data (N observations) to obtain B new samples. This is called "uniform" resampling because each observation has a uniform probability of 1/N of being selected at each step of the

My colleague Robert Allison recently blogged about using the diameter of Texas as a unit of measurement. The largest distance across Texas is about 801 miles, so Robert wanted to find the set of all points such that the distance from the point to Texas is less than or equal

In a previous article, I showed how to find the intersection (if it exists) between two line segments in the plane. There are some fun problems in probability theory that involve intersections of line segments. One is "What is the probability that two randomly chosen chords of a circle intersect?"

Back in high school, you probably learned to find the intersection of two lines in the plane. The intersection requires solving a system of two linear equations. There are three cases: (1) the lines intersect in a unique point, (2) the lines are parallel and do not intersect, or (3)

SAS enables you to evaluate a regression model at any location within the range of the data. However, sometimes you might be interested in how the predicted response is increasing or decreasing at specified locations. You can use finite differences to compute the slope (first derivative) of a regression model.

Which president of the United States is ranked the greatest by presidential historians? This article visualizes the results of the 2018 Presidential Greatness Survey, which was created and administered by B. Rottinghaus and J. Vaughn. They analyzed 166 responses from experts in political science who ranked the 44 US presidents

This article describes how to obtain an initial guess for nonlinear regression models, especially nonlinear mixed models. The technique is to first fit a simpler fixed-effects model by replacing the random effects with their expected values. The parameter estimates for the fixed-effects model are often good initial guesses for the

When you fit nonlinear fixed-effect or mixed models, it is difficult to guess the model parameters that fit the data. Yet, most nonlinear regression procedures (such as PROC NLIN and PROC NLMIXED in SAS) require that you provide a good guess! If your guess is not good, the fitting algorithm,

A previous article provides an example of using the BOOTSTRAP statement in PROC TTEST to compute bootstrap estimates of statistics in a two-sample t test. The BOOTSTRAP statement is new in SAS/STAT 14.3 (SAS 9.4M5). However, you can perform the same bootstrap analysis in earlier releases of SAS by using

Bootstrap resampling is a powerful way to estimate the standard error for a statistic without making any parametric assumptions about its sampling distribution. The bootstrap method is often implemented by using a sequence of calls to resample from the data, compute a statistic on each sample, and analyze the bootstrap

My 2018 SAS Global Forum paper was about "how to use the random-number generators (RNGs) in SAS." You can read the paper for details, but I recently recorded a short video that summarizes the main ideas in the paper. In particular, the video gives an overview of the new RNGs