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.
When you perform a linear regression, you can examine the R-square value, which is a goodness-of-fit statistic that indicates how well the response variable can be represented as a linear combination of the explanatory variables. But did you know that you can also go the other direction? Given a set
Do you know that you can create a vector that has a specific correlation with another vector? That is, given a vector, x, and a correlation coefficient, ρ, you can find a vector, y, such that corr(x, y) = ρ. The vectors x and y can have an arbitrary number
A segmented regression model is a piecewise regression model that has two or more sub-models, each defined on a separate domain for the explanatory variables. For simplicity, assume the model has one continuous explanatory variable, X. The simplest segmented regression model assumes that the response is modeled by one parametric
One purpose of principal component analysis (PCA) is to reduce the number of important variables in a data analysis. Thus, PCA is known as a dimension-reduction algorithm. I have written about four simple rules for deciding how many principal components (PCs) to keep. There are other methods for deciding how
"O Christmas tree, O Christmas tree, how lovely are your branches!" The idealized image of a Christmas tree is a perfectly straight conical tree with lush branches and no bare spots. Although this ideal exists only on Christmas cards, forest researchers are always trying to develop trees that approach the
A SAS customer asked a great question: "I have parameter estimates for a logistic regression model that I computed by using multiple imputations. How do I use these parameter estimates to score new observations and to visualize the model? PROC LOGISTIC can do the computation I want, but how do
Most games of skill are transitive. If Player A wins against Player B and Player B wins against Player C, then you expect Player A to win against Player C, should they play. Because of this, you can rank the players: A > B > C Interestingly, not all games
I previously showed how to create a decile calibration plot for a logistic regression model in SAS. A decile calibration plot (or "decile plot," for short) is used in some fields to visualize agreement between the data and a regression model. It can be used to diagnose an incorrectly specified
To help visualize regression models, SAS provides the EFFECTPLOT statement in several regression procedures and in PROC PLM, which is a general-purpose procedure for post-fitting analysis of linear models. When scoring and visualizing a model, it is important to use reasonable combinations of the explanatory variables for the visualization. When
I have previously written about how to plot a discontinuous function in SAS. That article shows how to use the GROUP= option on the SERIES statement to graph a discontinuous function. An alternative approach is to place a missing value for the Y variable at the locations at which the
The REFLINE statement in PROC SGPLOT is one of my favorite ways to augment statistical graphics such as scatter plots, series plots, and histograms. The REFLINE statement overlays a vertical or horizontal reference line on a graph. You can specify the location of the reference lines on the REFLINE statement.
Intuitively, the skewness of a unimodal distribution indicates whether a distribution is symmetric or not. If the right tail has more mass than the left tail, the distribution is "right skewed." If the left tail has more mass, the distribution is "left skewed." Thus, estimating skewness requires some estimates about
The expected value of a random variable is essentially a weighted mean over all possible values. You can compute it by summing (or integrating) a probability-weighted quantity over all possible values of the random variable. The expected value is a measure of the "center" of a probability distribution. You can
When there are two equivalent ways to do something, I advocate choosing the one that is simpler and more efficient. Sometimes, I encounter a SAS program that simulates random numbers in a way that is neither simple nor efficient. This article demonstrates two improvements that you can make to your
The skewness of a distribution indicates whether a distribution is symmetric or not. The Wikipedia article about skewness discusses two common definitions for the sample skewness, including the definition used by SAS. In the middle of the article, you will discover the following sentence: In general, the [estimators] are both
A fundamental principle of data analysis is that a statistic is an estimate of a parameter for the population. A statistic is calculated from a random sample. This leads to uncertainty in the estimate: a different random sample would have produced a different statistic. To quantify the uncertainty, SAS procedures
The triangulation theorem for polygons says that every simple polygon can be triangulated. In fact, if the polygon has V vertices, you can decompose it into V-2 non-overlapping triangles. In this article, a "polygon" always means a simple polygon. Also, a "random point" means one that is drawn at random
How can you efficiently generate N random uniform points in a triangular region of the plane? There is a very cool algorithm (which I call the reflection method) that makes the process easy. I no longer remember where I saw this algorithm, but it is different from the "weighted average"
A previous article discusses the confidence band for the mean predicted value in a regression model. The article shows a "graded confidence band plot," which I saw in Claus O. Wilke's online book, Fundamentals of Data Visualization (Section 16.3). It communicates uncertainty in the predictions. A graded band plot is
You've probably seen many graphs that are similar to the one at the right. This plot shows a regression line overlaid on a scatter plot of some data. Given a value for the independent variable (x), the regression line gives the best prediction for the mean of the response variable
A previous article shows how to use a recursive formula to compute exact probabilities for the Poisson-binomial distribution. The recursive formula is an O(N2) computation, where N is the number of parameters for the Poisson-binomial (PB) distribution. If you have a distribution that has hundreds (or even thousands) of parameters,
Finite-precision computations can be tricky. You might know, mathematically, that a certain result must be non-negative or must be within a certain interval. However, when you actually compute that result on a computer that uses finite-precision, you might observe that the value is slightly negative or slightly outside of the
When working with a probability distribution, it is useful to know how to compute four essential quantities: a random sample, the density function, the cumulative distribution function (CDF), and quantiles. I recently discussed the Poisson-binomial distribution and showed how to generate a random sample. This article shows how to compute
The Poisson-binomial distribution is a generalization of the binomial distribution. For the binomial distribution, you carry out N independent and identical Bernoulli trials. Each trial has a probability, p, of success. The total number of successes, which can be between 0 and N, is a binomial random variable. The distribution
Many textbooks and research papers present formulas that involve recurrence relations. Familiar examples include: The factorial function: Set Fact(0)=1 and define Fact(n) = n*Fact(n-1) for n > 0. The Fibonacci numbers: Set Fib(0)=1 and Fib(1)=1 and define Fib(n) = Fib(n-1) + Fib(n-2) for n > 1. The binomial coefficients (combinations
A previous article discussed how to solve regression problems in which the parameters are constrained to be a specified constant (such as B1 = 1) or are restricted to obey a linear equation such as B4 = –2*B2. In SAS, you can use the RESTRICT statement in PROC REG to
A data analyst recently asked a question about restricted least square regression in SAS. Recall that a restricted regression puts linear constraints on the coefficients in the model. Examples include forcing a coefficient to be 1 or forcing two coefficients to equal each other. Each of these problems can be
My 2020 SAS Global Forum paper was about how to write custom parallel programs by using the iml action in SAS Viya 3.5. My conference presentation was canceled because of the coronavirus pandemic, but I recently recorded a 15-minute video that summarizes the main ideas in the paper. One of
I previously wrote about the RAS algorithm, which is a simple algorithm that performs matrix balancing. Matrix balancing refers to adjusting the cells of a frequency table to match known values of the row and column sums. Ideally, the balanced matrix will reflect the structural relationships in the original matrix.
Matrix balancing is an interesting problem that has a long history. Matrix balancing refers to adjusting the cells of a frequency table to match known values of the row and column sums. One of the early algorithms for matrix balancing is known as the RAS algorithm, but it is also
