Longitudinal data are used in many health-related studies in which individuals are measured at multiple points in time to monitor changes in a response variable, such as weight, cholesterol, or blood pressure. There are many excellent articles and books that describe the advantages of a mixed model for analyzing longitudinal

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This article discusses how to restrict a multivariate function to a linear subspace. This is a useful technique in many situations, including visualizing an objective function that is constrained by linear equalities. For example, the graph to the right is from a previous article about how to evaluate quadratic polynomials.

What is an efficient way to evaluate a multivariate quadratic polynomial in p variables? The answer is to use matrix computations! A multivariate quadratic polynomial can be written as the sum of a purely quadratic term (degree 2), a purely linear term (degree 1), and a constant term (degree 0).

In a linear regression model, the predicted values are on the same scale as the response variable. You can plot the observed and predicted responses to visualize how well the model agrees with the data, However, for generalized linear models, there is a potential source of confusion. Recall that a

My colleague, Mike Drutar, recently showed how to create a "strip plot" that shows the distribution of temperatures for each calendar month at a particular location. Mike created the strip plot in SAS Visual Analytics by using a point-and-click interface. This article shows how to create a similar graph by

Biplots are two-dimensional plots that help to visualize relationships in high dimensional data. A previous article discusses how to interpret biplots for continuous variables. The biplot projects observations and variables onto the span of the first two principal components. The observations are plotted as markers; the variables are plotted as

In grade school, students learn how to round numbers to the nearest integer. In later years, students learn variations, such as rounding up and rounding down by using the greatest integer function and least integer function, respectively. My sister, who is an engineer, learned a rounding method that rounds half-integers

Principal component analysis (PCA) is an important tool for understanding relationships in continuous multivariate data. When the first two principal components (PCs) explain a significant portion of the variance in the data, you can visualize the data by projecting the observations onto the span of the first two PCs. In

Understanding multivariate statistics requires mastery of high-dimensional geometry and concepts in linear algebra such as matrix factorizations, basis vectors, and linear subspaces. Graphs can help to summarize what a multivariate analysis is telling us about the data. This article looks at four graphs that are often part of a principal

Every year at Halloween, I post an article that shows a SAS trick that is a real treat. This article shows how to use the INTNX function to find dates that are related to a specified date. The INTNX function is a sweet treat, indeed. I previously wrote an article

A common task in SAS programming is to specify a list of variables that satisfy some pattern. You can specify lists for the KEEP= or DROP= data set options, and you can use lists of variables on many SAS statements such as the VAR and MODEL statements. Although SAS has

In response to a recent article about how to compute the cosine similarity of observations, a reader asked whether it is practical (or even possible) to perform these types of computations on data sets that have many thousands of observations. The problem is that the cosine similarity matrix is an

Computing rates and proportions is a common task in data analysis. When you are computing several proportions, it is helpful to visualize how the rates vary among subgroups of the population. Examples of proportions that depend on subgroups include: Mortality rates for various types of cancers Incarceration rates by race

The EFFECT statement is supported by more than a dozen SAS/STAT regression procedures. Among other things, it enables you to generate spline effects that you can use to fit nonlinear relationships in data. Recently there was a discussion on the SAS Support Communities about how to interpret the parameter estimates

I recently wrote about how to use PROC TTEST in SAS/STAT software to compute the geometric mean and related statistics. This prompted a SAS programmer to ask a related question. Suppose you have dozens (or hundreds) of variables and you want to compute the geometric mean of each. What is

In a previous article, I mentioned that the VLINE statement in PROC SGPLOT is an easy way to graph the mean response at a set of discrete time points. I mentioned that you can choose three options for the length of the "error bars": the standard deviation of the data,

It is always great to read an old paper or blog post and think, "This task is so much easier in SAS 9.4!" I had that thought recently when I stumbled on a 2007 paper by Wei Cheng titled "Graphical Representation of Mean Measurement over Time." A substantial portion of

I frequently see questions on SAS discussion forums about how to compute the geometric mean and related quantities in SAS. Unfortunately, the answers to these questions are sometimes confusing or even wrong. In addition, some published papers and web sites that claim to show how to calculate the geometric mean

There are several different kinds of means. They all try to find an average value from among a set of numbers. Although the most popular mean is the arithmetic mean, the geometric mean can be useful for problems in statistics, finance, and biology. A common application of the geometric mean

One of the strengths of the SAS/IML language is its flexibility. Recently, a SAS programmer asked how to generalize a program in a previous article. The original program solved one optimization problem. The reader said that she wants to solve this type of problem 300 times, each time using a

Although I do not typically blog about undocumented SAS options, I'll make an exception this time. For many years, I have known that the CONTENTS and COMPARE procedures support the BRIEF and SHORT options, but I always forget which option goes with which procedure. For the record, here are the

In a scatter plot that displays many points, it can be important to visualize the density of the points. Scatter plots (indeed, all plots that show individual markers) can suffer from overplotting, which means that the graph does not indicate how many observations are at a specific (x, y) location.

A moving average is a statistical technique that is used to smooth a time series. My colleague, Cindy Wang, wrote an article about the Hull moving average (HMA), which is a time series smoother that is sometimes used as a technical indicator by stock market traders. Cindy showed how to

I often use axis tables in PROC SGPLOT in SAS to add a table of text to a graph so that the table values are aligned with the data. But axis tables are not the only way to display tabular data in a graph. You can also use the TEXT

The TEXT statement in PROC SGPLOT supports the ROTATE= option to rotate the specified text. It is worth knowing how the ROTATE= option interacts with the POSITION= option, which determines the anchor point at which the text is positioned. Briefly, the text is positioned FIRST, then the rotation occurs. The

When you order an item online, the website often recommends other items based on your purchase. In fact, these kinds of "recommendation engines" contributed to the early success of companies like Amazon and Netflix. SAS uses a recommender engine to suggest articles on the SAS Support Communities. Although recommender engines

An important application of the dot product (inner product) of two vectors is to determine the angle between the vectors. If u and v are two vectors, then cos(θ) = (u ⋅ v) / (|u| |v|) You could apply the inverse cosine function if you wanted to find θ in

A SAS programmer asked an intriguing question on the SAS Support Communities: Can you use SAS to create a graph that shows how the elements in a box-and-whiskers plot relate to the data? The SAS documentation has several examples that explain how to read a box plot. One of the

Most SAS programmers know how to use PROC APPEND or the SET statement in DATA step to unconditionally append new observations to an existing data set. However, sometimes you need to scan the data to determine whether or not to append observations. In this situation, many SAS programmers choose one

An important application of nonlinear optimization is finding parameters of a model that fit data. For some models, the parameters are constrained by the data. A canonical example is the maximum likelihood estimation of a so-called "threshold parameter" for the three-parameter lognormal distribution. For this distribution, the objective function is