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Rick WicklinJune 8, 2026 4

How to add confidence bands on the ECDF plot in SAS

A previous article shows how to construct an empirical cumulative distribution function (ECDF) for univariate data by using PROC UNIVARIATE or in the SAS IML language. The ECDF is a tool for visualizing the distribution of a sample and is helpful for estimating quantiles in the data. In statistics, we

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Analytics | Learn SAS | Programming Tips

Rick WicklinJune 1, 2026 1

How to visualize the variability in an ECDF

A previous article shows how to construct an empirical cumulative distribution function (ECDF) in SAS. The most common way is by using PROC UNIVARIATE, but you can also call the ECDF function in the SAS IML language. The ECDF is a tool for visualizing the distribution of a univariate sample

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Analytics | Learn SAS | Programming Tips

Rick WicklinMay 26, 2026 2

Create and graph the ECDF function in SAS

The empirical cumulative distribution function (ECDF) is an important tool in statistics. It is one of several plots you can use to visualize the shape of a data distribution. It is not used as often as the histogram, the kernel density estimate, and the box plot, but it is essential

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Analytics | Programming Tips

Rick WicklinMay 11, 2026 1

Compute multivariate normal orthant probabilities

In general, it is very difficult to compute a probability for a multivariate continuous distribution. For all continuous distributions, the probability requires solving a complicated multiple integral. For example, the probability for a bivariate normal distribution requires integrating the bivariate normal density over a two-dimensional (2-D) area. The probability for

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Rick WicklinMay 4, 2026 1

On standardizing multivariate normal probabilities

A course in elementary statistics always introduces the "Z-score." A Z-score is the result of standardizing a normally distributed random variable. By subtracting the distribution's mean and dividing by its standard deviation, you transform a general normal random variable into a standardized variable that has zero mean and unit standard

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Rick WicklinApril 27, 2026 0

Specify knots for a spline basis

A previous article discusses regression splines and how to use the EFFECT statement in SAS regression procedures to specify the location of knots for a regressor variable, X. Knots are breakpoints that partition the range of X into subintervals. The splines are defined on a set of adjacent subintervals. The

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Analytics | Data Visualization | Programming Tips

Rick WicklinApril 22, 2026 1

Visualize the placement of knots for regression splines

Splines are useful tools for fitting regression models to data. A spline replaces a single variable (call it X) with several other variables, which are a spline basis for X. When using a spline basis, the shape and location of the basis functions depend on the placement of knots. Knots

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Analytics | Learn SAS | Programming Tips

Rick WicklinApril 20, 2026 0

The sample median is a biased estimator for skewed distributions

Did you know that, for skewed distributions, the sample median is a biased estimator of the population median? You can show this in two ways. For any distribution, you can run a Monte Carlo simulation that reveals the bias. And, for some distributions (such as the exponential distribution), you can

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Analytics | Programming Tips

Rick WicklinApril 13, 2026 1

On monitoring the convergence of Monte Carlo simulations

A previous article discusses Welford's one-pass method for computing the sample mean and variance. The article mentions that a useful application of Welford's formulas is to monitor the convergence of a Monte Carlo simulation. This article shows how to implement that strategy in SAS. The Goldilocks Principle for Monte Carlo

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Analytics | Learn SAS

Geometric interpretation of the singular value decomposition (SVD) as the product of a rotation/reflection, followed by a scaling, followed by another rotation/reflection.

Rick WicklinMarch 16, 2026 4

On the nonuniqueness of low-rank matrix factors

A common task in statistics is to approximate a large data matrix by using a low-rank approximation. Low-rank approximations are used for reducing the dimension of a problem (for example, in principal component analysis), for image analysis via the singular value decomposition (SVD), and for decomposing large matrices that are

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