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Rick WicklinAugust 16, 2023 0

Add Unicode symbols and format text labels in SAS

It has been more than a decade since SAS 9.3 changed the default ODS destination from the old LISTING destination to more modern destinations such as HTML. One of the advantages of modern output destinations is support for Unicode symbols, superscripts, subscripts, and for formatting text by using boldface, italics,

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Rick WicklinAugust 14, 2023 2

Bootstrap confidence intervals for the predicted mean in a regression model

In ordinary least squares regression, there is an explicit formula for the confidence limit of the predicted mean. That is, for any observed value of the explanatory variables, you can create a 95% confidence interval (CI) for the predicted response. This formula assumes that the model is correctly specified and

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Rick WicklinAugust 9, 2023 1

Plot confidence intervals for a regression model in SAS

A SAS programmer wanted to use PROC SGPLOT in SAS to visualize a regression model. The programmer wanted to visualize confidence limits for the predicted mean at certain values of the explanatory variable. This article shows two options for adding confidence limits to a scatter plot. You can use a

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Rick WicklinAugust 7, 2023 0

Construct an envelope function for the acceptance-rejection method

The acceptance-rejection method (sometimes called rejection sampling) is a method that enables you to generate a random sample from an arbitrary distribution by using only the probability density function (PDF). This is in contrast to the inverse CDF method, which uses the cumulative distribution function (CDF) to generate a random

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Rick WicklinJuly 31, 2023 0

Create a probability distribution from almost any positive function

There are dozens of common probability distributions for a continuous univariate random variable. Familiar examples include the normal, exponential, uniform, gamma, and beta distributions. Where did these distributions come from? Well, some mathematician needed a model for a stochastic process and wrote down the equation for the distribution, typically by

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Programming Tips

Rick WicklinJuly 26, 2023 0

The trace of a matrix of crossproducts

Let X be any rectangular matrix. What is the trace of the crossproducts matrix, X'*X? Interestingly, you do not need to form the crossproducts matrix to compute the answer! It turns out that tr(X'*X) equals the sum of the squared elements of X. Theorem: For any matrix, X, the trace

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Rick WicklinJuly 24, 2023 4

Modifications of the Wilcoxon signed rank test and exact p-values

In a previous article, I discussed the Wilcoxon signed rank test, which is a nonparametric test for the location of the median. The Wikipedia article about the signed rank test mentions a variation of the test due to Pratt (1959). Whereas the standard Wilcoxon test excludes values that equal μ0

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Rick WicklinJuly 19, 2023 1

On the computation of the Wilcoxon signed rank statistic

Wilcoxon's signed rank test is a popular nonparametric alternative to a paired t test. In a paired t test, you analyze measurements for subjects before and after some treatment or intervention. You analyze the difference in the measurements for each subject, and test whether the mean difference is significantly different

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Rick WicklinJuly 10, 2023 2

Simulate from a Markov model

A previous article shows an example of a Markov chain model and computes the probability that the system ends up in a terminal state (called an absorbing state). As explained previously, you can often compute exact probabilities for questions about Markov chains. Nevertheless, it can be useful to know how

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Rick WicklinJuly 5, 2023 1

The probability of reaching a terminal state in a Markov chain

A previous article shows how to model the probabilities in a discrete-time Markov chain by using a Markov transition matrix. A Markov chain is a discrete-time stochastic process for which the current state of the system determines the probability of the next state. In this process, the probabilities for transitioning

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