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Rick WicklinJune 29, 2022 5

Compute the multivariate t density function

A previous article shows how to compute the probability density function (PDF) for the multivariate normal distribution. In a similar way, you can compute the density function for the multivariate t distribution. This article discusses the density function for the multivariate t distribution, shows how to compute it, and visualizes

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Analytics | Programming Tips

Rick WicklinJune 27, 2022 0

The derivative of a quantile function

Recently, I needed to solve an optimization problem in which the objective function included a term that involved the quantile function (inverse CDF) of the t distribution, which is shown to the right for DF=5 degrees of freedom. I casually remarked to my colleague that the optimizer would have to

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Analytics | Data Visualization | Learn SAS

Rick WicklinJune 20, 2022 0

Partial leverage plots

For a linear regression model, a useful but underutilized diagnostic tool is the partial regression leverage plot. Also called the partial regression plot, this plot visualizes the parameter estimates table for the regression. For each effect in the model, you can visualize the following statistics: The estimate for each regression

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Rick WicklinJune 8, 2022 0

The effect of weight functions in a robust regression method

M estimation is a robust regression technique that assigns a weight to each observation based on the magnitude of the residual for that observation. Large residuals are downweighted (assigned weights less than 1) whereas observations with small residuals are given weights close to 1. By iterating the reweighting and fitting

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Rick WicklinJune 6, 2022 1

Weights for residuals in robust regression

An early method for robust regression was iteratively reweighted least-squares regression (Huber, 1964). This is an iterative procedure in which each observation is assigned a weight. Initially, all weights are 1. The method fits a least-squares model to the weighted data and uses the size of the residuals to determine

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Analytics | Programming Tips

Rick WicklinMay 25, 2022 0

How much does a bootstrap estimate depend on the random number stream?

Many modern statistical techniques incorporate randomness: simulation, bootstrapping, random forests, and so forth. To use the technique, you need to specify a seed value, which determines pseudorandom numbers that are used in the algorithm. Consequently, the seed value also determines the results of the algorithm. In theory, if you know

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Rick WicklinMay 23, 2022 4

The balanced bootstrap in SAS

I have previously blogged about ways to perform balanced bootstrap resampling in SAS. I recently learned about an easier way: Since SAS/STAT 14.2 (SAS 9.4M4), the SURVEYSELECT procedure has supported balanced bootstrap sampling. This article reviews balanced bootstrap sampling and shows how to use the METHOD=BALBOOT option in PROC SURVEYSELECT

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Rick WicklinMay 9, 2022 2

The derivative of the determinant of a matrix

Did you know that there is a mathematical formula that simplifies finding the derivative of a determinant? You can compute the derivative of a determinant of an n x n matrix by using the sum of n other determinants. The n determinants are for matrices that are equal to the original matrix

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Analytics | Learn SAS | Programming Tips

Rick WicklinMay 2, 2022 0

Simulate the null distribution for a hypothesis test

Recently, I wrote about Bartlett's test for sphericity. The purpose of this hypothesis test is to determine whether the variables in the data are uncorrelated. It works by testing whether the sample correlation matrix is close to the identity matrix. Often statistics textbooks or articles include a statement such as

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Rick WicklinApril 27, 2022 0

On Bartlett's sphericity test for correlation

When you have many correlated variables, principal component analysis (PCA) is a classical technique to reduce the dimensionality of the problem. The PCA finds a smaller dimensional linear subspace that explains most of the variability in the data. There are many statistical tools that help you decide how many principal

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