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Analytics | Data Visualization | Programming Tips

Rick WicklinJuly 19, 2021 1

Copulas and multivariate distributions with normal marginals

After my recent articles on simulating data by using copulas, many readers commented about the power of copulas. Yes, they are powerful, and the geometry of copulas is beautiful. However, it is important to be aware of the limitations of copulas. This article creates a bizarre example of bivariate data,

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Analytics | Programming Tips

Rick WicklinJuly 14, 2021 0

Compare computational methods for least squares regression

In a previous article, I discussed various ways to solve a least-square linear regression model. I discussed the SWEEP operator (used by many SAS regression routines), the LU-based methods (SOLVE and INV in SAS/IML), and the QR decomposition (CALL QR in SAS/IML). Each method computes the estimates for the regression

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Analytics | Learn SAS

Rick WicklinJuly 12, 2021 1

The QR algorithm for least-squares regression

In computational statistics, there are often several ways to solve the same problem. For example, there are many ways to solve for the least-squares solution of a linear regression model. A SAS programmer recently mentioned that some open-source software uses the QR algorithm to solve least-squares regression problems and asked

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Analytics | Programming Tips

Rick WicklinJuly 7, 2021 10

Simulate multivariate correlated data by using PROC COPULA in SAS

In general, it is hard to simulate multivariate data that has a specified correlation structure. Copulas make that task easier for continuous distributions. A previous article presented the geometry behind a copula and explained copulas in an intuitive way. Although I strongly believe that statistical practitioners should be familiar with

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Analytics | Programming Tips

Rick WicklinJune 30, 2021 0

Compute 2-D cumulative sums and ogives

A recent article about how to estimate a two-dimensional distribution function in SAS inspired me to think about a related computation: a 2-D cumulative sum. Suppose you have numbers in a matrix, X. A 2-D cumulative sum is a second matrix, C, such that the C[p,q] gives the sum of

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Analytics | Learn SAS

Rick WicklinJune 28, 2021 16

Estimate a bivariate CDF in SAS

This article shows how to estimate and visualize a two-dimensional cumulative distribution function (CDF) in SAS. SAS has built-in support for this computation. Although the bivariate CDF is not used as much as the univariate CDF, the bivariate version is still a useful tool in understanding the probable values of

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Analytics | Data Visualization

Rick WicklinJune 16, 2021 3

The geometry of the Iman-Conover transformation

A previous article showed how to simulate multivariate correlated data by using the Iman-Conover transformation (Iman and Conover, 1982). The transformation preserves the marginal distributions of the original data but permutes the values (columnwise) to induce a new correlation among the variables. When I first read about the Iman-Conover transformation,

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Analytics | Programming Tips

Rick WicklinJune 14, 2021 5

Simulate correlated variables by using the Iman-Conover transformation

Simulating univariate data is relatively easy. Simulating multivariate data is much harder. The main difficulty is to generate variables that have given univariate distributions but also are correlated with each other according to a specified correlation matrix. However, Iman and Conover (1982, "A distribution-free approach to inducing rank correlation among

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Analytics | Learn SAS

Rick WicklinJune 9, 2021 0

Rank-based scores and tied values

Many nonparametric statistical methods use the ranks of observations to compute distribution-free statistics. In SAS, two procedures that use ranks are PROC NPAR1WAY and PROC CORR. Whereas the SPEARMAN option in PROC CORR (which computes rank correlation) uses only the "raw" tied ranks, PROC NPAR1WAY uses transformations of the ranks,

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Analytics | Programming Tips

Rick WicklinJune 7, 2021 1

Permutation tests and independent sorting of data

For many univariate statistics (mean, median, standard deviation, etc.), the order of the data is unimportant. If you sort univariate data, the mean and standard deviation do not change. However, you cannot sort an individual variable (independently) if you want to preserve its relationship with other variables. This statement is

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