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
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
In The Essential Guide to Bootstrapping in SAS, I note that there are many SAS procedures that support bootstrap estimates without requiring the analyst to write a program. I have previously written about using bootstrap options in the TTEST procedure. This article discusses the NLIN procedure, which can fit nonlinear
This article implements Passing-Bablok regression in SAS. Passing-Bablok regression is a one-variable regression technique that is used to compare measurements from different instruments or medical devices. The measurements of the two variables (X and Y) are both measured with errors. Consequently, you cannot use ordinary linear regression, which assumes that
On this blog, I write about a diverse set of topics that are relevant to statistical programming and data visualization. In a previous article, I presented some of the most popular blog posts from 2021. The most popular articles often deal with elementary or familiar topics that are useful to
It can be frustrating when the same probability distribution has two different parameterizations, but such is the life of a statistical programmer. I previously wrote an article about the gamma distribution, which has two common parameterizations: one that uses a scale parameter (β) and another that uses a rate parameter
I previously wrote about how to understand standardized regression coefficients in PROC REG in SAS. You can obtain the standardized estimates by using the STDB option on the MODEL statement in PROC REG. Several readers have written to ask whether I could write a similar article about the STDCOEF option
I recently learned about a new feature in PROC QUANTREG that was added in SAS/STAT 15.1 (part of SAS 9.4M6). Recall that PROC QUANTREG enables you to perform quantile regression in SAS. (If you are not familiar with quantile regression, see an earlier article that describes quantile regression and provides
A segmented regression model is a piecewise regression model that has two or more sub-models, each defined on a separate domain for the explanatory variables. For simplicity, assume the model has one continuous explanatory variable, X. The simplest segmented regression model assumes that the response is modeled by one parametric
This resource is designed primarily for beginner to intermediate data scientists or analysts who are interested in identifying and applying machine learning algorithms to address the problems of their interest. A typical question asked by a beginner, when facing a wide variety of machine learning algorithms, is “which algorithm should
A SAS customer asked a great question: "I have parameter estimates for a logistic regression model that I computed by using multiple imputations. How do I use these parameter estimates to score new observations and to visualize the model? PROC LOGISTIC can do the computation I want, but how do
A previous article discussed how to solve regression problems in which the parameters are constrained to be a specified constant (such as B1 = 1) or are restricted to obey a linear equation such as B4 = –2*B2. In SAS, you can use the RESTRICT statement in PROC REG to
When you write a program that simulates data from a statistical model, you should always check that the simulation code is correct. One way to do this is to generate a large simulated sample, estimate the parameters in the simulated data, and make sure that the estimates are close to
A SAS customer asked how to specify interaction effects between a classification variable and a spline effect in a SAS regression procedure. There are at least two ways to do this. If the SAS procedure supports the EFFECT statement, you can build the interaction term in the MODEL statement. For
This article shows how to find local maxima and maxima on a regression curve, which means finding points where the slope of the curve is zero. An example appears at the right, which shows locations where the loess smoother in a scatter plot has local minima and maxima. Except for
Did you add "learn something new" to your list of New Year's resolutions? Last week, I wrote about the most popular articles from The DO Loop in 2019. The most popular articles are about elementary topics in SAS programming or univariate statistics because those topics have broad appeal. Advanced topics
Many SAS procedures can automatically create a graph that overlays multiple prediction curves and their prediction limits. This graph (sometimes called a "fit plot" or a "sliced fit plot") is useful when you want to visualize a model in which a continuous response variable depends on one continuous explanatory variable
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
SAS/STAT software contains a number of so-called HP procedures for training and evaluating predictive models. ("HP" stands for "high performance.") A popular HP procedure is HPLOGISTIC, which enables you to fit logistic models on Big Data. A goal of the HP procedures is to fit models quickly. Inferential statistics such
Knowing how to visualize a regression model is a valuable skill. A good visualization can help you to interpret a model and understand how its predictions depend on explanatory factors in the model. Visualization is especially important in understanding interactions between factors. Recently I read about work by Jacob A.
Here's a simulation tip: When you simulate a fixed-effect generalized linear regression model, don't add a random normal error to the linear predictor. Only the response variable should be random. This tip applies to models that apply a link function to a linear predictor, including logistic regression, Poisson regression, and
SAS regression procedures support several parameterizations of classification variables. When a categorical variable is used as an explanatory variable in a regression model, the procedure generates dummy variables that are used to construct a design matrix for the model. The process of forming columns in a design matrix is called
I've previously written about how to deal with nonconvergence when fitting generalized linear regression models. Most generalized linear and mixed models use an iterative optimization process, such as maximum likelihood estimation, to fit parameters. The optimization might not converge, either because the initial guess is poor or because the model
An analyst was using SAS to analyze some data from an experiment. He noticed that the response variable is always positive (such as volume, size, or weight), but his statistical model predicts some negative responses. He posted the data and asked if it is possible to modify the graph so
Maybe if we think and wish and hope and pray It might come true. Oh, wouldn't it be nice? The Beach Boys Months ago, I wrote about how to use the EFFECT statement in SAS to perform regression with restricted cubic splines. This is the modern way to use splines
When you use maximum likelihood estimation (MLE) to find the parameter estimates in a generalized linear regression model, the Hessian matrix at the optimal solution is very important. The Hessian matrix indicates the local shape of the log-likelihood surface near the optimal value. You can use the Hessian to estimate
I previously discussed how you can use validation data to choose between a set of competing regression models. In that article, I manually evaluated seven models for a continuous response on the training data and manually chose the model that gave the best predictions for the validation data. Fortunately, SAS
This article shows how to use SAS to simulate data that fits a linear regression model that has categorical regressors (also called explanatory or CLASS variables). Simulating data is a useful skill for both researchers and statistical programmers. You can use simulation for answering research questions, but you can also
Recently I was asked to explain the result of an ANOVA analysis that I posted to a statistical discussion forum. My program included some simulated data for an ANOVA model and a call to the GLM procedure to estimate the parameters. I was asked why the parameter estimates from PROC
If you want to bootstrap the parameters in a statistical regression model, you have two primary choices. The first, case resampling, is discussed in a previous article. This article describes the second choice, which is resampling residuals (also called model-based resampling). This article shows how to implement residual resampling in
