## Tag: linear regression

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Add loess smoothers to residual plots

When fitting a least squares regression model to data, it is often useful to create diagnostic plots of the residuals versus the explanatory variables. If the model fits the data well, the plots of the residuals should not display any patterns. Systematic patterns can indicate that you need to include

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Influential observations in a linear regression model: The DFFITS and Cook's D statistics

A previous article describes the DFBETAS statistics for detecting influential observations, where "influential" means that if you delete the observation and refit the model, the estimates for the regression coefficients change substantially. Of course, there are other statistics that you could use to measure influence. Two popular ones are the

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Influential observations in a linear regression model: The DFBETAS statistics

My article about deletion diagnostics investigated how influential an observation is to a least squares regression model. In other words, if you delete the i_th observation and refit the model, what happens to the statistics for the model? SAS regression procedures provide many tables and graphs that enable you to

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Leave-one-out statistics and a formula to update a matrix inverse

For linear regression models, there is a class of statistics that I call deletion diagnostics or leave-one-out statistics. These observation-wise statistics address the question, "If I delete the i_th observation and refit the model, what happens to the statistics for the model?" For example: The PRESS statistic is similar to

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The Theil-Sen robust estimator for simple linear regression

Modern statistical software provides many options for computing robust statistics. For example, SAS can compute robust univariate statistics by using PROC UNIVARIATE, robust linear regression by using PROC ROBUSTREG, and robust multivariate statistics such as robust principal component analysis. Much of the research on robust regression was conducted in the

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4 reasons to use PROC PLM for linear regression models in SAS

Have you ever run a regression model in SAS but later realize that you forgot to specify an important option or run some statistical test? Or maybe you intended to generate a graph that visualizes the model, but you forgot? Years ago, your only option was to modify your program

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Deming regression for comparing different measurement methods

Deming regression (also called errors-in-variables regression) is a total regression method that fits a regression line when the measurements of both the explanatory variable (X) and the response variable (Y) are assumed to be subject to normally distributed errors. Recall that in ordinary least squares regression, the explanatory variable (X)

Programming Tips
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Singular parameterizations, generalized inverses, and regression estimates

I remember the first time I used PROC GLM in SAS to include a classification effect in a regression model. I thought I had done something wrong because the parameter estimates table was followed by a scary-looking note: Note: The X'X matrix has been found to be singular, and a

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Generalized inverses for matrices

A data analyst asked how to compute parameter estimates in a linear regression model when the underlying data matrix is rank deficient. This situation can occur if one of the variables in the regression is a linear combination of other variables. It also occurs when you use the GLM parameterization

Programming Tips
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On the assumptions (and misconceptions) of linear regression

A frequent topic on SAS discussion forums is how to check the assumptions of an ordinary least squares linear regression model. Some posts indicate misconceptions about the assumptions of linear regression. In particular, I see incorrect statements such as the following: Help! A histogram of my variables shows that they

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An easier way to run thousands of regressions

SAS programmers on SAS discussion forums sometimes ask how to run thousands of regressions of the form Y = B0 + B1*X_i, where i=1,2,.... A similar question asks how to solve thousands of regressions of the form Y_i = B0 + B1*X for thousands of response variables. I have previously

Programming Tips
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The sweep operator: A fundamental operation in regression

The sweep operator performs elementary row operations on a system of linear equations. The sweep operator enables you to build regression models by "sweeping in" or "sweeping out" particular rows of the X`X matrix. As you do so, the estimates for the regression coefficients, the error sum of squares, and

Analytics
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Should you use principal component regression?

This article describes the advantages and disadvantages of principal component regression (PCR). This article also presents alternative techniques to PCR. In a previous article, I showed how to compute a principal component regression in SAS. Recall that principal component regression is a technique for handling near collinearities among the regression

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Principal component regression in SAS

A common question on discussion forums is how to compute a principal component regression in SAS. One reason people give for wanting to run a principal component regression is that the explanatory variables in the model are highly correlated which each other, a condition known as multicollinearity. Although principal component

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SAS Viyaで線形回帰

SAS Viyaで線形回帰を行う方法を紹介します。 言語はPythonを使います。 SAS Viyaで線形回帰を行う方法には大きく以下の手法が用意されています。 多項回帰：　simpleアクションセットで提供。 一般化線形回帰または一般線形回帰：　regressionアクションセットで提供。 機械学習で回帰：　各種機械学習用のアクションセットで提供。 今回は単純なサインカーブを利用して、上記3種類の回帰モデルを作ってみます。   【サインカーブ】 -4≦x<4の範囲でサインカーブを作ります。 普通に \$\$y = sin(x) \$\$を算出しても面白みがないので、乱数を加減して以下のようなデータを作りました。これをトレーニングデータとします。 青い点線が \$\$y=sin(x)\$\$ の曲線、グレーの円は \$\$y=sin(x)\$\$ に乱数を加減したプロットです。 グレーのプロットの中心を青い点線が通っていることがわかります。 今回はグレーのプロットをトレーニングデータとして線形回帰を行います。グレーのプロットはだいぶ散らばって見えますが、回帰モデルとしては青い点線のように中心を通った曲線が描けるはずです。 トレーニングデータのデータセット名は "sinx" とします。説明変数は "x"、ターゲット変数は "y" になります。 各手法で生成したモデルで回帰を行うため、-4≦x<4 の範囲で0.01刻みで"x" の値をとった "rangex" というデータセットも用意します。 まずはCASセッションを生成し、それぞれのデータをCASにアップロードします。 import swat host = "localhost" port = 5570 user = "cas" password = "p@ssw0rd"

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Visualize a design matrix

Most SAS regression procedures support a CLASS statement which internally generates dummy variables for categorical variables. I have previously described what dummy variables are and how are they used. I have also written about how to create design matrices that contain dummy variables in SAS, and in particular how to

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An easy way to run thousands of regressions in SAS

A common question on SAS discussion forums is how to repeat an analysis multiple times. Most programmers know that the most efficient way to analyze one model across many subsets of the data (perhaps each country or each state) is to sort the data and use a BY statement to

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Simulate many samples from a linear regression model

In a previous article, I showed how to simulate data for a linear regression model with an arbitrary number of continuous explanatory variables. To keep the discussion simple, I simulated a single sample with N observations and p variables. However, to use Monte Carlo methods to approximate the sampling distribution

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Simulate data for a linear regression model

This article shows how to simulate a data set in SAS that satisfies a least squares regression model for continuous variables. When you simulate to create "synthetic" (or "fake") data, you (the programmer) control the true parameter values, the form of the model, the sample size, and magnitude of the

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Dummy variables in SAS/IML

Last week I showed how to create dummy variables in SAS by using the GLMMOD procedure. The procedure enables you to create design matrices that encode continuous variables, categorical variables, and their interactions. You can use dummy variables to replace categorical variables in procedures that do not support a CLASS

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Four ways to create a design matrix in SAS

SAS programmers sometimes ask, "How do I create a design matrix in SAS?" A design matrix is a numerical matrix that represents the explanatory variables in regression models. In simple models, the design matrix contains one column for each continuous variable and multiple columns (called dummy variables) for each classification

Learn SAS
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Create dummy variables in SAS

A dummy variable (also known as indicator variable) is a numeric variable that indicates the presence or absence of some level of a categorical variable. The word "dummy" does not imply that these variables are not smart. Rather, dummy variables serve as a substitute or a proxy for a categorical

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How to interpret a residual-fit spread plot

In a previous blog post, I described how to use a spread plot to compare the distributions of several variables. Each spread plot is a graph of centered data values plotted against the estimated cumulative probability. Thus, spread plots are similar to a (rotated) plot of the empirical cumulative distribution