Modeling and Multivariate Methods - SAS

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Contents Introduction to Logistic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199 The Logistic Fit Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Logistic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Iteration History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 Whole Model Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 Lack of Fit Test (Goodness of Fit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Likelihood Ratio Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205 Logistic Fit Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Plot Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Likelihood Ratio Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Wald Tests for Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Odds Ratios (Nominal Responses Only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Inverse Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Save Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 ROC Curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Lift Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Confusion Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Example of a Nominal Logistic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Example of an Ordinal Logistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Example of a Quadratic Ordinal Logistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224 Stacking Counts in Multiple Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226
Chapter 7 Performing Logistic Regression on Nominal and Ordinal Responses 199 Introduction to Logistic Models Introduction to Logistic Models Logistic regression fits nominal Y responses to a linear model of X terms. To be more precise, it fits probabilities for the response levels using a logistic function. For two response levels, the function is: PY ( = r 1 ) = ( 1 + e – Xb ) – 1 where r 1 is the first response or equivalently: PY ( = r 1 ) log----------------------- PY ( = r 2 ) = Xb where r 1 and r 2 are the two responses For r nominal responses, where r > 2 , it fits r – 1 sets of linear model parameters of the following form: PY ( = j) log-------------------- PY ( = r) = Xb j The fitting principal of maximum likelihood means that the βs are chosen to maximize the joint probability attributed by the model to the responses that did occur. This fitting principal is equivalent to minimizing the negative log-likelihood (–LogLikelihood) as attributed by the model: n Loss = – logLikelihood = –log( Prob( ith row has the y j th response) ) i = 1 For example, consider an experiment that was performed on metal ingots prepared with different heating and soaking times. The ingots were then tested for readiness to roll. See Cox (1970). The Ingots.jmp data table in the Sample Data folder has the experimental results. The categorical variable called ready has values 1 and 0 for readiness and not readiness to roll, respectively. The Fit Model platform fits the probability of the not readiness (0) response to a logistic cumulative distribution function applied to the linear model with regressors heat and soak: 1 Probability (not ready to roll) = --------------------------------------------------------------- –( β 0 + β 1 heat + β 2 soak) 1 + e The parameters are estimated by minimizing the sum of the negative logs of the probabilities attributed to the observations by the model (maximum likelihood). To analyze this model, select Analyze > Fit Model. The ready variable is Y, the response, and heat and soak are the model effects. The count column is the Freq variable. When you click Run, iterative calculations take place. When the fitting process converges, the nominal or ordinal regression report appears. The following sections discuss the report layout and statistical tables, and show examples.
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280 Creating Neural Networks Chapte
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296 Modeling Relationships With Gau
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306 Fitting Dispersion Effects with
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Contents The Categorical Platform .
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386 Performing Categorical Response
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Contents Introduction to Choice Mod
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Contents Principal Components. . .
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538 Plotting Surfaces Chapter 23 La
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640 References Becker, R.A. and Cle
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Contents The Response Models . . .
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696 Index Customizing 267 Nonlinear
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698 Index reliability analysis 453-
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702 Index
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