SAS/STAT 14.1 User’s Guide

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® SAS/STAT 14.1User’sGuide TheNLMIXED Procedure® This document is an individual chapter from SAS/STAT 14.1 User’s Guide. ® The correct bibliographic citation for this manual is as follows: SAS Institute Inc. 2015. SAS/STAT 14.1 User’s Guide. Cary, NC: SAS Institute Inc. ® SAS/STAT 14.1 User’s Guide Copyright © 2015, SAS Institute Inc., Cary, NC, USA All Rights Reserved. Produced in the United States of America. For a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. For a web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. 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Other brand and product names are trademarks of their respective companies.Chapter 82 The NLMIXED Procedure Contents Overview: NLMIXED Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518 Literature on Nonlinear Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . . 6518 PROC NLMIXED Compared with Other SAS Procedures and Macros . . . . . . . . 6519 Getting Started: NLMIXED Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520 Nonlinear Growth Curves with Gaussian Data . . . . . . . . . . . . . . . . . . . . . 6520 Logistic-Normal Model with Binomial Data . . . . . . . . . . . . . . . . . . . . . . 6523 Syntax: NLMIXED Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527 PROC NLMIXED Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527 ARRAY Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6545 BOUNDS Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6546 BY Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6546 CONTRAST Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6547 ESTIMATE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6547 ID Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6547 MODEL Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6548 PARMS Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6548 PREDICT Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6549 RANDOM Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6550 REPLICATE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6551 Programming Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6551 Details: NLMIXED Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6553 Modeling Assumptions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 6553 Integral Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6555 Built-in Log-Likelihood Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6557 Hierarchical Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6559 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6561 Finite-Difference Approximations of Derivatives . . . . . . . . . . . . . . . . . . . . 6566 Hessian Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6568 Active Set Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6568 Line-Search Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6570 Restricting the Step Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6571 Computational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6572 Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6575 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6576 Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65766518 F Chapter 82: The NLMIXED Procedure Displayed Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6577 ODS Table Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6580 Examples: NLMIXED Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6580 Example 82.1: One-Compartment Model with Pharmacokinetic Data . . . . . . . . . 6580 Example 82.2: Probit-Normal Model with Binomial Data . . . . . . . . . . . . . . . 6584 Example 82.3: Probit-Normal Model with Ordinal Data . . . . . . . . . . . . . . . . 6587 Example 82.4: Poisson-Normal Model with Count Data . . . . . . . . . . . . . . . . 6591 Example 82.5: Failure Time and Frailty Model . . . . . . . . . . . . . . . . . . . . . 6594 Example 82.6: Simulated Nested Linear Random-Effects Model . . . . . . . . . . . . 6604 Example 82.7: Overdispersion Hierarchical Nonlinear Mixed Model . . . . . . . . . . 6607 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613 Overview: NLMIXED Procedure Introduction The NLMIXED procedure fits nonlinear mixed models—that is, models in which both fixed and random effects enter nonlinearly. These models have a wide variety of applications, two of the most common being pharmacokinetics and overdispersed binomial data. PROC NLMIXED enables you to specify a conditional distribution for your data (given the random effects) having either a standard form (normal, binomial, Poisson) or a general distribution that you code using SAS programming statements. PROC NLMIXED fits nonlinear mixed models by maximizing an approximation to the likelihood integrated over the random effects. Different integral approximations are available, the principal ones being adaptive Gaussian quadrature and a first-order Taylor series approximation. A variety of alternative optimization techniques are available to carry out the maximization; the default is a dual quasi-Newton algorithm. Successful convergence of the optimization problem results in parameter estimates along with their approxi- mate standard errors based on the second derivative matrix of the likelihood function. PROC NLMIXED enables you to use the estimated model to construct predictions of arbitrary functions by using empirical Bayes estimates of the random effects. You can also estimate arbitrary functions of the nonrandom parameters, and PROC NLMIXED computes their approximate standard errors by using the delta method. Literature on Nonlinear Mixed Models Davidian and Giltinan (1995) and Vonesh and Chinchilli (1997) provide good overviews as well as general theoretical developments and examples of nonlinear mixed models. Pinheiro and Bates (1995) is a primary reference for the theory and computational techniques of PROC NLMIXED. They describe and compare several different integrated likelihood approximations and provide evidence that adaptive Gaussian quadrature is one of the best methods. Davidian and Gallant (1993) also use Gaussian quadrature for nonlinear mixed models, although the smooth nonparametric density they advocate for the random effects is currently not available in PROC NLMIXED.PROC NLMIXED Compared with Other SAS Procedures and Macros F 6519 Traditional approaches to fitting nonlinear mixed models involve Taylor series expansions, expanding around either zero or the empirical best linear unbiased predictions of the random effects. The former is the basis for the well-known first-order method (Beal and Sheiner 1982, 1988; Sheiner and Beal 1985), and it is optionally available in PROC NLMIXED. The latter is the basis for the estimation method of Lindstrom and Bates (1990), and it is not available in PROC NLMIXED. However, the closely related Laplacian approximation is an option; it is equivalent to adaptive Gaussian quadrature with only one quadrature point. The Laplacian approximation and its relationship to the Lindstrom-Bates method are discussed by: Beal and Sheiner (1992); Wolfinger (1993); Vonesh (1992, 1996); Vonesh and Chinchilli (1997); Wolfinger and Lin (1997). A parallel literature exists in the area of generalized linear mixed models, in which random effects appear as a part of the linear predictor inside a link function. Taylor-series methods similar to those just described are discussed in articles such as: Harville and Mee (1984); Stiratelli, Laird, and Ware (1984); Gilmour, Anderson, and Rae (1985); Goldstein (1991); Schall (1991); Engel and Keen (1992); Breslow and Clayton (1993); Wolfinger and O’Connell (1993); McGilchrist (1994), but such methods have not been implemented in PROC NLMIXED because they can produce biased results in certain binary data situations (Rodriguez and Goldman 1995; Lin and Breslow 1996). Instead, a numerical quadrature approach is available in PROC NLMIXED, as discussed in: Pierce and Sands (1975); Anderson and Aitkin (1985); Hedeker and Gibbons (1994); Crouch and Spiegelman (1990); Longford (1994); McCulloch (1994); Liu and Pierce (1994); Diggle, Liang, and Zeger (1994). Nonlinear mixed models have important applications in pharmacokinetics, and Roe (1997) provides a wide- ranging comparison of many popular techniques. Yuh et al. (1994) provide an extensive bibliography on nonlinear mixed models and their use in pharmacokinetics. PROC NLMIXED Compared with Other SAS Procedures and Macros The models fit by PROC NLMIXED can be viewed as generalizations of the random coefficient models fit by the MIXED procedure. This generalization allows the random coefficients to enter the model nonlinearly, whereas in PROC MIXED they enter linearly. With PROC MIXED you can perform both maximum likelihood and restricted maximum likelihood (REML) estimation, whereas PROC NLMIXED implements only maximum likelihood. This is because the analog to the REML method in PROC NLMIXED would involve a high-dimensional integral over all of the fixed-effects parameters, and this integral is typically not available in closed form. Finally, PROC MIXED assumes the data to be normally distributed, whereas PROC NLMIXED enables you to analyze data that are normal, binomial, or Poisson or that have any likelihood programmable with SAS statements. PROC NLMIXED does not implement the same estimation techniques available with the NLINMIX macro or the default estimation method of the GLIMMIX procedure. These are based on the estimation methods of: Lindstrom and Bates (1990); Breslow and Clayton (1993); Wolfinger and O’Connell (1993), and they iteratively fit a set of generalized estimating equations (see Chapters 14 and 15 of Littell et al. 2006; Wolfinger 1997). In contrast, PROC NLMIXED directly maximizes an approximate integrated likelihood. This remark also applies to the SAS/IML macros MIXNLIN (Vonesh and Chinchilli 1997) and NLMEM (Galecki 1998). The GLIMMIX procedure also fits mixed models for nonnormal data with nonlinearity in the conditional mean function. In contrast to the NLMIXED procedure, PROC GLIMMIX assumes that the model contains a linear predictor that links covariates to the conditional mean of the response. The NLMIXED procedure is designed to handle general conditional mean functions, whether they contain a linear component or not. As mentioned earlier, the GLIMMIX procedure by default estimates parameters in generalized linear mixed models by pseudo-likelihood techniques, whereas PROC NLMIXED by default performs maximum6520 F Chapter 82: The NLMIXED Procedure likelihood estimation by adaptive Gauss-Hermite quadrature. This estimation method is also available with the GLIMMIX procedure (METHOD=QUAD in the PROC GLIMMIX statement). PROC NLMIXED has close ties with the NLP procedure in SAS/OR software. PROC NLMIXED uses a subset of the optimization code underlying PROC NLP and has many of the same optimization-based options. Also, the programming statement functionality used by PROC NLMIXED is the same as that used by PROC NLP and the MODEL procedure in SAS/ETS software. Getting Started: NLMIXED Procedure Nonlinear Growth Curves with Gaussian Data As an introductory example, consider the orange tree data of Draper and Smith (1981). These data consist of seven measurements of the trunk circumference (in millimeters) on each of five orange trees. You can input these data into a SAS data set as follows: data tree; input tree day y; datalines; 1 118 30 1 484 58 1 664 87 ... more lines ... 5 1582 177 ; Lindstrom and Bates (1990) and Pinheiro and Bates (1995) propose the following logistic nonlinear mixed model for these data: b Cu 1 i1 y D Ce ij ij 1CexpŒ.d b /=b  ij 2 3 Here,y represents the jth measurement on the ith tree (iD1;:::;5;jD1;:::;7),d is the corresponding ij ij day,b ;b ;b are the fixed-effects parameters,u are the random-effect parameters assumed to be iid 1 2 3 i1 2 2 N.0; /, ande are the residual errors assumed to be iidN.0; / and independent of theu . This model ij i1 u e has a logistic form, and the random-effect parametersu enter the model linearly. i1 The statements to fit this nonlinear mixed model are as follows: proc nlmixed data=tree; parms b1=190 b2=700 b3=350 s2u=1000 s2e=60; num = b1+u1; ex = exp(-(day-b2)/b3); den = 1 + ex; model y normal(num/den,s2e); random u1 normal(0,s2u) subject=tree; run;Nonlinear Growth Curves with Gaussian Data F 6521 The PROC NLMIXED statement invokes the procedure and inputs the tree data set. The PARMS statement identifies the unknown parameters and their starting values. Here there are three fixed-effects parameters (b1, b2, b3) and two variance components (s2u, s2e). The next three statements are SAS programming statements specifying the logistic mixed model. A new variable u1 is included to identify the random effect. These statements are evaluated for every observation in the data set when the NLMIXED procedure computes the log likelihood function and its derivatives. The MODEL statement defines the dependent variable and its conditional distribution given the random effects. Here a normal (Gaussian) conditional distribution is specified with mean num/den and variance s2e. The RANDOM statement defines the single random effect to be u1, and specifies that it follow a normal distribution with mean 0 and variance s2u. The SUBJECT= argument in the RANDOM statement defines a variable indicating when the random effect obtains new realizations; in this case, it changes according to the values of the tree variable. PROC NLMIXED assumes that the input data set is clustered according to the levels of the tree variable; that is, all observations from the same tree occur sequentially in the input data set. The output from this analysis is as follows. Figure 82.1 Model Specifications The The NLMIXED NLMIXED Procedure Procedure Specifications Data Set WORK.TREE Dependent Variable y Distribution for Dependent Variable Normal Random Effects u1 Distribution for Random Effects Normal Subject Variable tree Optimization Technique Dual Quasi-Newton Integration Method Adaptive Gaussian Quadrature The “Specifications” table lists basic information about the nonlinear mixed model you have specified (Figure 82.1). Included are the input data set, the dependent and subject variables, the random effects, the relevant distributions, and the type of optimization. The “Dimensions” table lists various counts related to the model, including the number of observations, subjects, and parameters (Figure 82.2). These quantities are useful for checking that you have specified your data set and model correctly. Also listed is the number of quadrature points that PROC NLMIXED has selected based on the evaluation of the log likelihood at the starting values of the parameters. Here, only one quadrature point is necessary because the random-effect parametersu enter the model linearly. (The Gauss-Hermite quadrature with a single quadrature point i1 results in the Laplace approximation of the log likelihood.)6522 F Chapter 82: The NLMIXED Procedure Figure 82.2 Dimensions Table for Growth Curve Model Dimensions Observations Used 35 Observations Not Used 0 Total Observations 35 Subjects 5 Max Obs per Subject 7 Parameters 5 Quadrature Points 1 Figure 82.3 Starting Values of Parameter Estimates and Negative Log Likelihood Initial Parameters Negative Log b1 b2 b3 s2u s2e Likelihood 190 700 350 1000 60 132.491787 The “Parameters” table lists the parameters to be estimated, their starting values, and the negative log likelihood evaluated at the starting values (Figure 82.3). Figure 82.4 Iteration History for Growth Curve Model Iteration History Negative Log Maximum Iteration Calls Likelihood Difference Gradient Slope 1 8 131.6867 0.805045 0.010269 -0.63300 2 12 131.6447 0.042082 0.014783 -0.01820 3 16 131.6141 0.030583 0.009809 -0.02796 4 20 131.5725 0.041555 0.001186 -0.01344 5 22 131.5719 0.000627 0.000200 -0.00121 6 25 131.5719 5.549E-6 0.000092 -7.68E-6 7 28 131.5719 1.096E-6 6.097E-6 -1.29E-6 NOTE: GCONV convergence criterion satisfied. The “Iteration History” table records the history of the minimization of the negative log likelihood (Fig- ure 82.4). For each iteration of the quasi-Newton optimization, values are listed for the number of function calls, the value of the negative log likelihood, the difference from the previous iteration, the absolute value of the largest gradient, and the slope of the search direction. The note at the bottom of the table indicates that the algorithm has converged successfully according to the GCONV convergence criterion, a standard criterion computed using a quadratic form in the gradient and the inverse Hessian. The final maximized value of the log likelihood as well as the information criterion of Akaike (AIC), its small sample bias corrected version (AICC), and the Bayesian information criterion (BIC) in the “smaller is better” form appear in the “Fit Statistics” table (Figure 82.5). These statistics can be used to compare different nonlinear mixed models.Logistic-Normal Model with Binomial Data F 6523 Figure 82.5 Fit Statistics for Growth Curve Model Fit Statistics -2 Log Likelihood 263.1 AIC (smaller is better) 273.1 AICC (smaller is better) 275.2 BIC (smaller is better) 271.2 Figure 82.6 Parameter Estimates at Convergence Parameter Estimates 95% Standard Confidence Parameter Estimate Error DF t Value Pr t Limits Gradient b1 192.05 15.6473 4 12.27 0.0003 148.61 235.50 1.154E-6 b2 727.90 35.2474 4 20.65 .0001 630.04 825.76 5.289E-6 b3 348.07 27.0793 4 12.85 0.0002 272.88 423.25 -6.1E-6 s2u 999.88 647.44 4 1.54 0.1974 -797.71 2797.46 -3.84E-6 s2e 61.5139 15.8832 4 3.87 0.0179 17.4150 105.61 2.892E-6 The maximum likelihood estimates of the five parameters and their approximate standard errors computed using the final Hessian matrix are displayed in the “Parameter Estimates” table (Figure 82.6). Approximate t-values and Wald-type confidence limits are also provided, with degrees of freedom equal to the number of subjects minus the number of random effects. You should interpret these statistics cautiously for variance parameters like s2u and s2e. The final column in the output shows the gradient vector at the optimization solution. Each element appears to be sufficiently small to indicate a stationary point. Since the random-effect parametersu enter the model linearly, you can obtain equivalent results by using i1 the first-order method (specify METHOD=FIRO in the PROC NLMIXED statement). Logistic-Normal Model with Binomial Data This example analyzes the data from Beitler and Landis (1985), which represent results from a multi-center clinical trial investigating the effectiveness of two topical cream treatments (active drug, control) in curing an infection. For each of eight clinics, the number of trials and favorable cures are recorded for each treatment. The SAS data set is as follows.6524 F Chapter 82: The NLMIXED Procedure data infection; input clinic t x n; datalines; 1 1 11 36 1 0 10 37 2 1 16 20 2 0 22 32 3 1 14 19 3 0 7 19 4 1 2 16 4 0 1 17 5 1 6 17 5 0 0 12 6 1 1 11 6 0 0 10 7 1 1 5 7 0 1 9 8 1 4 6 8 0 6 7 ; Supposen denotes the number of trials for the ith clinic and the jth treatment (iD1;:::;8IjD0;1), and ij x denotes the corresponding number of favorable cures. Then a reasonable model for the preceding data is ij the following logistic model with random effects: x ju  Binomial.n ;p / ij i ij ij and   p ij  Dlog D C t Cu ij 0 1 j i 1p ij 2 The notationt indicates the jth treatment, and theu are assumed to be iidN.0; /. j i u The PROC NLMIXED statements to fit this model are as follows: proc nlmixed data=infection; parms beta0=-1 beta1=1 s2u=2; eta = beta0 + beta1 t + u; expeta = exp(eta); p = expeta/(1+expeta); model x binomial(n,p); random u normal(0,s2u) subject=clinic; predict eta out=eta; estimate '1/beta1' 1/beta1; run; The PROC NLMIXED statement invokes the procedure, and the PARMS statement defines the parameters and their starting values. The next three statements definep , and the MODEL statement defines the ij conditional distribution ofx to be binomial. The RANDOM statement defines u to be the random effect ij with subjects defined by the clinic variable.Logistic-Normal Model with Binomial Data F 6525 The PREDICT statement constructs predictions for each observation in the input data set. For this example, predictions of and approximate standard errors of prediction are output to a data set named eta. These ij predictions include empirical Bayes estimates of the random effectsu . i The ESTIMATE statement requests an estimate of the reciprocal of . 1 The output for this model is as follows. Figure 82.7 Model Information and Dimensions for Logistic-Normal Model The The NLMIXED NLMIXED Procedure Procedure Specifications Data Set WORK.INFECTION Dependent Variable x Distribution for Dependent Variable Binomial Random Effects u Distribution for Random Effects Normal Subject Variable clinic Optimization Technique Dual Quasi-Newton Integration Method Adaptive Gaussian Quadrature Dimensions Observations Used 16 Observations Not Used 0 Total Observations 16 Subjects 8 Max Obs per Subject 2 Parameters 3 Quadrature Points 5 The “Specifications” table provides basic information about the nonlinear mixed model (Figure 82.7). For example, the distribution of the response variable, conditional on normally distributed random effects, is binomial. The “Dimensions” table provides counts of various variables. You should check this table to make sure the data set and model have been entered properly. PROC NLMIXED selects five quadrature points to achieve the default accuracy in the likelihood calculations. Figure 82.8 Starting Values of Parameter Estimates Initial Parameters Negative Log beta0 beta1 s2u Likelihood -1 1 2 37.5945925 The “Parameters” table lists the starting point of the optimization and the negative log likelihood at the starting values (Figure 82.8).6526 F Chapter 82: The NLMIXED Procedure Figure 82.9 Iteration History and Fit Statistics for Logistic-Normal Model Iteration History Negative Log Maximum Iteration Calls Likelihood Difference Gradient Slope 1 4 37.3622692 0.232323 2.88208 -19.3762 2 6 37.1460375 0.216232 0.92193 -0.82852 3 9 37.0300936 0.115944 0.31590 -0.59175 4 11 37.0223017 0.007792 0.019060 -0.01615 5 13 37.0222472 0.000054 0.001743 -0.00011 6 16 37.0222466 6.57E-7 0.000091 -1.28E-6 7 19 37.0222466 5.38E-10 2.078E-6 -1.1E-9 NOTE: GCONV convergence criterion satisfied. Fit Statistics -2 Log Likelihood 74.0 AIC (smaller is better) 80.0 AICC (smaller is better) 82.0 BIC (smaller is better) 80.3 The “Iteration History” table indicates successful convergence in seven iterations (Figure 82.9). The “Fit Statistics” table lists some useful statistics based on the maximized value of the log likelihood. Figure 82.10 Parameter Estimates for Logistic-Normal Model Parameter Estimates 95% Standard Confidence Parameter Estimate Error DF t Value Pr t Limits Gradient beta0 -1.1974 0.5561 7 -2.15 0.0683 -2.5123 0.1175 -3.1E-7 beta1 0.7385 0.3004 7 2.46 0.0436 0.02806 1.4488 -2.08E-6 s2u 1.9591 1.1903 7 1.65 0.1438 -0.8555 4.7737 -2.48E-7 The “Parameter Estimates” table indicates marginal significance of the two fixed-effects parameters (Fig- ure 82.10). The positive value of the estimate of indicates that the treatment significantly increases the 1 chance of a favorable cure. Figure 82.11 Table of Additional Estimates Additional Estimates Standard Label Estimate Error DF t Value Pr t Alpha Lower Upper 1/beta1 1.3542 0.5509 7 2.46 0.0436 0.05 0.05146 2.6569 The “Additional Estimates” table displays results from the ESTIMATE statement (Figure 82.11). The estimate 2 of1= equals1=0:7385D 1:3542 and its standard error equals0:3004=0:7385 D 0:5509 by the delta 1 method (Billingsley 1986; Cox 1998). Note that this particular approximation produces a t-statistic identical to that for the estimate of . Not shown is the eta data set, which contains the original 16 observations and 1 predictions of the . ijSyntax: NLMIXED Procedure F 6527 Syntax: NLMIXED Procedure The following statements are available in the NLMIXED procedure: PROC NLMIXED options ; ARRAY array-specification ; BOUNDS boundary-constraints ; BY variables ; CONTRAST 'label' expression , expression options ; ESTIMATE 'label' expression options ; ID names ; MODEL model-specification ; PARMS parameters-and-starting-values ; PREDICT expression OUT=SAS-data-set options ; RANDOM random-effects-specification ; REPLICATE variable ; Programming statements ; The following sections provide a detailed description of each of these statements. PROC NLMIXED Statement PROC NLMIXED options ; The PROC NLMIXED statement invokes the NLMIXED procedure. Table 82.1 summarizes the options available in the PROC NLMIXED statement. Table 82.1 PROC NLMIXED Statement Options Option Description Basic Options DATA= Specifies the input data set METHOD= Specifies the integration method NOSORTSUB Requests that the unique SUBJECT= variable values not be used NTHREADS= Specifies the number of threads to use Displayed Output Specifications ALPHA= Specifies for confidence limits CORR Requests the correlation matrix COV Requests the covariance matrix DF= Specifies the degrees of freedom for p-values and confidence limits ECORR Requests the correlation matrix of additional estimates ECOV Requests the covariance matrix of additional estimates EDER Requests derivatives of additional estimates EMPIRICAL Requests the empirical (“sandwich”) estimator of covariance matrix HESS Requests the Hessian matrix ITDETAILS Requests iteration details START Specifies the gradient at starting values6528 F Chapter 82: The NLMIXED Procedure Table 82.1 continued Option Description Debugging Output FLOW Displays the model execution messages LISTCODE Displays compiled model program LISTDEP Produces a model dependency listing LISTDER Displays the model derivatives LIST Displays the model program and variables TRACE Displays detailed model execution messages XREF Displays the model cross references Quadrature Options NOADSCALE Requests no adaptive scaling NOAD Requests no adaptive centering OUTQ= Displays output data set QFAC= Specifies the search factor QMAX= Specifies the maximum points QPOINTS= Specifies the number of points QSCALEFAC= Specifies the scale factor QTOL= Specifies the tolerance Empirical Bayes Options EBOPT Requests comprehensive optimization EBSSFRAC= Specifies the step-shortening fraction EBSSTOL= Specifies the step-shortening tolerance EBSTEPS= Specifies the number of Newton steps EBSUBSTEPS= Specifies the number of substeps EBTOL= Specifies the convergence tolerance EBZSTART Requests zero as the starting values OUTR= Displays an output data set that contains empirical Bayes estimates of random effects and their approximate standard errors Optimization Specifications HESCAL= Specifies the type of Hessian scaling INHESSIAN= Specifies the start for approximated Hessian LINESEARCH= Specifies the line-search method LSPRECISION= Specifies the line-search precision OPTCHECK= Checks optimality in a neighborhood RESTART= Specifies the iteration number for update restart TECHNIQUE= Specifies the minimization technique UPDATE= Specifies the update technique Derivatives Specifications DIAHES Uses only the diagonal of Hessian FDHESSIAN= Specifies the finite-difference second derivatives FD= Specifies the finite-difference derivativesPROC NLMIXED Statement F 6529 Table 82.1 continued Option Description Constraint Specifications LCDEACT= Specifies the Lagrange multiplier tolerance for deactivating LCEPSILON= Specifies the range for active constraints LCSINGULAR= Specifies the tolerance for dependent constraints Termination Criteria Specifications ABSCONV= Specifies the absolute function convergence criterion ABSFCONV= Specifies the absolute function difference convergence criterion ABSGCONV= Specifies the absolute gradient convergence criterion ABSXCONV= Specifies the absolute parameter convergence criterion FCONV= Specifies the relative function convergence criterion FCONV2= Specifies another relative function convergence criterion FDIGITS= Specifies the number accurate digits in objective function FSIZE= Specifies the FSIZE parameter of the relative function and relative gradient termination criteria GCONV= Specifies the relative gradient convergence criterion MAXFUNC= Specifies the maximum number of function calls MAXITER= Specifies the maximum number of iterations MAXTIME= Specifies the upper limit seconds of CPU time MINITER= Specifies the minimum number of iterations XCONV= Specifies the relative parameter convergence criterion XSIZE= Used in XCONV criterion Step Length Specifications DAMPSTEP= Specifies the damped steps in line search INSTEP= Specifies the initial trust-region radius MAXSTEP= Specifies the maximum trust-region radius Singularity Tolerances SINGCHOL= Specifies the tolerance for Cholesky roots SINGHESS= Specifies the tolerance for Hessian SINGSWEEP= Specifies the tolerance for sweep SINGVAR= Specifies the tolerance for variances Covariance Matrix Tolerances ASINGULAR= Specifies the absolute singularity for inertia CFACTOR= Specifies the multiplication factor for COV matrix COVSING= Specifies the tolerance for singular COV matrix G4= Specifies the threshold for Moore-Penrose inverse MSINGULAR= Specifies the relative M singularity for inertia VSINGULAR= Specifies the relative V singularity for inertia These options are described in alphabetical order. For a description of the mathematical notation used in the following sections, see the section “Modeling Assumptions and Notation” on page 6553.6530 F Chapter 82: The NLMIXED Procedure ABSCONV=r ABSTOL=r .k/ specifies an absolute function convergence criterion. For minimization, termination requiresf. / r. The default value of r is the negative square root of the largest double-precision value, which serves only as a protection against overflows. ABSFCONV=r n ABSFTOL=r n specifies an absolute function difference convergence criterion. For all techniques except NMSIMP, termination requires a small change of the function value in successive iterations: .k1/ .k/ jf. /f. /jr .k/ The same formula is used for the NMSIMP technique, but is defined as the vertex with the lowest .k1/ function value, and is defined as the vertex with the highest function value in the simplex. The default value is r = 0. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated. ABSGCONV=r n ABSGTOL=r n specifies an absolute gradient convergence criterion. Termination requires the maximum absolute gradient element to be small: .k/ maxjg . /jr j j This criterion is not used by the NMSIMP technique. The default value is r = 1E–5. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated. If you specify more than one RANDOM statement, the default value is r = 1E–3. ABSXCONV=r n ABSXTOL=r n specifies an absolute parameter convergence criterion. For all techniques except NMSIMP, termination requires a small Euclidean distance between successive parameter vectors, .k/ .k1/ k  kr 2 .k/ For the NMSIMP technique, termination requires either a small length of the vertices of a restart simplex, .k/ r or a small simplex size, .k/  r .k/ .k/ where the simplex size is defined as the L1 distance from the simplex vertex with the smallest .k/ .k/ function value to the other n simplex points ¤ : l X .k/ .k/ .k/  D k  k 1 l ¤y l The default is r = 1E–8 for the NMSIMP technique and r = 0 otherwise. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.PROC NLMIXED Statement F 6531 ALPHA= specifies the alpha level to be used in computing confidence limits. The default value is 0.05. ASINGULAR=r ASING=r specifies an absolute singularity criterion for the computation of the inertia (number of positive, negative, and zero eigenvalues) of the Hessian and its projected forms. The default value is the square root of the smallest positive double-precision value. CFACTOR=f specifies a multiplication factor f for the estimated covariance matrix of the parameter estimates. COV requests the approximate covariance matrix for the parameter estimates. CORR requests the approximate correlation matrix for the parameter estimates. COVSING=r0 specifies a nonnegative threshold that determines whether the eigenvalues of a singular Hessian matrix are considered to be zero. DAMPSTEP =r DS =r .0/ specifies that the initial step-size value for each line search (used by the QUANEW, CONGRA, or NEWRAP technique) cannot be larger than r times the step-size value used in the former iteration. If you specify the DAMPSTEP option without factor r, the default value is r = 2. The DAMPSTEP=r option can prevent the line-search algorithm from repeatedly stepping into regions where some objective functions are difficult to compute or where they could lead to floating-point overflows during the computation of objective functions and their derivatives. The DAMPSTEP=r option can save time-costly function calls that result in very small step sizes . For more details on setting the start values of each line search, see the section “Restricting the Step Length” on page 6571. DATA=SAS-data-set specifies the input data set. Observations in this data set are used to compute the log likelihood function that you specify with PROC NLMIXED statements. NOTE: In SAS/STAT 12.3 and previous releases, if you are using a RANDOM statement, the input data set must be clustered according to the SUBJECT= variable. One easy way to accomplish this is to sort your data by the SUBJECT= variable before calling the NLMIXED procedure. PROC NLMIXED does not sort the input data set for you. DF=d specifies the degrees of freedom to be used in computing p values and confidence limits. PROC NLMIXED calculates the default degrees of freedom as follows:  When there is no RANDOM statement in the model, the default value is the number of observa- tions.  When only one RANDOM statement is specified, the default value is the number of subjects minus the number of random effects for random-effects models.6532 F Chapter 82: The NLMIXED Procedure  When multiple RANDOM statements are specified, the default degrees of freedom is the number of subjects in the lowest nested level minus the total number of random effects. For example, if the highest level of hierarchy is specified by SUBJECT=S1 and the next level of hierarchy (nested within S1) is specified by SUBJECT=S2(S1), then the degrees of freedom is computed as the total number of subjects from S2(S1) minus the total number of random-effects variables in the model. If the degrees of freedom computation leads to a nonpositive value, then the default value is the total number of observations. DIAHES specifies that only the diagonal of the Hessian be used. EBOPT requests that a more comprehensive optimization be carried out if the default empirical Bayes opti- mization fails to converge. If you specify more than one RANDOM statement, this option is ignored. EBSSFRAC=r0 specifies the step-shortening fraction to be used while computing empirical Bayes estimates of the random effects. The default value is 0.8. If you specify more than one RANDOM statement, this option is ignored. EBSSTOL=r0 specifies the objective function tolerance for determining the cessation of step-shortening while computing empirical Bayes estimates of the random effects. The default value is r = 1E–8. If you specify more than one RANDOM statement, this option is ignored. EBSTEPS=n0 specifies the maximum number of Newton steps for computing empirical Bayes estimates of random effects. The default value is n = 50. If you specify more than one RANDOM statement, this option is ignored. EBSUBSTEPS=n0 specifies the maximum number of step-shortenings for computing empirical Bayes estimates of random effects. The default value is n = 20. If you specify more than one RANDOM statement, this option is ignored. EBTOL=r0 specifies the convergence tolerance for empirical Bayes estimation. The default value isrDE4, where is the machine precision. This default value equals approximately 1E–12 on most machines. If you specify more than one RANDOM statement, this option is ignored. EBZSTART requests that a zero be used as starting values during empirical Bayes estimation. By default, the starting values are set equal to the estimates from the previous iteration (or zero for the first iteration). ECOV requests the approximate covariance matrix for all expressions specified in ESTIMATE statements. ECORR requests the approximate correlation matrix for all expressions specified in ESTIMATE statements.PROC NLMIXED Statement F 6533 EDER requests the derivatives of all expressions specified in ESTIMATE statements with respect to each of the model parameters. EMPIRICAL requests that the covariance matrix of the parameter estimates be computed as a likelihood-based empirical (“sandwich”) estimator (White 1982). Iff./Dlogfm./g is the objective function for the optimization andm./ denotes the marginal log likelihood (see the section “Modeling Assumptions and Notation” on page 6553 for notation and further definitions) the empirical estimator is computed as s X 1 0 1 O O O O H./ g ./g ./ H./ i i iD1 whereH is the second derivative matrix of f andg is the first derivative of the contribution to f by i the ith subject. If you choose the EMPIRICAL option, this estimator of the covariance matrix of the 1 O parameter estimates replaces the model-based estimatorH./ in subsequent calculations. You can output the subject-specific gradients g to a SAS data set with the SUBGRADIENT option in the i PROC NLMIXED statement. The EMPIRICAL option requires the presence of a RANDOM statement and is available for METHOD=GAUSS and METHOD=ISAMP only. If you specify more than one RANDOM statement, this option is ignored. FCONV=r n FTOL=r n specifies a relative function convergence criterion. For all techniques except NMSIMP, termination requires a small relative change of the function value in successive iterations, .k/ .k1/ jf. /f. /j r .k1/ max.jf. /j; FSIZE/ where FSIZE is defined by the FSIZE= option. The same formula is used for the NMSIMP technique, .k/ .k1/ but is defined as the vertex with the lowest function value, and is defined as the vertex FDIGITS with the highest function value in the simplex. The default isrD10 , where FDIGITS is the value of the FDIGITS= option. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate. FCONV2=r n FTOL2=r n specifies another function convergence criterion. For all techniques except NMSIMP, termination requires a small predicted reduction .k/ .k/ .k/ .k/ df f. /f. Cs / of the objective function. The predicted reduction 1 .k/ .k/0 .k/ .k/0 .k/ .k/ df Dg s s H s 2 1 .k/0 .k/ D s g 2 r6534 F Chapter 82: The NLMIXED Procedure is computed by approximating the objective function f by the first two terms of the Taylor series and substituting the Newton step: .k/ .k/1 .k/ s DŒH  g For the NMSIMP technique, termination requires a small standard deviation of the function values of .k/ thenC1 simplex vertices ,lD0;:::;n, l s h i X 2 1 .k/ .k/ f. /f. / r l nC1 l P .k/ .k/ 1 .k/ wheref. /D f. /. If there aren boundary constraints active at , the mean and act l nC1 l standard deviation are computed only for thenC1n unconstrained vertices. The default value act is r = 1E–6 for the NMSIMP technique and r = 0 otherwise. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate. FD = FORWARD CENTRAL r specifies that all derivatives be computed using finite difference approximations. The following specifications are permitted: FD is equivalent to FD=100. FD=CENTRAL uses central differences. FD=FORWARD uses forward differences. FD=r uses central differences for the initial and final evaluations of the gradient and for the Hessian. During iteration, start with forward differences and switch to a corresponding central-difference formula during the iteration process when one of the following two criteria is satisfied:  The absolute maximum gradient element is less than or equal to r times the ABSGCONV= threshold.  The normalized predicted function reduction (see the GTOL option) is less than or equal tomax.1E6; rGTOL/. The 1E–6 ensures that the switch is done, even if you set the GTOL threshold to zero. Note that the FD and FDHESSIAN options cannot apply at the same time. The FDHESSIAN option is ignored when only first-order derivatives are used. See the section “Finite-Difference Approximations of Derivatives” on page 6566 for more information. FDHESSIAN =FORWARD CENTRAL FDHES =FORWARD CENTRAL FDH =FORWARD CENTRAL specifies that second-order derivatives be computed using finite difference approximations based on evaluations of the gradients. FDHESSIAN=FORWARD uses forward differences. FDHESSIAN=CENTRAL uses central differences. FDHESSIAN uses forward differences for the Hessian except for the initial and final output. Note that the FD and FDHESSIAN options cannot apply at the same time. See the section “Finite- Difference Approximations of Derivatives” on page 6566 for more information.

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