/****************************************************************** CHAPTER 12, EXAMPLE 1 Example: Clinical trial of anti-epileptic drug progabide (Thall and Vail, Biometrics, 1990) Randomized, placebo-controlled study of treatment of epileptic seizures with progabide. Patients were randomized to treatment with progabide, or to placebo in addition to standard chemotherapy. Response variable: Count of number of seizures Measurement schedule: Baseline measurement during 8 weeks prior to randomization. Four measurements during consecutive two-week intervals. Interested in the effect of treatment with progabide on changes in an individual's rate of seizures? ******************************************************************/ options ls=80 ps=59 nodate; run; /****************************************************************** ... column 1 subject column 2 number of seizures column 3 visit (baseine (0) and 1--4 biweekly visits) column 4 =0 if placebo, = 1 if progabide column 5 baseline number of seizures in 8 weeks prior to study column 6 age ******************************************************************/ data seizure; infile "seize.dat.txt"; input subject seize visit trt base age; run; data seizure; set seizure; o=2; v=1; if visit=0 then o=8; if visit=0 then v=0; logo=log(o); run; proc print data=seizure; run; /****************************************************************** Delete the unusual subject and run again; we only use the compound symmetric covariance for the rest of the analyses. ******************************************************************/ data seizure; set seizure; if subject=207 then delete; run; title "INDEPENDENCE CORRELATION"; proc genmod data=seizure; class subject; model seize = v trt trt*v / dist = poisson link = log offset=logo; repeated subject=subject / corrw covb modelse; run; /****************************************************************** Output from PROC GENMOD NDEPENDENCE CORRELATION 15 The GENMOD Procedure Model Information Data Set WORK.SEIZURE Distribution Poisson Link Function Log Dependent Variable seize Offset Variable logo Number of Observations Read 290 Number of Observations Used 290 Class Level Information Class Levels Values subject 58 101 102 103 104 106 107 108 110 111 112 113 114 116 117 118 121 122 123 124 126 128 129 130 135 137 139 141 143 145 147 201 202 203 204 205 206 208 209 210 211 213 214 215 217 218 219 220 221 222 225 226 227 228 230 232 234 236 238 Parameter Information Parameter Effect Prm1 Intercept Prm2 v Prm3 trt Prm4 v*trt Algorithm converged. GEE Model Information Correlation Structure Independent Subject Effect subject (58 levels) Number of Clusters 58 Correlation Matrix Dimension 5 Maximum Cluster Size 5 Minimum Cluster Size 5 INDEPENDENCE CORRELATION 16 The GENMOD Procedure Covariance Matrix (Model-Based) Prm1 Prm2 Prm3 Prm4 Prm1 0.01223 -0.01223 -0.01223 0.01223 Prm2 -0.01223 0.02318 0.01223 -0.02318 Prm3 -0.01223 0.01223 0.02495 -0.02495 Prm4 0.01223 -0.02318 -0.02495 0.05129 Covariance Matrix (Empirical) Prm1 Prm2 Prm3 Prm4 Prm1 0.02476 -0.001152 -0.02476 0.001152 Prm2 -0.001152 0.01348 0.001152 -0.01348 Prm3 -0.02476 0.001152 0.03751 -0.002999 Prm4 0.001152 -0.01348 -0.002999 0.02931 Algorithm converged. Working Correlation Matrix Col1 Col2 Col3 Col4 Col5 Row1 1.0000 0.0000 0.0000 0.0000 0.0000 Row2 0.0000 1.0000 0.0000 0.0000 0.0000 Row3 0.0000 0.0000 1.0000 0.0000 0.0000 Row4 0.0000 0.0000 0.0000 1.0000 0.0000 Row5 0.0000 0.0000 0.0000 0.0000 1.0000 GEE Fit Criteria QIC -1052.5376 QICu -1060.3906 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z| Intercept 1.3476 0.1574 1.0392 1.6560 8.56 <.0001 v 0.1108 0.1161 -0.1168 0.3383 0.95 0.3399 trt -0.1080 0.1937 -0.4876 0.2716 -0.56 0.5770 v*trt -0.3016 0.1712 -0.6371 0.0339 -1.76 0.0781 Analysis Of GEE Parameter Estimates Model-Based Standard Error Estimates Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z| Intercept 1.3476 0.1106 1.1309 1.5644 12.19 <.0001 v 0.1108 0.1522 -0.1876 0.4092 0.73 0.4668 trt -0.1080 0.1579 -0.4176 0.2015 -0.68 0.4940 v*trt -0.3016 0.2265 -0.7455 0.1423 -1.33 0.1829 Scale 3.2469 . . . . . NOTE: The scale parameter for GEE estimation was computed as the square root of the normalized Pearson's chi-square. ******************************************************************/ /****************************************************************** Assume conditional rate of seizures follows the following mixed effects loglinear model, ******************************************************************/ proc nlmixed data=seizure qpoints=50; parms beta0=1.347 beta1=.1108 beta2=-.1080 beta3=-.3016 s2u1=0.5 s2u2=0.25 cu12=0.01; eta = logo + beta0 + beta1*v + beta2*trt + beta3*v*trt + u1 + u2*v; mu=exp(eta); model seize ~ poisson(mu); random u1 u2 ~ normal([0,0], [s2u1,cu12,s2u2]) subject=subject; run; /***************************************************************** Output from PROC NLMIXED The NLMIXED Procedure Specifications Data Set WORK.SEIZURE Dependent Variable seize Distribution for Dependent Variable Poisson Random Effects u1 u2 Distribution for Random Effects Normal Subject Variable subject Optimization Technique Dual Quasi-Newton Integration Method Adaptive Gaussian Quadrature Dimensions Observations Used 290 Observations Not Used 0 Total Observations 290 Subjects 58 Max Obs Per Subject 5 Parameters 7 Quadrature Points 50 Parameters beta0 beta1 beta2 beta3 s2u1 s2u2 cu12 NegLogLike 1.347 0.1108 -0.108 -0.3016 0.5 0.25 0.01 897.968374 Fit Statistics -2 Log Likelihood 1786.1 AIC (smaller is better) 1800.1 AICC (smaller is better) 1800.5 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > |t| Alpha Lower beta0 1.0696 0.1343 56 7.96 <.0001 0.05 0.8005 beta1 0.005862 0.1070 56 0.05 0.9565 0.05 -0.2085 beta2 -0.00967 0.1860 56 -0.05 0.9587 0.05 -0.3823 beta3 -0.3471 0.1489 56 -2.33 0.0233 0.05 -0.6453 s2u1 0.4528 0.09354 56 4.84 <.0001 0.05 0.2654 s2u2 0.2161 0.05864 56 3.69 0.0005 0.05 0.09862 cu12 0.01725 0.05287 56 0.33 0.7454 0.05 -0.08867 Parameter Estimates Parameter Upper Gradient beta0 1.3387 0.000721 beta1 0.2202 -0.0008 beta2 0.3630 0.000983 beta3 -0.04888 -0.00059 s2u1 0.6402 0.000191 s2u2 0.3336 -0.00083 cu12 0.1232 0.00075 *************************************************************************/ /****************************************************************** Results of the analysis suggest: 1. A patient treated with placebo has the same expected seizure rate before and after randomization [exp(0.0059) apprx = 1]. 2. A patient treated with progabide has expected seizure rate reduced after treatment by approximately 28% [1- exp(0.0059 - 0.3471)= 0.28]. 3. Estimated variance of the random intercepts and slopes in relatively large 4. Heterogeneity should not be ignored *************************************************************************/