The NLMIXED Procedure

## Example 61.5 Failure Time and Frailty Model

In this example an accelerated failure time model with proportional hazard is fitted with and without random effects. The data are from the "Getting Started" example of PROC LIFEREG; see Chapter 48, The LIFEREG Procedure. Thirty-eight patients are divided into two groups of equal size, and different pain relievers are assigned to each group. The outcome reported is the time in minutes until headache relief. The variable censor indicates whether relief was observed during the course of the observation period (censor = 0) or whether the observation is censored (censor = 1). The SAS DATA step for these data is as follows:

```   data headache;
input minutes group censor @@;
patient = _n_;
datalines;
11  1  0   12  1  0   19  1  0   19  1  0
19  1  0   19  1  0   21  1  0   20  1  0
21  1  0   21  1  0   20  1  0   21  1  0
20  1  0   21  1  0   25  1  0   27  1  0
30  1  0   21  1  1   24  1  1   14  2  0
16  2  0   16  2  0   21  2  0   21  2  0
23  2  0   23  2  0   23  2  0   23  2  0
25  2  1   23  2  0   24  2  0   24  2  0
26  2  1   32  2  1   30  2  1   30  2  0
32  2  1   20  2  1
;
```

In modeling survival data, censoring of observations must be taken into account carefully. In this example, only right censoring occurs. If , , and denote the density of failure, hazard function, and survival distribution function at time , respectively, the log likelihood can be written as

(See Cox and Oakes 1984, Ch. 3.) In these expressions is the set of uncensored observations, is the set of censored observations, and denotes the total sample size.

The proportional hazards specification expresses the hazard in terms of a baseline hazard, multiplied by a constant. In this example the hazard is that of a Weibull model and is parameterized as and .

The linear predictor is set equal to the intercept in the reference group (group = 2); this defines the baseline hazard. The corresponding distribution of survival past time is . See Cox and Oakes (1984, Table 2.1) and the section "Supported Distributions" in Chapter 48, The LIFEREG Procedure, for this and other survival distribution models and various parameterizations.

The following NLMIXED statements fit this accelerated failure time model and estimate the cumulative distribution function of time to headache relief:

```   proc nlmixed data=headache;
bounds gamma > 0;
linp  = b0 - b1*(group-2);
alpha = exp(-linp);
G_t   = exp(-(alpha*minutes)**gamma);
g     = gamma*alpha*((alpha*minutes)**(gamma-1))*G_t;
ll    = (censor=0)*log(g) + (censor=1)*log(G_t);
model minutes ~ general(ll);
predict 1-G_t out=cdf;
run;
```

Output 61.5.1 Specifications Table for Fixed-Effects Failure Time Model
The NLMIXED Procedure

Specifications
Dependent Variable minutes
Distribution for Dependent Variable General
Optimization Technique Dual Quasi-Newton
Integration Method None

The "Specifications" table shows that no integration is required, since the model does not contain random effects (Output 61.5.1).

Output 61.5.2 Negative Log Likelihood with Default Starting Values
Parameters
gamma b0 b1 NegLogLike
1 1 1 263.990327

No starting values were given for the three parameters. The NLMIXED procedure assigns the default value of 1.0 in this case. The negative log likelihood based on these starting values is shown in Output 61.5.2.

Output 61.5.3 Iteration History for Fixed-Effects Failure Time Model
Iteration History
Iter   Calls NegLogLike Diff MaxGrad Slope
1   2 169.244311 94.74602 22.5599 -2230.83
2   4 142.873508 26.3708 14.88631 -3.64643
3   6 140.633695 2.239814 11.25234 -9.49454
4   8 122.890659 17.74304 19.44959 -2.50807
5   9 121.396959 1.493699 13.85584 -4.55427
6   11 120.623843 0.773116 13.67062 -1.38064
7   12 119.278196 1.345647 15.78014 -1.69072
8   14 116.271325 3.006871 26.94029 -3.2529
9   16 109.427401 6.843925 19.88382 -6.9289
10   19 103.298102 6.129298 12.15647 -4.96054
11   22 101.686239 1.611863 14.24868 -4.34059
12   23 100.027875 1.658364 11.69853 -13.2049
13   26 99.9189048 0.108971 3.602552 -0.55176
14   28 99.8738836 0.045021 0.170712 -0.16645
15   30 99.8736392 0.000244 0.050822 -0.00041
16   32 99.8736351 4.071E-6 0.000705 -6.9E-6
17   34 99.8736351 6.1E-10 4.768E-6 -1.23E-9

 NOTE: GCONV convergence criterion satisfied.

The "Iteration History" table shows that the procedure converges after 17 iterations and 34 evaluations of the objective function (Output 61.5.3).

Output 61.5.4 Fit Statistics and Parameter Estimates
Fit Statistics
-2 Log Likelihood 199.7
AIC (smaller is better) 205.7
AICC (smaller is better) 206.5
BIC (smaller is better) 210.7

Parameter Estimates
Parameter Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper Gradient
gamma 4.7128 0.6742 38 6.99 <.0001 0.05 3.3479 6.0777 5.327E-8
b0 3.3091 0.05885 38 56.23 <.0001 0.05 3.1900 3.4283 -4.77E-6
b1 -0.1933 0.07856 38 -2.46 0.0185 0.05 -0.3523 -0.03426 -1.22E-6

The parameter estimates and their standard errors shown in Output 61.5.4 are identical to those obtained with the LIFEREG procedure and the following statements:

```   proc lifereg data=headache;
class group;
model minutes*censor(1) = group / dist=weibull;
output out=new cdf=prob;
run;
```

The t statistic and confidence limits are based on 38 degrees of freedom. The LIFEREG procedure computes z-intervals for the parameter estimates.

For the two groups you obtain

The probabilities of headache relief by minutes are estimated as

These probabilities, calculated at the observed times, are shown for the two groups in Output 61.5.5 and printed with the following statements:

```   proc print data=cdf;
var group censor patient minutes pred;
run;
```

Since the slope estimate is negative with p-value of 0.0185, you can infer that pain reliever 1 leads to overall significantly faster relief, but the estimated probabilities give no information about patient-to-patient variation within and between groups. For example, while pain reliever 1 provides faster relief overall, some patients in group 2 might respond more quickly than some patients in group 1. A frailty model enables you to accommodate and estimate patient-to-patient variation in health status by introducing random effects into a subject’s hazard function.

Output 61.5.5 Estimated Cumulative Distribution Function
Obs group censor patient minutes Pred
1 1 0 1 11 0.03336
2 1 0 2 12 0.04985
3 1 0 3 19 0.35975
4 1 0 4 19 0.35975
5 1 0 5 19 0.35975
6 1 0 6 19 0.35975
7 1 0 7 21 0.51063
8 1 0 8 20 0.43325
9 1 0 9 21 0.51063
10 1 0 10 21 0.51063
11 1 0 11 20 0.43325
12 1 0 12 21 0.51063
13 1 0 13 20 0.43325
14 1 0 14 21 0.51063
15 1 0 15 25 0.80315
16 1 0 16 27 0.90328
17 1 0 17 30 0.97846
18 1 1 18 21 0.51063
19 1 1 19 24 0.73838
20 2 0 20 14 0.04163
21 2 0 21 16 0.07667
22 2 0 22 16 0.07667
23 2 0 23 21 0.24976
24 2 0 24 21 0.24976
25 2 0 25 23 0.35674
26 2 0 26 23 0.35674
27 2 0 27 23 0.35674
28 2 0 28 23 0.35674
29 2 1 29 25 0.47982
30 2 0 30 23 0.35674
31 2 0 31 24 0.41678
32 2 0 32 24 0.41678
33 2 1 33 26 0.54446
34 2 1 34 32 0.87656
35 2 1 35 30 0.78633
36 2 0 36 30 0.78633
37 2 1 37 32 0.87656
38 2 1 38 20 0.20414

The following statements model the hazard for patient in terms of , where is a (normal) random patient effect. Notice that the only difference from the previous NLMIXED statements are the RANDOM statement and the addition of z in the linear predictor. The empirical Bayes estimates of the random effect (RANDOM statement), the parameter estimates (ODS OUTPUT statement), and the estimated cumulative distribution function (PREDICT statement) are saved to subsequently graph the patient-specific distribution functions.

```   ods output ParameterEstimates=est;
bounds gamma > 0;
linp  = b0 - b1*(group-2) + z;
alpha = exp(-linp);
G_t   = exp(-(alpha*minutes)**gamma);
g     = gamma*alpha*((alpha*minutes)**(gamma-1))*G_t;
ll = (censor=0)*log(g) + (censor=1)*log(G_t);
model minutes ~ general(ll);
random z ~ normal(0,exp(2*logsig)) subject=patient out=EB;
predict 1-G_t out=cdf;
run;
```

Output 61.5.6 Specifications for Random Frailty Model
The NLMIXED Procedure

Specifications
Dependent Variable minutes
Distribution for Dependent Variable General
Random Effects z
Distribution for Random Effects Normal
Subject Variable patient
Optimization Technique Dual Quasi-Newton

The "Specifications" table shows that the objective function is computed by adaptive Gaussian quadrature because of the presence of random effects (compare Output 61.5.6 and Output 61.5.1). The "Dimensions" table reports that nine quadrature points are being used to integrate over the random effects (Output 61.5.7).

Output 61.5.7 Dimensions Table for Random Frailty Model
Dimensions
Observations Used 38
Observations Not Used 0
Total Observations 38
Subjects 38
Max Obs Per Subject 1
Parameters 4

Output 61.5.8 Iteration History for Random Frailty Model
Iteration History
Iter   Calls NegLogLike Diff MaxGrad Slope
1   5 142.121411 28.82225 12.14484 -88.8664
2   7 136.440369 5.681042 25.93096 -65.7217
3   9 122.972041 13.46833 46.56546 -146.887
4   11 120.904825 2.067216 23.77936 -94.2862
5   13 109.224144 11.68068 57.65493 -92.4075
6   15 105.064733 4.159411 4.824649 -19.5879
7   16 101.902207 3.162526 14.1287 -6.33767
8   18 99.6907395 2.211468 7.676822 -3.42364
9   20 99.3654033 0.325336 5.689204 -0.93978
10   22 99.2602178 0.105185 0.317643 -0.23408
11   24 99.254434 0.005784 1.17351 -0.00556
12   25 99.2456973 0.008737 0.247412 -0.00871
13   27 99.2445445 0.001153 0.104942 -0.00218
14   29 99.2444958 0.000049 0.005646 -0.0001
15   31 99.2444957 9.147E-8 0.000271 -1.84E-7

 NOTE: GCONV convergence criterion satisfied.

The procedure converges after 15 iterations (Output 61.5.8). The achieved log likelihood is only 1.2 less than that in the model without random effects (compare Output 61.5.9 and Output 61.5.4). Compared to a chi-square distribution with one degree of freedom, the addition of the random effect appears not to improve the model significantly. You must exercise care, however, in interpreting likelihood ratio tests when the value under the null hypothesis falls on the boundary of the parameter space (see, for example, Self and Liang 1987).

Output 61.5.9 Fit Statistics for Random Frailty Model
Fit Statistics
-2 Log Likelihood 198.5
AIC (smaller is better) 206.5
AICC (smaller is better) 207.7
BIC (smaller is better) 213.0

Output 61.5.10 Parameter Estimates
Parameter Estimates
Parameter Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper Gradient
gamma 6.2867 2.1334 37 2.95 0.0055 0.05 1.9641 10.6093 -1.89E-7
b0 3.2786 0.06576 37 49.86 <.0001 0.05 3.1453 3.4118 0.000271
b1 -0.1761 0.08264 37 -2.13 0.0398 0.05 -0.3436 -0.00868 0.000111
logsig -1.9027 0.5273 37 -3.61 0.0009 0.05 -2.9711 -0.8343 0.000027

The estimate of the Weibull parameter has changed drastically from the model without random effects (compare Output 61.5.10 and Output 61.5.4). The variance of the patient random effect is . The listing in Output 61.5.11 shows the empirical Bayes estimates of the random effects. These are the adjustments made to the linear predictor in order to obtain a patient’s survival distribution. The listing is produced with the following statements:

```   proc print data=eb;
var Patient Effect Estimate StdErrPred;
run;
```

Output 61.5.11 Empirical Bayes Estimates of Random Effects
Obs patient Effect Estimate StdErrPred
1 1 z -0.13597 0.23249
2 2 z -0.13323 0.22793
3 3 z -0.06294 0.13813
4 4 z -0.06294 0.13813
5 5 z -0.06294 0.13813
6 6 z -0.06294 0.13813
7 7 z -0.02568 0.11759
8 8 z -0.04499 0.12618
9 9 z -0.02568 0.11759
10 10 z -0.02568 0.11759
11 11 z -0.04499 0.12618
12 12 z -0.02568 0.11759
13 13 z -0.04499 0.12618
14 14 z -0.02568 0.11759
15 15 z 0.05980 0.11618
16 16 z 0.10458 0.12684
17 17 z 0.17147 0.14550
18 18 z 0.06471 0.13807
19 19 z 0.11157 0.14604
20 20 z -0.13406 0.22899
21 21 z -0.12698 0.21667
22 22 z -0.12698 0.21667
23 23 z -0.08506 0.15701
24 24 z -0.08506 0.15701
25 25 z -0.05797 0.13294
26 26 z -0.05797 0.13294
27 27 z -0.05797 0.13294
28 28 z -0.05797 0.13294
29 29 z 0.06420 0.13956
30 30 z -0.05797 0.13294
31 31 z -0.04266 0.12390
32 32 z -0.04266 0.12390
33 33 z 0.07618 0.14132
34 34 z 0.16292 0.16460
35 35 z 0.13193 0.15528
36 36 z 0.06327 0.12124
37 37 z 0.16292 0.16460
38 38 z 0.02074 0.14160

The predicted values and patient-specific survival distributions can be plotted with the SAS code that follows:

```   proc transpose data=est(keep=estimate) out=trest;
data trest; set trest; rename col1=gamma col2=b0 col3=b1;
data pred; merge eb(keep=estimate)
array pp{2} pred1-pred2;
if _n_ = 1 then merge trest(keep=gamma b0 b1);
do time=11 to 32;
linp      = b0 - b1*(group-2) + estimate;
pp{group} = 1-exp(- (exp(-linp)*time)**gamma);
symbolid  = patient+1;
output;
end;
keep pred1 pred2 time patient;
run;
data pred;
merge pred
cdf(where  = (group=1)
rename = (pred=pcdf1 minutes=minutes1)
keep   = pred minutes group)
cdf(where  = (group=2)
rename = (pred=pcdf2 minutes=minutes2)
keep   = pred minutes group);
drop group;
run;

proc sgplot data=pred noautolegend;
pcdf1   ='Estimated Patient-specific CDF';
series  x=time     y=pred1  /
group=patient
lineattrs=(pattern=solid color=black);
series  x=time     y=pred2  /
group=patient
lineattrs=(pattern=dash color=black);
scatter x=minutes1 y=pcdf1  /
markerattrs=(symbol=CircleFilled size=9);
scatter x=minutes2 y=pcdf2  /
markerattrs=(symbol=Circle       size=9);
run;
```

The separation of the distribution functions by groups is evident in Output 61.5.12. Most of the distributions of patients in the first group are to the left of the distributions in the second group. The separation is not complete, however. Several patients who are assigned the second pain reliever experience headache relief more quickly than patients assigned to the first group.

Output 61.5.12 Patient-Specific CDFs and Predicted Values. Pain Reliever 1: Solid Lines, Closed Circles; Pain Reliever 2: Dashed Lines, Open Circles.

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