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 The PHREG Procedure

## Example 64.14 Bayesian Analysis of Piecewise Exponential Model

This example illustrates using a piecewise exponential model in a Bayesian analysis. Consider the Rats data set in the section Getting Started: PHREG Procedure. In the following statements, PROC PHREG is used to carry out a Bayesian analysis for the piecewise exponential model. In the BAYES statement, the option PIECEWISE stipulates a piecewise exponential model, and PIECEWISE=HAZARD requests that the constant hazards be modeled in the original scale. By default, eight intervals of constant hazards are used, and the intervals are chosen such that each has roughly the same number of events.

```   data Rats;
label Days  ='Days from Exposure to Death';
input Days Status Group @@;
datalines;
143 1 0   164 1 0   188 1 0   188 1 0
190 1 0   192 1 0   206 1 0   209 1 0
213 1 0   216 1 0   220 1 0   227 1 0
230 1 0   234 1 0   246 1 0   265 1 0
304 1 0   216 0 0   244 0 0   142 1 1
156 1 1   163 1 1   198 1 1   205 1 1
232 1 1   232 1 1   233 1 1   233 1 1
233 1 1   233 1 1   239 1 1   240 1 1
261 1 1   280 1 1   280 1 1   296 1 1
296 1 1   323 1 1   204 0 1   344 0 1
;
run;
```
```proc phreg data=Rats;
model Days*Status(0)=Group;
bayes seed=1 piecewise=hazard;
run;
```

The "Model Information" table in Output 64.14.1 shows that the piecewise exponential model is being used.

Output 64.14.1 Model Information
The PHREG Procedure

Bayesian Analysis

Model Information
Data Set WORK.RATS
Dependent Variable Days Days from Exposure to Death
Censoring Variable Status
Censoring Value(s) 0
Model Piecewise Exponential
Sampling Algorithm ARMS
Burn-In Size 2000
MC Sample Size 10000
Thinning 1

By default the time axis is partitioned into eight intervals of constant hazard. Output 64.14.2 details the number of events and observations in each interval. Note that the constant hazard parameters are named Lambda1,..., Lambda8. You can supply your own partition by using the INTERVALS= suboption within the PIECEWISE=HAZARD option.

Output 64.14.2 Interval Partition
Constant Hazard Time Intervals
Interval N Event Hazard
Parameter
[Lower, Upper)
0 176 5 5 Lambda1
176 201.5 5 5 Lambda2
201.5 218 7 5 Lambda3
218 232.5 5 5 Lambda4
232.5 233.5 4 4 Lambda5
233.5 253.5 5 4 Lambda6
253.5 288 4 4 Lambda7
288 Infty 5 4 Lambda8

The model parameters consist of the eight hazard parameters Lambda1, ..., Lambda8, and the regression coefficient Group. The maximum likelihood estimates are displayed in Output 64.14.3. Again, these estimates are used as the starting values for simulation of the posterior distribution.

Output 64.14.3 Maximum Likelihood Estimates
Maximum Likelihood Estimates
Parameter DF Estimate Standard
Error
95% Confidence Limits
Lambda1 1 0.000953 0.000443 0.000084 0.00182
Lambda2 1 0.00794 0.00371 0.000672 0.0152
Lambda3 1 0.0156 0.00734 0.00120 0.0300
Lambda4 1 0.0236 0.0115 0.00112 0.0461
Lambda5 1 0.3669 0.1959 -0.0172 0.7509
Lambda6 1 0.0276 0.0148 -0.00143 0.0566
Lambda7 1 0.0262 0.0146 -0.00233 0.0548
Lambda8 1 0.0545 0.0310 -0.00626 0.1152
Group 1 -0.6223 0.3468 -1.3020 0.0573

Without using the PRIOR= suboption within the PIECEWISE=HAZARD option to specify the prior of the hazard parameters, the default is to use the noninformative and improper prior displayed in Output 64.14.4.

Output 64.14.4 Hazard Prior
Improper Prior for Hazards
Parameter Prior
Lambda1 1 / Lambda1
Lambda2 1 / Lambda2
Lambda3 1 / Lambda3
Lambda4 1 / Lambda4
Lambda5 1 / Lambda5
Lambda6 1 / Lambda6
Lambda7 1 / Lambda7
Lambda8 1 / Lambda8

The noninformative uniform prior is used for the regression coefficient Group (Output 64.14.5), as in the section Bayesian Analysis.

Output 64.14.5 Coefficient Prior
Uniform Prior for Regression
Coefficients
Parameter Prior
Group Constant

Summary statistics for all model parameters are shown in Output 64.14.6 and Output 64.14.7.

Output 64.14.6 Summary Statistics
The PHREG Procedure

Bayesian Analysis

Posterior Summaries
Parameter N Mean Standard
Deviation
Percentiles
25% 50% 75%
Lambda1 10000 0.000945 0.000444 0.000624 0.000876 0.00118
Lambda2 10000 0.00782 0.00363 0.00519 0.00724 0.00979
Lambda3 10000 0.0155 0.00735 0.0102 0.0144 0.0195
Lambda4 10000 0.0236 0.0116 0.0152 0.0217 0.0297
Lambda5 10000 0.3634 0.1965 0.2186 0.3266 0.4685
Lambda6 10000 0.0278 0.0153 0.0166 0.0249 0.0356
Lambda7 10000 0.0265 0.0151 0.0157 0.0236 0.0338
Lambda8 10000 0.0558 0.0323 0.0322 0.0488 0.0721
Group 10000 -0.6154 0.3570 -0.8569 -0.6186 -0.3788

Output 64.14.7 Interval Statistics
Posterior Intervals
Parameter Alpha Equal-Tail Interval HPD Interval
Lambda1 0.050 0.000289 0.00199 0.000208 0.00182
Lambda2 0.050 0.00247 0.0165 0.00194 0.0152
Lambda3 0.050 0.00484 0.0331 0.00341 0.0301
Lambda4 0.050 0.00699 0.0515 0.00478 0.0462
Lambda5 0.050 0.0906 0.8325 0.0541 0.7469
Lambda6 0.050 0.00676 0.0654 0.00409 0.0580
Lambda7 0.050 0.00614 0.0648 0.00421 0.0569
Lambda8 0.050 0.0132 0.1368 0.00637 0.1207
Group 0.050 -1.3190 0.0893 -1.3379 0.0652

The default diagnostics—namely, lag1, lag5, lag10, lag50 autocorrelations (Output 64.14.8), the Geweke diagnostics (Output 64.14.9), and the effective sample size diagnostics (Output 64.14.10)—show a good mixing of the Markov chain.

Output 64.14.8 Autocorrelations
The PHREG Procedure

Bayesian Analysis

Posterior Autocorrelations
Parameter Lag 1 Lag 5 Lag 10 Lag 50
Lambda1 0.0705 0.0015 0.0017 -0.0076
Lambda2 0.0909 0.0206 -0.0013 -0.0039
Lambda3 0.0861 -0.0072 0.0011 0.0002
Lambda4 0.1447 -0.0023 0.0081 0.0082
Lambda5 0.1086 0.0072 -0.0038 -0.0028
Lambda6 0.1281 0.0049 -0.0036 0.0048
Lambda7 0.1925 -0.0011 0.0094 -0.0011
Lambda8 0.2128 0.0322 -0.0042 -0.0045
Group 0.5638 0.0410 -0.0003 -0.0071

Output 64.14.9 Geweke Diagnostics
Geweke Diagnostics
Parameter z Pr > |z|
Lambda1 -0.0705 0.9438
Lambda2 -0.4936 0.6216
Lambda3 0.5751 0.5652
Lambda4 1.0514 0.2931
Lambda5 0.8910 0.3729
Lambda6 0.2976 0.7660
Lambda7 1.6543 0.0981
Lambda8 0.6686 0.5038
Group -1.2621 0.2069

Output 64.14.10 Effective Sample Size
Effective Sample Sizes
Parameter ESS Autocorrelation
Time
Efficiency
Lambda1 7775.3 1.2861 0.7775
Lambda2 6874.8 1.4546 0.6875
Lambda3 7655.7 1.3062 0.7656
Lambda4 6337.1 1.5780 0.6337
Lambda5 6563.3 1.5236 0.6563
Lambda6 6720.8 1.4879 0.6721
Lambda7 5968.7 1.6754 0.5969
Lambda8 5137.2 1.9466 0.5137
Group 2980.4 3.3553 0.2980

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