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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.
| Model Information | ||
|---|---|---|
| Data Set | WORK.RATS | |
| Dependent Variable | Days | Days from Exposure to Death |
| Censoring Variable | Status | |
| Censoring Value(s) | 0 | |
| Model | Piecewise Exponential | |
| 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.
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.
| 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.
The noninformative uniform prior is used for the regression coefficient Group (Output 64.14.5), as in the section Bayesian Analysis.
Summary statistics for all model parameters are shown in Output 64.14.6 and Output 64.14.7.
| 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 |
| 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.
| 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 |
| Effective Sample Sizes | |||
|---|---|---|---|
| Parameter | ESS | Correlation 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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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.