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Discrete Time Survival Analysis

Patient 5 (56 years old, did not receive a drug treatment) was observed for one time period, died. So, the observation for patient was not censored. Patient 10 (58, no drug) was observed for two time periods did not die, i.e., observation was censored. Finally, patient 20 (52, no drug) was observed for four time periods, died (not censored).use http://www.ats.ucla.edu/stat/stata/library/cancer describeContains data from cancer.dta obs: 48 Patient Survival in Drug Trial vars: 7 2 Jan 1904 13:58 size: 1,248 (99.1% of memory free) ------------------------------------------------------------------------------- storage display value variable name type format label variable label ------------------------------------------------------------------------------- id float %9.0g studytime int %8.0g Months to death or end of exp. died int %8.0g 1 if patient died drug float %9.0g age int %8.0g Patient's age at start of exp. distime float %9.0g censor float %9.0g -------------------------------------------------------------------------------tab distimedistime | Freq. Percent Cum. ------------+----------------------------------- 1 | 11 22.92 22.92 2 | 13 27.08 50.00 3 | 6 12.50 62.50 4 | 8 16.67 79.17 5 | 4 8.33 87.50 6 | 6 12.50 100.00 ------------+----------------------------------- Total | 48 100.00univar age-------------- Quantiles -------------- Variable n Mean S.D. Min .25 Mdn .75 Max ------------------------------------------------------------------------------- age 48 55.88 5.66 47.00 50.50 56.00 60.00 67.00 -------------------------------------------------------------------------------list distime drug age died censor if id==5distime drug age died censor 5. 1 0 56 1 0list distime drug age died censor if id==10distime drug age died censor 10. 2 0 58 0 1list distime drug age died censor if id==20distime drug age died censor 20. 4 0 52 1 0

In this dataset there is one observation for each patient. In order to do discrete time survival analysis we to have as many observations as there are time periods for each patient. For patients that die we need a response variable that is zero until the last time period when it is coded one. For patients that don't die the response variable will be zero for every observation.

A collection of Stata commands written by Alexis Dinno (Harvard School of Public Health) will help us with the analysis. You can download this family of commands from within Stata by typingThe command that we are interested in is
**prsnperd** which creates the type of dataset that we
want. **prsnperd** wants a variable that indicates whether the observation is censored or not
which in our dataset is the variable **censor**.
**prsnperd** creates the following variables: **_period** which is the time period,
**_Y** which is the response variable and _d1 through _d6 which are the dummy coded time periods.
Here is what it looks like.

Now we can actually do the discrete time survival analysis using theprsnperd id distime censor list id _period _Y if id==5id _period _Y 5. 5 1 1list id _period _Y if id==10id _period _Y 11. 10 1 0 12. 10 2 0list id _period _Y if id==20id _period _Y 35. 20 1 0 36. 20 2 0 37. 20 3 0 38. 20 4 1

Bothlogit _Y drug age _d1-_d6, cluster(id) noconsLogit estimates Number of obs = 143 Wald chi2(8) = 45.39 Log likelihood = -55.65503 Prob > chi2 = 0.0000 (standard errors adjusted for clustering on id) ------------------------------------------------------------------------------ | Robust _Y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- drug | -3.024052 .6859866 -4.41 0.000 -4.368561 -1.679543 age | .1607128 .0497324 3.23 0.001 .063239 .2581866 _d1 | -9.309867 2.754574 -3.38 0.001 -14.70873 -3.911001 _d2 | -8.335442 2.641359 -3.16 0.002 -13.51241 -3.158473 _d3 | -8.326742 2.533321 -3.29 0.001 -13.29196 -3.361525 _d4 | -7.071942 2.564526 -2.76 0.006 -12.09832 -2.045564 _d5 | -7.19799 2.490689 -2.89 0.004 -12.07965 -2.316328 _d6 | -7.622593 2.722941 -2.80 0.005 -12.95946 -2.285726 ------------------------------------------------------------------------------logit _Y drug age _d1-_d6, noconsLogit estimates Number of obs = 143 LR chi2(8) = . Log likelihood = -55.65503 Prob > chi2 = . ------------------------------------------------------------------------------ _Y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- drug | -3.024052 .6347086 -4.76 0.000 -4.268058 -1.780046 age | .1607128 .051414 3.13 0.002 .0599433 .2614823 _d1 | -9.309867 2.922645 -3.19 0.001 -15.03815 -3.581589 _d2 | -8.335442 2.780394 -3.00 0.003 -13.78491 -2.885969 _d3 | -8.326742 2.823744 -2.95 0.003 -13.86118 -2.792306 _d4 | -7.071942 2.734906 -2.59 0.010 -12.43226 -1.711624 _d5 | -7.19799 2.811519 -2.56 0.010 -12.70847 -1.687513 _d6 | -7.622593 2.988678 -2.55 0.011 -13.48029 -1.764892 ------------------------------------------------------------------------------logit, orLogit estimates Number of obs = 143 LR chi2(8) = . Log likelihood = -55.65503 Prob > chi2 = . ------------------------------------------------------------------------------ _Y | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- drug | .0486039 .0308493 -4.76 0.000 .014009 .1686304 age | 1.174348 .0603779 3.13 0.002 1.061776 1.298854 _d1 | .0000905 .0002646 -3.19 0.001 2.94e-07 .0278315 _d2 | .0002399 .0006669 -3.00 0.003 1.03e-06 .0558007 _d3 | .000242 .0006832 -2.95 0.003 9.55e-07 .0612797 _d4 | .0008486 .0023208 -2.59 0.010 3.99e-06 .1805723 _d5 | .0007481 .0021033 -2.56 0.010 3.03e-06 .184979 _d6 | .0004893 .0014623 -2.55 0.011 1.40e-06 .1712052 ------------------------------------------------------------------------------

Notice that the hazard maxes out at time period four and then declines.dthaz drug age, specify(1 56)Discrete-Time Estimation of Conditional Hazard and Survival Probabilities ------------------------------------------------------------------------------ Time Parameterization: Fully Discrete Additional predictors specified as: drug = 1 age = 56 ----------------------------------------- Period p(Hazard) p(Survival) (T_j) ^H(T_j) ^S(T_j) ----------------------------------------- 0 -- 1 1 0.0344 0.9656 2 0.0863 0.8822 3 0.0870 0.8055 4 0.2505 0.6037 5 0.2276 0.4663 6 0.1616 0.3910 ----------------------------------------- Logit Link (assumes proportional odds)

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