# Tied time data → Untied time data

The codes used in this tutorial are available below.
rs.surv
stns, stpp, strs, stnet (original tied data)
stns, stpp, strs, stnet (untied data)

### Settings

This tutorial first used the colon.dta, which is originally a tied survival time data, to generate the relative survival estimates of 1-, 5-, and 10-year relative survival using the Pohar-Perme estimator (1), by rs.surv() in R, stpp, strs, stnet in Stata. In the second part, an small epislon ~ $Norm(0, 0.01)$ was added into each individual’s survival time to make no individual have exactly the same survival time (to untie the data).

• Survival time was calculated by (date of exit - date of diagnosis).
• We define failure as status == 1 2  (died from cancer or other causes).
• In strs, ht (hazard transformation) must be specified to command the survival be calculated by transforming cumulative excess hazard ($\lambda$) instead of an actuarial approach (default) (2). However, in stnet, by default, survival is calculated by using hazard transformation.
• In strs and stnet, monthly intervals were calculated up to ten years by specifying br(0(=1/12')10).

### Results

To make the tables look tidier, here we dismissed the 95% CI, which however can be found in the outputs of the syntax. The esimates of 1-, 5-, and 10-year relative survival by each program are shown below:

#### Using colon.dta (original tied time data)

t rs.surv stns stpp strs stnet
1 0.676 0.677 0.676 0.677 0.677
5 0.473 0.474 0.473 0.474 0.474
10 0.433 0.437 0.434 0.434 0.434

#### Using colon.dta (untied time data)

t rs.surv stns stpp strs stnet
1 0.676 0.677 0.677 0.677 0.677
5 0.472 0.474 0.473 0.474 0.474
10 0.431 0.435 0.433 0.434 0.434

### Explanation

• By default, both rs.surv(), stns, stpp calculate survival using the product integral method on the hazard level, whereas in strs and stnet time-scale is split into numbers of intervals (using actuarial life-table approach).
• In stpp, the Fleming-Harrington estimator (using the expoential of the negative cumulative (excess) hazard), which appears to be more sensitive to ties, is also eligible to be applied (3).
• stnet should generate the identical estimates as strs, given that ht is specified in strs.
• Removing ties did not have an effect with discrete time estimators using life-table framework. (strs and stnet).
• In strs and stnet, life-table framework is implemented to estimate relative survival. However, should the cutpoints be in months br(0(=1/12')10) or in years br(0(1)10)?
A: Monthly estimate is more accurate. Both strs and stnet - calculates the attained age and attained year at the beginning of each interval and takes the floor() of these values from the popmort file to obtain the expected mortality rate. However, typically the survivaly probility from the popmort file (calculated from 1-year probability of death, by using by $-exp(H)$) is the probability of surviving 1 year, $p$. If it is monthly interval, we take the twelth root of the survival probability, $p^{1/12}$. Calculating by month literally means we do it 12 times to calculate the survival from an $x$ year-old person until he turns $(x+1)$ years old, but if using annual interval, we do it once instead.

It is worth paying attention to how age and year are managed in each program.

• stns: age of dianosis in the cohort data and age in the popmort file are needed in age(); year of dianosis date of diagnosis and the calendar year in the popmort file should be specified in period() .
• stpp: age and date of diagnosis should be specified in agediag() and datediag().
• strs: age and year of diagnosis are required in diagage() and diagyear().
• stnet: date of dianosis diagdate() and date of birth birthdate() are required.

### References

1. Pohar Perme M, Stare J, Estève J. On Estimation in Relative Survival. Biometrics. 2012;68:113-120
2. Dickman P, Coviello E. Estimating and modelling relative survival. The Stata Journal. 2015;15:186-215.
3. Fleming TR, and Harrington DP. Nonparametric Estimation of the Survival Distribution in Censored Data.Communications in Statistics—Theory and Methods. 1984;13:2469–2486.

### Acknowledgement

We especially want to say thank-you to Paul Lambert for offering his scenario2_1.dta as an example dataset and substantial insights on the syntaxes of stnsand stnet.