# RMST of Standardized Survival Functions

Here I will show another useful measure from standardized survival functions. There have been several papers promoting the use of restricted mean survival time (RMST) in clinical trials. The arguments are (i) ease of interpretation (though I am not convinced a restricted mean is that easy to explain) and (ii) providing a simple summary in the presence of non-proportional hazards. See Royston and Parmar (2013) for a description of the use of the measure in RCTs.

The restricted mean survival time at time $t^*$ is defined as, $$ E\left[min(t,t^*)\right] $$ i.e. it is the mean up to some point $t^*$. The treatment effect in a RCT can be defined as the difference in RMST between the randomized arms at time $t^*$. The RMST can be estimated by calculating the area under the survival curve between 0 and $t^*$. In an observational study where we need to take account of potential confounders, we can define the RMST of the standardized survival function as

$$ RMST(t^*|X=x,Z) = \int_0^{t^*} E\left[S(t|X=x,Z)\right] $$

and is estimated by

$$ \widehat{RMST}(t^*|X=x,Z) = \int_0^{t^*} \frac{1}{N}\sum_{i=1}^{N}S(t|X=x,Z=z_i)] $$

Contrasts between exposure groups can either be differences or ratios,

$$ \widehat{RMST}(t^*|X=1,Z) - \widehat{RMST}(t^*|X=0,Z) $$

$$ \frac{\widehat{RMST}(t^*|X=1,Z)}{\widehat{RMST}(t^*|X=0,Z)} $$

Standardized RMST and contrasts is implemented in `standsurv`

using the `rmst`

option.

## Example

I will use the Rotterdam Breast cancer data. The code below loads and `stset`

’s the data and then fits a model using `stpm2`

.

```
. use https://www.pclambert.net/data/rott2b, clear
(Rotterdam breast cancer data (augmented with cause of death))
. stset os, f(osi==1) scale(12) exit(time 120)
failure event: osi == 1
obs. time interval: (0, os]
exit on or before: time 120
t for analysis: time/12
------------------------------------------------------------------------------
2,982 total observations
0 exclusions
------------------------------------------------------------------------------
2,982 observations remaining, representing
1,171 failures in single-record/single-failure data
20,002.424 total analysis time at risk and under observation
at risk from t = 0
earliest observed entry t = 0
last observed exit t = 10
. stpm2 hormon age enodes pr_1, scale(hazard) df(4) eform nolog tvc(hormon) dftvc(3)
Log likelihood = -2666.5999 Number of obs = 2,982
--------------------------------------------------------------------------------
| exp(b) Std. Err. z P>|z| [95% Conf. Interval]
---------------+----------------------------------------------------------------
xb |
hormon | .8019893 .0741703 -2.39 0.017 .6690322 .961369
age | 1.013249 .0024115 5.53 0.000 1.008534 1.017987
enodes | .1132406 .011008 -22.41 0.000 .0935961 .1370082
pr_1 | .9061179 .0119267 -7.49 0.000 .883041 .9297979
_rcs1 | 2.644573 .0814503 31.58 0.000 2.489656 2.809129
_rcs2 | 1.209479 .0379393 6.06 0.000 1.13736 1.286172
_rcs3 | 1.014 .0162037 0.87 0.384 .9827339 1.046262
_rcs4 | .9961807 .0072731 -0.52 0.600 .9820273 1.010538
_rcs_hormon1 | 1.003465 .0756175 0.05 0.963 .8656822 1.163176
_rcs_hormon2 | .891054 .056664 -1.81 0.070 .786637 1.009331
_rcs_hormon3 | 1.025052 .0390804 0.65 0.516 .9512477 1.104583
_cons | 1.103353 .1771893 0.61 0.540 .8054133 1.511508
--------------------------------------------------------------------------------
Note: Estimates are transformed only in the first equation.
```

I have made the effect of our exposure, `hormon`

, time-dependent using the `tvc`

option to illustrate that we can have interactions etc with our exposure in our model. This is an interaction with time, i.e. non proportional hazards.

I first calculate the standardized survival curves where everyone is forced to be exposed and then unexposed.

```
. range timevar 0 10 100
(2,882 missing values generated)
. standsurv, at1(hormon 0) at2(hormon 1) timevar(timevar) ci atvar(S_hormon0 S_hormon1)
.
. twoway (area S_hormon0 timevar, sort fcolor(red%30) lcolor(red)) ///
> , legend(off) ///
> ylabel(0(0.1)1, format(%3.1f)) ///
> ytitle("S(t)") ///
> xtitle("Years from surgery") ///
> title("No treatment") ///
> name(hormon0, replace)
.
. twoway (area S_hormon1 timevar, sort fcolor(blue%30) lcolor(blue)) ///
> , legend(off) ///
> ylabel(0(0.1)1, format(%3.1f)) ///
> ytitle("S(t)") ///
> xtitle("Years from surgery") ///
> title("Treatment") ///
> name(hormon1, replace)
.
. graph combine hormon0 hormon1, nocopies
```

The RMST at 10 years for each of the standardized survival functions is the area under the standardized survival curve, shown by the shaded areas in the graphs above.

I will now run `standsurv`

again with the `rmst`

option to estimate these togther with the difference in RMST. I only want the RMST at
10 years so create a variable `t_rmst10`

with only one observation, equal to 10.

```
. gen t_rmst10 = 10 in 1
(2,981 missing values generated)
. standsurv, at1(hormon 0) at2(hormon 1) timevar(t_rmst10) rmst ci contrast(difference) ///
> atvar(rmst_h0 rmst_h1) contrastvar(rmstdiff)
```

I will first list the standardized RMST in both treatment groups.

```
. list t_rmst10 rmst_h0* rmst_h1* in 1, noobs abb(12)
+------------------------------------------------------------------------------------------+
| t_rmst10 rmst_h0 rmst_h0_lci rmst_h0_uci rmst_h1 rmst_h1_lci rmst_h1_uci |
|------------------------------------------------------------------------------------------|
| 10 7.5444214 7.4318182 7.6587307 7.9386152 7.6870964 8.1983635 |
+------------------------------------------------------------------------------------------+
```

The RMST at 10 years is 7.54 years in those not taking treatment and 7.94 years in those taking treatment. The 95% confidence
intervals are also shown. As I used the `contrast(difference)`

option I can look at the difference in RMST at 10 years.

```
. list t_rmst rmstdiff* in 1, noobs abb(12)
+---------------------------------------------------+
| t_rmst10 rmstdiff rmstdiff_lci rmstdiff_uci |
|---------------------------------------------------|
| 10 .3941938 .11623628 .67215133 |
+---------------------------------------------------+
```

The difference is 0.39 years (95% CI 0.12 to 0.67).

The RMST will vary by the choice of $t^*$. A range of values of $t^*$ can be given and then plotted.

```
. range t_rmst 0 10 50
(2,932 missing values generated)
. standsurv, at1(hormon 0) at2(hormon 1) timevar(t_rmst) rmst ci contrast(difference) ///
> atvar(rmst_h0b rmst_h1b) contrastvar(rmstdiffb)
```

We can plot how the RMST changes and the difference in RMST changes as a function of $t^*$.

```
. twoway (line rmst_h0b rmst_h1b t_rmst) ///
> , legend(order(1 "No treatment" 2 "Treatment") cols(1) pos(11)) ///
> ytitle("RMST (years)") ///
> xtitle("Years from surgery") ///
> name(RMST,replace)
.
. twoway (rarea rmstdiffb_lci rmstdiffb_uci t_rmst, color(blue%20)) ///
> (line rmstdiffb t_rmst, lcolor(blue)) ///
> , legend(off) ///
> ylabel(, format(%3.1f)) ///
> ytitle("Difference in RMST (years)") ///
> xtitle("Years from surgery") ///
> name(RMSTdiff, replace)
.
. graph combine RMST RMSTdiff, nocopies
```

## References

Royston, P. Parmar, M. K. B. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome.
*BMC medical research methodology* 2013;**13**:152