# stpp - Estimating marginal relative (net) survival

All examples below use the colon cancer data available with strs. I first load and stset the data.

. use "https://pclambert.net/data/colon.dta", clear
(Colon carcinoma, diagnosed 1975-94, follow-up to 1995)

. stset surv_mm,f(status=1,2) id(id) scale(12) exit(time 120.5)

id:  id
failure event:  status == 1 2
obs. time interval:  (surv_mm[_n-1], surv_mm]
exit on or before:  time 120.5
t for analysis:  time/12

------------------------------------------------------------------------------
15,564  total observations
0  exclusions
------------------------------------------------------------------------------
15,564  observations remaining, representing
15,564  subjects
10,459  failures in single-failure-per-subject data
51,685.667  total analysis time at risk and under observation
at risk from t =         0
earliest observed entry t =         0
last observed exit t =  10.04167



I have restricted follow-up time to 120.5 months (just over 10 years). Survival time information is available in completed months, so is 0.5 if someone died in the first month after diagnosis etc. stpp requires survival time in years, so I have used the scale(12) option to transform from months to years.

### Marginal relative survival in the study population

I will first estimate marginal relative survival in the study population as a whole.

. stpp R_pp1 using "https://pclambert.net/data/popmort.dta", ///
>                 agediag(age) datediag(dx)                  ///
>                 pmother(sex) list(1 5 10)

Pohar Perme Estimates of Marginal Relative Survival

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.677 (0.669 to 0.685)
5    | 0.475 (0.464 to 0.486)
10    | 0.436 (0.417 to 0.457)
--------+--------------------------



This creates a new variable R_pp1 containing the marginal relative survival evaluated at each value of _t. Confidence limits are stored in R_pp1_lci and R_pp1_uci.

A filename that stores the expected mortality rates needs to be given. In addition options the age at diagnosis (agediag()) and the date of diagnosis (datediag()) are required. The age at diagnosis should be in years, but it is best to avoid using truncated (integer) age as this assumes that each person was diagnosed on their birthday. The pmother(sex) option is required as the expected rates vary by sex. If the expected rates vary by other factors, for example region, deprivation etc, then these should be added to the pmother() option, and these variables should exist in both the data and the population mortality file.

There are options to define the names of the attained age and attained calendar year variables as well as the name of the rate variable in the population mortality file. My syntax is simple here as I have relying on the default names. See the help file for more details.

The list() option lists the times at which estimates are to be displayed on screen. Note that when using the list() option, the results are saved to a matrix, so that these can be accessed for table creation etc.

. matrix list r(PP)

r(PP)[3,4]
c1         c2         c3         c4
r1          1  .67692218  .66924135  .68469116
r2          5  .47485589  .46431329  .48563786
r3         10  .43646685  .41730935  .45650382



Alternatively the marginal relative survival can be plotted against _t.

. twoway (rarea R_pp1_lci R_pp1_uci _t, sort color(red%30) connect(stairstep)) ///
>       (line R_pp1 _t, sort lcolor(red) connect(stairstep))                   ///
>       ,legend(off)                                                           ///
>       xtitle("Years from diagnosis")                                         ///
>       ytitle(Marginal relative survival)                                     ///
>       ylabel(,format(%3.1f))                                                 ///
>       name(R_pp1, replace)



### Marginal relative survival stratified by sex

Using the by option will give estimates separately by sex.

. stpp R_pp2 using "https://pclambert.net/data/popmort.dta", ///
>       agediag(age) datediag(dx)                            ///
>       pmother(sex) list(1 5 10)                            ///
>       by(sex)

Pohar Perme Estimates of Marginal Relative Survival

-> sex = 1

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.689 (0.677 to 0.701)
5    | 0.483 (0.466 to 0.500)
10    | 0.423 (0.394 to 0.454)
--------+--------------------------

-> sex = 2

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.669 (0.659 to 0.679)
5    | 0.469 (0.456 to 0.483)
10    | 0.445 (0.420 to 0.472)
--------+--------------------------



And can be plotted as deparate lines through use of if statements.

. twoway (rarea R_pp2_lci R_pp2_uci _t if sex==1, sort color(red%30) connect(stairstep)) ///
>       (line R_pp2 _t if sex==1, sort lcolor(red) connect(stairstep))                   ///
>       (rarea R_pp2_lci R_pp2_uci _t if sex==2, sort color(blue%30) connect(stairstep)) ///
>       (line R_pp2 _t if sex==2, sort lcolor(blue) connect(stairstep))                  ///
>       ,legend(order(2 "males" 4 "females") ring(0) pos(1))                             ///
>       xtitle("Years from diagnosis")                                                   ///
>       ytitle(Marginal relative survival)                                               ///
>       ylabel(,format(%3.1f))                                                           ///
>       name(R_pp2, replace)



What is important to note here is that the marginal relative survival estimates may not be comparable due to differences in the age distribution. Each line gives an estimate of marginal net survival which can be considerd an average over each groups own age distribution (strictly is is an estimate averaged over other variables in the population mortality file too.) Although, the age distributions do not differ much in this case, in general it is sensible to age standardize so that differences are not due to differences in the age distribution.

There are two ways to age standardize using stpp, which are explained in the next two sections.

Traditional age standardization first obtains estimates separately with each age group and then obtains a weighted average. The weights for each age group could be derived from the study data, e.g. the combined age distribution of males and females or the age distribution of one of the sexes. Alternatively, a reference age distibution could be used, such as the International Cancer Survival Standard (ICSS) (Corazziari 2004).

I will use the ICSS weights. First the same agegroups used in ICSS are created

. recode age (min/44=1) (45/54=2) (55/64=3) (65/74=4) (75/max=5), gen(ICSSagegrp)
(15564 differences between age and ICSSagegrp)

. tab ICSSagegrp

RECODE of |
age (Age at |
diagnosis) |      Freq.     Percent        Cum.
------------+-----------------------------------
1 |        735        4.72        4.72
2 |      1,243        7.99       12.71
3 |      2,767       17.78       30.49
4 |      4,951       31.81       62.30
5 |      5,868       37.70      100.00
------------+-----------------------------------
Total |     15,564      100.00



Then stpp can be used with the standstrata() and standweight() options.

. stpp R_pp3 using "https://pclambert.net/data/popmort.dta", ///
>       agediag(age) datediag(dx)                            ///
>       pmother(sex) list(1 5 10)                            ///
>       by(sex)                                              ///
>       standstrata(ICSSagegrp)                              ///
>       standweight(0.07 0.12 0.23 0.29 0.29)

Pohar Perme Estimates of Marginal Relative Survival
(Standardized by ICSSagegrp)

-> sex = 1

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.697 (0.685 to 0.709)
5    | 0.490 (0.472 to 0.507)
10    | 0.424 (0.394 to 0.453)
--------+--------------------------

-> sex = 2

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.701 (0.691 to 0.711)
5    | 0.490 (0.477 to 0.503)
10    | 0.457 (0.435 to 0.479)
--------+--------------------------



Note that the weights given in the standweight() option give the ICSS weights for each age group.

These estimates can now be plotted.

. twoway (rarea R_pp3_lci R_pp3_uci _t if sex==1, sort color(red%30) connect(stairstep)) ///
>       (line R_pp3 _t if sex==1, sort lcolor(red) connect(stairstep))                   ///
>       (rarea R_pp3_lci R_pp3_uci _t if sex==2, sort color(blue%30) connect(stairstep)) ///
>       (line R_pp3 _t if sex==2, sort lcolor(blue) connect(stairstep))                  ///
>       ,legend(order(2 "males" 4 "females") ring(0) pos(1))                             ///
>       xtitle("Years from diagnosis")                                                   ///
>       ytitle(Marginal relative survival)                                               ///
>       ylabel(,format(%3.1f))                                                           ///
>       name(R_pp3, replace)



## Age standardization using individual weights

An alternative way to age standardize is to upweight or downweight individuals relative to the reference population (Rutherford et al 2020). This avoids the need to estimate separately in each if the age groups.

First the weights need to be stored in a variable and then weights are calulated as the ratio of the proportion in the reference population to the proportion in the age group to which an individual belongs.

. recode ICSSagegrp (1=0.07) (2=0.12) (3=0.23) (4=0.29) (5=0.29), gen(ICSSwt)
(15564 differences between ICSSagegrp and ICSSwt)

. label define ICSSlab 1 "<45" 2 "45-54" 3 "55-64" 4 "65-75" 5 "75+"

. label values ICSSagegrp ICSSlab

. bysort sex: gen sextotal= _N

. bysort ICSSagegrp sex:gen a_age = _N/sextotal

. gen double wt_age = ICSSwt/a_age



The weights for males are shown below

. by ICSSagegrp sex: gen firstrow = _n==1

. list ICSSagegrp ICSSwt a_age wt_age if firstrow & sex==1, noobs ab(12)

+--------------------------------------------+
| ICSSagegrp   ICSSwt      a_age      wt_age |
|--------------------------------------------|
|        <45      .07   .0605678   1.1557292 |
|      45-54      .12   .0932177   1.2873096 |
|      55-64      .23   .2157729   1.0659357 |
|      65-75      .29   .3337539   .86890357 |
|        75+      .29   .2966877   .97745879 |
+--------------------------------------------+



The variable ICSSwt shows the proportion in each age group in the reference population and the a_age variables gives the proportion observed in the study popopulation. For the youngest age group there are slightly smaller proportion in the study population and so each individual in this group is upweighted by 1.156. In the oldest age group there is a slighly higher proportion in the study population compared to the reference population and so each individual is slightly downweighted by 0.977.

Now the weights have been calculated these can be passed to stpp using the indweights() option.

. stpp R_pp4 using "https://pclambert.net/data/popmort.dta", ///
>       agediag(age) datediag(dx)                            ///
>       pmother(sex) list(1 5 10)                            ///
>       by(sex)                                              ///
>       indweights(wt_age)

Pohar Perme Estimates of Marginal Relative Survival

-> sex = 1

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.692 (0.680 to 0.704)
5    | 0.484 (0.468 to 0.502)
10    | 0.425 (0.397 to 0.455)
--------+--------------------------

-> sex = 2

Time    |   PP (95% CI)
--------+--------------------------
1    | 0.696 (0.686 to 0.706)
5    | 0.485 (0.472 to 0.498)
10    | 0.452 (0.432 to 0.473)
--------+--------------------------



And then plotted

. twoway (rarea R_pp4_lci R_pp4_uci _t if sex==1, sort color(red%30) connect(stairstep)) ///
>       (line R_pp4 _t if sex==1, sort lcolor(red) connect(stairstep))                   ///
>       (rarea R_pp4_lci R_pp4_uci _t if sex==2, sort color(blue%30) connect(stairstep)) ///
>       (line R_pp4 _t if sex==2, sort lcolor(blue) connect(stairstep))                  ///
>       ,legend(order(2 "males" 4 "females") ring(0) pos(1))                             ///
>       xtitle("Years from diagnosis")                                                   ///
>       ytitle(Marginal relative survival)                                               ///
>       ylabel(,format(%3.1f))                                                           ///
>       name(R_pp4, replace)



## References

I. Corazziari, M. Quinn, R. Capocaccia, R. Standard cancer patient population for age standardising survival ratios. Eur J Cancer 2004:40:2307-2316

E. Coviello, P.W. Dickman, K. Seppä, A. Pokhrel. Estimating net survival using a life table approach. The Stata Journal 2015;15:173-185

P.W. Dickman, E. Coviello, M.Hills, M. Estimating and modelling relative survival. The Stata Journal 2015;15:186-215

M. Pohar Perme, J. Stare, J. Estève. On Estimation in Relative Survival Biometrics 2012;68:113-120

Rutherford, M.J., Dickman, P.W., Coviello, E. & Lambert, P.C. Estimation of age-standardized net survival, even when age-specific data are sparse. Cancer Epidemiology 2020, 67, 101745.