# Proper way to match a reference population for survival analysis

I am performing survival analysis on a specific group under study containing thousands of individuals. I am looking for the best way to obtain a reference population from the total population (millions of individuals).

The reference population must be matched on age, e.g. have the same age distribution as the study population. Since the total population is much larger than the study population, the matched reference population can also be much larger.

The easiest way to proceed would be to simply draw a subsample from the total population of the same size as the study population (matched for age). Doing so would mean that the vast majority of the total population is never used, which may introduce bias and surely widens confidence intervals on the reference population.

To avoid this issue, I am currently considering two resampling approaches. I use the Kaplan-Meier estimator for the survivor function $\hat{S}(t)$.

First approach

1. bootstrap the total population to draw a large number of (matched) subsamples (lets say $n_{boot}$ samples of $n_{study}$ individuals, matched for age),

2. aggregate the subsamples into a large population $\mathcal{P}_{aggr}$ which contains duplicates ($| \mathcal{P}_{aggr}| = n_{boot} \times n_{study}$ non-unique individuals),

3. compute reference survivor function based on $\mathcal{P}_{aggr}$,

4. use exponential Greenwood confidence intervals (log-log transform).

Second approach

1. bootstrap the total population to draw a large number of (matched) subsamples (lets say $n_{boot}$ samples of $n_{study}$ individuals, matched for age),

2. compute survivor function per subsample: $\hat{S}_i(t), i=1..n_{boot}$,

3. aggregate survivor function estimates, global survivor estimate being $\hat{S}(t)=\frac{\sum_{i=1}^{n_{boot}} \hat{S}_i(t)}{n_{boot}}$,

4. confidence intervals can be obtained directly from the set of survivor function estimates.

Are there any strong statistical factors I must consider in favor of/against either approach? Is there a better way to obtain a matched reference population? I currently lean towards using the first approach as it is somewhat easier to implement and has a commonly used closed form for the confidence intervals.

• I may be missing something, but I'm not sure what's wrong with a random subsample here (simple random sampling). Just because most of the population is never used doesn't mean you've introduced bias. Jul 19, 2013 at 1:34
• I have all the data, I can use much more individuals at no extra cost (apart from some effort on my behalf). Surely using more data will yield more reliable results, e.g. less (risk for) bias and narrower confidence intervals. I want to get the best possible result with my resources. Jul 19, 2013 at 6:29

I will try to provide answer that is relevant to your second question: is there a better [or at least more mainstream] way to obtain a matched reference population?

### Relative survival analysis

The standard approach to comparing the survival in a certain subgroup to that in a wider (and much larger) population is to use relative survival analysis. An accessible introduction is available in this article, which is focused on explaining the implementation in the R package relsurv.

The essential idea is that the hazard experienced in the specific group under study can be decomposed into two components: one of which is contributed by the experience of the wider population and the other is contributed by the characteristics unique to the group under study.

One common approach is to view the effect of these components as additive:

$$\lambda_O = \lambda_P + \lambda_E$$

$$\lambda_P$$ is the baseline hazard experienced by the population and $$\lambda_E$$ is the excess hazard attributable to the particular exposures of the group under study. Regression can be performed which allows the difference between $$\lambda_O$$ and $$\lambda_E$$ to vary with age, sex, year of birth etc. I won't reiterate more from the article linked, because I can explain it no better.