# Proportion of controls in case-cohort design for survival analysis

I'm working on a study that is drawing from another very large study, and proposing a case-cohort design in order to reduce the overall number of samples we use (primarily due to cost of sample analysis). Briefly, the main cohort consists of ~40,000 individuals, ~1250 of whom had an event over the duration of the study. The plan is to use all 1250 event samples (cases) and select a random sample from the non-event samples (controls). The intent is to perform survival analysis (Cox Proportional Hazards or Accelerated Failure Time) to generate a probability of event within a given time period.

The question that has come up is the number of controls to choose. To me, a 1:1 balance is OK, but perhaps even better if we had more than one control per case. It was suggested, however, that this was unnecessary due to the cases being the samples which are affecting the calculation of event times. There was even a suggestion of 2 cases per 1 control.

I'm not sure how the optimization process operates and haven't found much description of the overall role of non-event controls in survival analysis, but it seems odd to me that there might be a suggestion of fewer that 1 control per case. I would assume that the controls are quite important to establish the distributions of the co-variates used in the model, and the more information we have about those distribution in controls, the better we can determine how cases differ.

What is an appropriate sample size for such a study - rules of thumb, of course, and how do the non-event samples affect the procedure? Are they of limited value as suggested or is my perception of their importance correct?

• Is there some reason why you want to limit the number of controls you use? It sounds like you are making this into a case-control study when you don't need to (and shouldn't). But perhaps you don't have the resources to measure all 40,000 subjects? – Cliff AB Oct 1 '15 at 18:13
• Cost is exactly the answer. We can't process all 40,000 subjects/samples from the study. Regarding making it a case-control (or nested case-control), my assumption was that this was more akin to a case-cohort design. Please correct me if I'm wrong. – KirkD_CO Oct 1 '15 at 18:25

Based on your comments, I would say you're actually in a better situation than a generic case-control scenario. In a standard case-control study, you sample $n_1$ cases and $n_2$ controls, with no knowledge of the proportion of cases and controls in the true population.

However, in your case, you know exactly how many cases and controls you have your cohort, you just don't have the resources to measure all of them. As such, you can use the following analysis scheme:

• Measure all cases
• Randomly sample and measure $p$ proportion of your controls
• Run a Cox-PH model in which cases have weight = 1 and controls have weight = $p^{-1}$

This will give you unbiased estimates of your survival curve, unlike a standard case-control study.

This does not yet answer your question of interest: what case:control ratio should you use (reparameterized as what $p$ should you use?). There's not really a good rule of thumb, as far as I know. Clearly, more data is better (ignoring cost) than less data. But this benefit drops off quickly; the more controls you sample, the more you will know about what doesn't case the event, but not about what does.

As such, I think a 1:1 matching really should be fine given that you want to keep costs down. But there could be some short comes in the following situations: you have lots of covariates and/or some of your covariates are extremely skewed (i.e. 99.9% of the values of $x_1$ = 0 and 0.1% = 1).

• Thanks for the response. I hadn't considered the weighting scheme you mention, and that helps a lot. Certainly something to consider. What of the second part of my question, sp., the effects of controls. Are they as relevant to the modelling process as the cases or do they serve less of a role. The supposition from a colleague is that we can have fewer controls than cases as they don't impact the process of fitting the survival curve. I'm not quite on board with that, but I don't have a rationale for it. – KirkD_CO Oct 1 '15 at 19:04
• In general, your colleague is correct; uncensored observations are more informative than censored observations (in the extreme: if all your data is censored, you have no information!). There can be some exceptions; suppose you had a rare covariate that was 100% protective. If you had few uncensored values, you would have very little to no information about this effect. But that's somewhat of an extreme case. And of course ignoring the censored cases causes bias. But yes, you are likely to get little return in using 2:1 over 1:1 controls:cases. – Cliff AB Oct 1 '15 at 19:36
• Thank you again. This has been tremendously helpful! I have another related but different question about this same project - please watch for that one to come up. I decided to separate things into unique questions. – KirkD_CO Oct 1 '15 at 20:45
• @CliffAB, if your primary interest is in estimating covariate effects, can you omit the weights? Similar to how you can estimate covariate effects in logistic regression with unequal probability samples? – not_bonferroni Jan 19 '17 at 20:00
• @not_bonferroni: correct – Cliff AB Jan 20 '17 at 15:08

This accepted answer is actually incorrect and confused with the concept of case-control and case-cohort.

1. In case cohort, one select a subcohort $S$ of certain size $k$ (mainly due to financial or logistic constraints that the full cohort cannot be measured completely) by either simply randomization (c.f. Prentice 1986) or stratified randomization. Case-cohort is a unmatched design and sampling should be done a priori without knowledge of case status or time. But depending on the design itself, all the cases (even outside the sampled subcohort $S$ can be included (see point 2 below).

2. In case-cohort a pseudo partial likelihood is used: $$\mathcal{L} = \prod_{t_j}\frac{\exp\bigl(\beta^Tz_i(t_j)\bigr)w_i}{\sum_{l\in R(t_j)} \exp(\beta^Tz_l(t_j))w_l}$$ There are several choices of weights $w_i$, $w_l$ and risk set $R$.

• For example one can choose $R=S \cup \{\text{cases only in } S\}$ and all $w_l=1$.
• An alternative one is to also include non-subcohort cases in $R$ and use inverse probability weighting. But the weights is $w_l=1/p_l$ where $p_l$ is the probability that $l$ is included in the case-cohort sample ($p_l=1$ for cases in this setting because we always include them).
3. Back to the original question, it is not clear what the design really is. If there is matching and only a fraction of the whole cohort is sampled, I would guess that could be a nested case-control study. But before I have more information, I cannot tell for sure.