# What is the best way to establish correlation between two time-to-event analysis?

Hi,
There has been a recent emergence of interest in surrogate endpoint analysis. For example, to determine if progression-free survival is a surrogate for overall survival. Current literature uses Kendall’s tau method or Spearman’s correlation tests…I find them hard to be interpretable… Is there a way that we can establish direct correlation between two right censored time-to-event analysis using any other modelling approach? And is there a package in R for that?

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Although some prefer Pearson’s correlation to Spearman’s, Pearson’s correlation 1) might not be appropriate for not-normally distributed data, 2) it may vary with monotonic transformations of the variables, and 3) is sensitive to outliers.
Spearman’s correlation is preferred because 1) its values are very similar to Pearson’s (on normally distributed data), which is familiar to researchers, 2) it is invariant to monotonic transformations and 3) not sensitive to outliers.
When data are right-censored (time-to-event data, for example), computing correlation is not straightforward, requiring special methods. For unadjusted correlation, one can use packages SurvCorr and survSpearman. Package SurvCorr uses a semiparametric approach. It assumes a correlation structure using copulas and deals with censored observations by imputing them. Package survSpearman computes Spearman’s correlation nonparametrically and deals with censored observations by computing the joint distribution of two survival variables. Advantages of SurvCorr: 1) faster than survSpearman (especially for large data), 2) gives confidence intervals (CI). The disadvantages of SurvCorr: it makes assumptions about the correlation structure that might not be true. The advantage of survSpearman: makes no assumptions. The Disadvantage of survSpearman: 1) the user has to compute the CI using the bootstrap, 2) it has quadratic computational complexity (which means, for example, that if your data grows twice as much, the computational time increases four-fold).

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Thank you very much…

I’m not qualified to say, but believe it’s important to also compare QoL-related-PROs as a secondary endpoint when a surrogate like PFS is the primary.

Doing so can help make a regulatory decision when an improvement in the primary endpoint is a close call (arguably, even if not). This comparison also provides information needed for patients to make an informed choice if the study drug protocol gains regulatory approval. This is especially important when the interventions are continuous until progression or until unacceptable toxicity – this to judge if the gain in time to event is worth the continuous pain.

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That is a great point…I am also thinking how to statistically establish a correlation between PFS and changes in PRO…

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Thanks you. I’m thinking it’s important for the FDA to provide industry guidance - a mandate of sorts - so that pairing PROs (as a secondary) with primary PFS is not done selectively – when the study drug sponsor feels it will win, and not otherwise.

A few quick thoughts on this interesting topic:

1. Correlating a surrogate endpoint (such as PFS) with overall survival is a common practice in oncology but often answers the wrong question when the structural causal relationships between PFS and overall survival are not taken into account. Surrogate endpoints may be more reliable than overall survival when evaluating RCT results and when transporting RCT inferences to our patients in clinic. Articles describing this accessibly for clinicians and methodologists are forthcoming. Interestingly, this is something that patients themselves often intuit. For example, patients with localized kidney cancer considered disease recurrence (corresponding to the surrogate endpoint of disease-free survival or recurrence-free survival) to be at least as important as their overall survival.

2. An efficient way to integrate PROs with other endpoints is to estimate their joint probabilities and corresponding utilities as was done for example here. Towards this goal, one may use something like the adverse event burden score adapted for PROs.

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