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Is there an equivalent of the Mann-Whitney U test, but for right-censored data?

I have right-censored numerical data from two different groups. The mean for one group is higher than the other. I'd like a hypothesis test to check whether this difference is statistically significant. In other words, I'd like a statistical test where the null hypothesis is that the two groups come from the same distribution, and the alternative hypothesis is that the distribution for one group is somehow larger than the distribution for the other group. My data does not look normal, so I need a non-parametric test.

In more detail, I asked a bunch of people to perform a task. Each sample represents how long the person took to complete the task. I asked them to keep trying until they completed the task, but after 1 hour, I stopped them, so for those folks, I have right-censored data: I don't know how long it would have taken them if they had kept trying for longer. Also, the endpoint is the same for all censored data points. The completion times for one group seems higher than the completion times for the other, but I'd like a hypothesis test that will let me assess statistical significance.

Is there a way to do a hypothesis test for this kind of right-censored data?


Approaches I've looked at: Should I be using the log-rank test? (Most of the completion times are small, in case that is relevant.) A Cox proportional hazards model doesn't seem appropriate in my setting, as the data doesn't look exponentially distributed. Is there something else from survival analysis that would solve my problem?

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    $\begingroup$ Cox proportional hazards doesn't assume constant hazards, so there's no need for the lifetimes to look exponential. If I understand the situation correctly the log-rank should be okay. $\endgroup$
    – Glen_b
    Apr 15, 2016 at 6:43
  • $\begingroup$ Ahh, thank you, @Glen_b! I see that I read the material about Cox proportional hazards too quickly. Anyway, thanks again. $\endgroup$
    – D.W.
    Apr 15, 2016 at 8:20
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    $\begingroup$ If nobody is observed for >1h and nobody stops before 1h without completing, then you could consider all censored observations as tied in rank, if all you need is a hypothesis test. Otherwise log-rank seems like an initial default choice. $\endgroup$
    – Björn
    Apr 15, 2016 at 8:46

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To summarize the comments I've received: yes, use the log-rank test. It does exactly this.

Alternatively, use Mann-Whitney but consider all the people who didn't finish within 1 hour as tied in rank (but larger than anyone who finished before then); this works if the threshold for right censoring is the same for everyone.

Cox proportional hazards doesn't assume constant hazards -- the hazard can change over time -- so a Cox proportional hazards model should be OK.

Thanks to Glen_b and Björn for their help.

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