ASA Connect

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

Original Message------

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

I estimated the survival distribution using SAS LIFEREG and decided to use Weibull. I don't think that assuming an exponential distribution would help solve the problem. I think that the biggest stumbling block is that there is staggered entry, i.e. different "Day 0" for each patient.

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

Original Message:
Sent: 08-21-2017 00:51
From: Eric Siegel
Subject: Survival analysis for ongoing study

Regarding the deaths thus far: Do they look like they are consistent with coming from an exponential distribution?

---

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Original Message------

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

Because there is staggered entry, I assume that the study had an accrual period. I have two questions:
(1) has accrual stopped?
(2) was the accrual rate roughly constant over the accrual period?
If the answers to both questions are Yes, then I think your problem is tractable: simply add half the duration of the accrual period to your Weibull estimate of when the 75th patient is supposed to die. It's not pretty, but it will give a serviceable answer. An adjustment much like that is done "under the hood" by power-calculation software modules for log-rank tests that allow you to enter an accrual period with a follow-up period. 
What to do about the standard error of the estimate? Um, excellent question: that's where I do my deer-in-the-headlights look. But half the duration of the accrual period is the mean of a uniform distribution (if the accrual rate is roughly constant over the accrual period), and that mean has a standard error. Hopefully the Weibull estimate of when the 75th patient is supposed to die also has a standard error; if so, then the two standard errors are combinable.

------------------------------
Eric Siegel, MS
Research Associate
Department of Biostatistics
Univ. Arkansas Medical Sciences
------------------------------

Original Message:
Sent: 08-21-2017 02:02
From: David Bristol
Subject: Survival analysis for ongoing study

I estimated the survival distribution using SAS LIFEREG and decided to use Weibull. I don't think that assuming an exponential distribution would help solve the problem. I think that the biggest stumbling block is that there is staggered entry, i.e. different "Day 0" for each patient.

------------------------------
David Bristol
Statistical Consulting Services, Inc.

Original Message:
Sent: 08-21-2017 00:51
From: Eric Siegel
Subject: Survival analysis for ongoing study

Regarding the deaths thus far: Do they look like they are consistent with coming from an exponential distribution?

---

Confidentiality Notice: This e-mail message, including any attachments, is for the sole use of the intended recipient(s) and may contain confidential and privileged information. Any unauthorized review, use, disclosure or distribution is prohibited. If you are not the intended recipient, please contact the sender by reply e-mail and destroy all copies of the original message.

Original Message------

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

David,

Offhand I would think that it is not the distribution of deaths over time that is of interest here but rather the distrubion of calendar date of death. We can assume that is uniform with gap K. The gap being the number of calendar days between  deaths. K can be estimated from the data you already have. Now assume a uniform distribution for remaining calendar dates as if  only those K days apart exist.  You can now count to 75. For interval estimation you can estimate an interquartile range of K from existing data, 

I would rather do something like this than get into simulation. I hope this helps. 

Jay Herson

------------------------------
Jay Herson
Johns Hopkins Bloomberg School of Public Health
------------------------------

Original Message:
Sent: 08-20-2017 16:58
From: David Bristol
Subject: Survival analysis for ongoing study

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

Original Message------

David,

Offhand I would think that it is not the distribution of deaths over time that is of interest here but rather the distrubion of calendar date of death. We can assume that is uniform with gap K. The gap being the number of calendar days between  deaths. K can be estimated from the data you already have. Now assume a uniform distribution for remaining calendar dates as if  only those K days apart exist.  You can now count to 75. For interval estimation you can estimate an interquartile range of K from existing data, 

I would rather do something like this than get into simulation. I hope this helps. 

Jay Herson

------------------------------
Jay Herson
Johns Hopkins Bloomberg School of Public Health
------------------------------

​How can there be death times for any of the remaining 25 patients?  They're still alive.

------------------------------
Emil M Friedman, PhD
emilfriedman@gmail.com
http://www.statisticalconsulting.org
------------------------------

Original Message:
Sent: 08-20-2017 16:58
From: David Bristol
Subject: Survival analysis for ongoing study

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

As a follow-up to the previous posts, I share the results I sent to the client.
All available data was used to estimate the parameters of a Weibull distribution (using SAS PROC LIFETEST).
Time to death was simulated from this distribution for the subjects who are currently censored.
For those with simulated time to death less than the censoring time, the simulation was discarded and another simulation performed for that subject until a simulated time to death was obtained exceeding the censoring time (an ad hoc conditional distribution).
For specified dates (31Dec2017, 30June2018,...), the distribution of the number of deaths was presented.
I assume that the client wasn't too happy to see how long it would be before the last two deaths were predicted to occur, but I think that it would be unrealistic not to use a long-tailed distribution; I haven't received a response since I sent the results a week ago.
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

Original Message:
Sent: 08-22-2017 11:26
From: Emil Friedman
Subject: Survival analysis for ongoing study

​How can there be death times for any of the remaining 25 patients?  They're still alive.

------------------------------
Emil M Friedman, PhD
emilfriedman@gmail.com
http://www.statisticalconsulting.org

Original Message:
Sent: 08-20-2017 16:58
From: David Bristol
Subject: Survival analysis for ongoing study

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

​It might be most interesting to investigate how long it would take for the study results to be available. For those who died after censoring, information may never be available, so what happens after censoring doesn't seem particularly relevant. I would focus on the time to death or censoring, whatever comes first, since that is what drives availability of the study's results. The distribution of time to death or censoring can be estimated using a Kaplan Meier approach, or even the empirical cumulative distribution function, as this is an uncensored outcome (censoring is included in this outcome). Then, for each participant still alive and in the study NOW, the distribution of the residual on-study time can be estimated from the Kaplan Meier result by conditioning on being alive and in the study now. This will for each person provide the estimated distribution of being alive and on study starting NOW and moving forward. Sum these, and you will obtain the estimated number of patients on study starting now and moving forward. It is unusual in survival analysis settings to wait until all participants have died though. One usually sets a pre-determined end-of-study time before the study starts enrolling.

Hope this helps, Judith.

------------------------------
Judith Lok
Associate Professor of Biostatistics
Harvard TH Chan School of Public Health
------------------------------

Original Message:
Sent: 08-31-2017 18:23
From: David Bristol
Subject: Survival analysis for ongoing study

As a follow-up to the previous posts, I share the results I sent to the client.
All available data was used to estimate the parameters of a Weibull distribution (using SAS PROC LIFETEST).
Time to death was simulated from this distribution for the subjects who are currently censored.
For those with simulated time to death less than the censoring time, the simulation was discarded and another simulation performed for that subject until a simulated time to death was obtained exceeding the censoring time (an ad hoc conditional distribution).
For specified dates (31Dec2017, 30June2018,...), the distribution of the number of deaths was presented.
I assume that the client wasn't too happy to see how long it would be before the last two deaths were predicted to occur, but I think that it would be unrealistic not to use a long-tailed distribution; I haven't received a response since I sent the results a week ago.
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.

Original Message:
Sent: 08-22-2017 11:26
From: Emil Friedman
Subject: Survival analysis for ongoing study

​How can there be death times for any of the remaining 25 patients?  They're still alive.

------------------------------
Emil M Friedman, PhD
emilfriedman@gmail.com
http://www.statisticalconsulting.org

Original Message:
Sent: 08-20-2017 16:58
From: David Bristol
Subject: Survival analysis for ongoing study

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

I used LIFEREG, not LIFETEST. The former provides parametric models, the latter is based on nonparametric KM estimate.

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------

Original Message:
Sent: 08-31-2017 18:23
From: David Bristol
Subject: Survival analysis for ongoing study

As a follow-up to the previous posts, I share the results I sent to the client.
All available data was used to estimate the parameters of a Weibull distribution (using SAS PROC LIFETEST).
Time to death was simulated from this distribution for the subjects who are currently censored.
For those with simulated time to death less than the censoring time, the simulation was discarded and another simulation performed for that subject until a simulated time to death was obtained exceeding the censoring time (an ad hoc conditional distribution).
For specified dates (31Dec2017, 30June2018,...), the distribution of the number of deaths was presented.
I assume that the client wasn't too happy to see how long it would be before the last two deaths were predicted to occur, but I think that it would be unrealistic not to use a long-tailed distribution; I haven't received a response since I sent the results a week ago.
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.

Original Message:
Sent: 08-22-2017 11:26
From: Emil Friedman
Subject: Survival analysis for ongoing study

​How can there be death times for any of the remaining 25 patients?  They're still alive.

------------------------------
Emil M Friedman, PhD
emilfriedman@gmail.com
http://www.statisticalconsulting.org

Original Message:
Sent: 08-20-2017 16:58
From: David Bristol
Subject: Survival analysis for ongoing study

I have a survival problem that's bugging me.
My client has an ongoing single-sample survival study in advanced cancer with 75 patients initially.
There have been 50 deaths. The death times and the censoring times for the remaining 25 patients are available.
The client wants to know when there will be 75 deaths (calendar time). I know that I can't actually know the death times for the patients who are currently censored, but I should be able to provide a probabilistic statement such as a probability of 75 deaths by calendar time T*.
The survival distribution for time to death from Day 0 can be estimated using a non-parametric approach such as KM or a parametric approach, but I don't see how to use that.
I can estimate the distribution of time to T* from censoring date for currently censored patients, but that doesn't seem to helpful for answering the question either.
I am considering simulation; that seems unnecessarily complicated, but can't think of a less complicated solution.
Any thoughts?
Thanks,
David

------------------------------
David Bristol
Statistical Consulting Services, Inc.
------------------------------