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I would like to know if there is a statistical model to analyze my problem:

I want to test if the location of the tumor is related to the level of a particular biomarker

  • A patient may have multiple tumours (tumor sites can be correlated but let's ignore this point to start).
  • It is not a competitive model, since the appearance of a tumor does not prevent the appearance of another in another location (but of course death prevents the appearance of a tumor).
  • These are survival data: we have the time from diagnosis until the appearance of the tumour.

More details: These tumors are the relapse of a primary tumor located in the breast. All patients included have a primary breast tumour. Some of them will develop secondary tumors in other locations. There are about ten possible locations. The question that arises: is the level of the biomarker related to the location? We already know that a high biomarker level decreases the risk of developing secondary tumors

I have no idea if there is a model to analyze such data! And if a model exists, an implemented one on R would be ideal :-)

Thanks for any suggestion

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  • $\begingroup$ Are these tumors recurrences of a previous primary tumor, or something else? Is there just a small number of potential locations for the (recurrent?) tumors? Please edit the question to provide more details, as the best approach might depend on those details. $\endgroup$
    – EdM
    May 4, 2023 at 14:31
  • $\begingroup$ @EdM: thanks for your comment, I've added details on the problematic $\endgroup$ May 5, 2023 at 8:08

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The most straightforward analysis would be of the time and location of the first recurrence. That would be a competing-risks model, with the locations (and death without recurrence) coded as separate types of event, and allowing for different regression coefficients for the biomarker among the different types of event. The competing risks vignette of the R survival package shows how to set up not only such a simple competing-risks model but also more complicated models that allow for transitions among multiple states.

You could extend that approach to the multiple locations over time, but you would then have to be very specific about your assumptions.

At the simplest level, you could set up a model of all recurrences by having a separate data row for each recurrence within an individual, with the time to recurrence and an event indicator coding the recurrence location or death as above, along with an ID for the individual. You could then model similarly to the competing-risks model, using a cluster term (based on the ID values) to account for within-individual correlations while modeling the different locations in parallel.

That would, however, ignore some very important aspects of cancer biology. In particular, the correlations over time and among tumor sites are likely to be very important. Once there's a recurrence at one location, the probability of metastasis to multiple locations increases greatly. Furthermore, different organs tend to support metastases having different biological properties, so once there is a metastasis to one organ the probability will increase of further metastasis to an organ that supports a biologically similar metastasis.

At the other extreme of modeling, you thus could have a complicated multi-state model that allows for each possible combination of recurrence locations over time. But with 10 possible locations, the number of coefficients to estimate would grow quickly with the number of recurrences: 10 possibilities for the first recurrence location, with each location then having 9 possibilities for the next recurrence (90 combinations at that level), and each combination of second recurrences over time having 8 possibilities for the third recurrence (720 possibilities at that third level).

You might be able to apply your understanding of the subject matter to find a model intermediate in complexity between the completely parallel/independent model and the all-possible-combinations model. I'd recommend getting some experienced statistical consultation if you decide to do anything beyond the competing-risk model for the location and time of the first recurrence.

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  • $\begingroup$ Thank you very much for your very detailed and useful comment. The clinicians decided to look at the effect of the biomarker level (continuous variable) only at the site of the FIRST tumor recurrence. In this case, what do you recommend: a competitive risk model with ten possible sites and of course death? $\endgroup$ May 9, 2023 at 11:22
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    $\begingroup$ @FloraGrappelli that would be a straightforward approach, but you need to allow the regression coefficients to differ among the recurrence sites. Apply your understanding of the subject matter to decide if you need to evaluate all 10 sites individually, or if some might be grouped together. The more sites you evaluate the more coefficients you need to estimate and the more recurrence events you need. That becomes even more of a problem if you (as you should) model your continuous biomarker flexibly, e.g. with a regression spline, which involves more coefficients per recurrence site. $\endgroup$
    – EdM
    May 9, 2023 at 14:01

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