Dear Dr. Beaumont,

Not sure I understand your task correctly, but for me  "to detect changes", actually means to use some receiver with a properly selected thresold. To compare two different receiver it's preferable to use  so called "receiver operating characteristic curves".   See the classical book of Van Trees "Detection, Estimation, and Modulation Theory, Part I"

or, maybe more specifically, this  article of  Dr. Florkowski : 

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2556590/

Best regards

Dear Marcus and Michael,

Thanks for your interest in that thread. Here is my problem statement: I want to know if, using imaging, the size of X patients tumor are changing. The change are evaluated between two time points and I calculate the relative change ( as % change).

Now, I am considering two ways of measuring the changes: One is the diameter of tumor, the second is based on tumor volume measurement. For these two measurements I report the % change of all tumors using waterfall plots.

My question is: Are these waterfall comparable? They are both expressed as % but they don't derive from the same metrics (Diameters(mm) Vs. Volume (mm3).

What do you feel? Do you have in mind a reference paper documenting comparisons of waterfall with different metrics?

Dear Hubert,

thanks for the clarifications.

How do you measure and estimate the diameter and volume?

Dear Hubert,

Diameter and volume of a same entity (tumour) are obviously interdependent. This can realized by simply arguing that for any sane geometric formation, the variables which have been used to explain the geometry are directly proportional (e.g diameter==volume).

Note: However, there are contradictory geometric figures like the four dimension cubes, but for this particular case, these contradictions can be ignored. 

Following up, I think there are two types of shape associated with tumours:

1: Benign: These tumors cells are smooth and round with a surrounding fibrous capsule. Thus, they expand almost equally in all directions. Given this fact, we can say that the shape of these tumours would be spherical. 

2: Malignant:  These tumour cells are Irregular shaped with no capsule. Thus, they will expand the overall tumour mass irregularly. 

Now that we have understood the tumour tissue geometry, we can easily say that 

1. for benign tumours, since they are spherical, the best metric to measure the size is diameter AND volume. Thus, in this case, the waterfall plots will be comparable.

2. for malignant tumours, since they are irregularly shaped, the best metric to measure is the volume. Thus, in this case, the waterfall plots won't be comparable and may result in confusion.

Dear Asheesh,

Thanks a lot, your perspective is particularly interesting. I will test your points with my data, i will let you know the results by the way.

Actually what I had in mind was more statistics oriented as Volume is connected to diameter by a power of 3, consequently standard deviation of relative change (SD expressed as %) can be expected to be higher for volume and also the magnitude of the response. At the end of the day, I am not sure to compare performance of the biomarker volume against biomarker diameter in considering metrics like the signed integral of waterfall plot with attached confidence interval. I am not able to find any paper that compared two different imaging biomarkers using waterfall plots (and signed integral of the plots)

Dear Hubert,

I was AFK, so couldn't reply.

Surely, the magnitudes will be of the order of three in this case. This remains a poorly staged problem in statistics. Also, I don't know of any paper which compares two waterplots which represent different matrices.

BUT, there is a way to evaluate the biomarkers using rank correlation (particularly Kendall rank correlation). I hope you are aware of it, but here is a nice explanation. 

https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient

So, after you have read it (or already know it), here is the how:

first let's define the variables and assumptions:

Assumption: If both the markers are in the same time space (I mean the timestamps of those readings are same, or both the markers have been measured simultaneously),

Variables: at least for benign tumours, we know the volume would be 2/3*Pi*r

1. At least for benign tumours, we know the volume would be 2/3*Pi*r3. let this volume be V* 

2. There is the original biomarker V.

if V*  is comparable to V, let's call it a concordant pair. A discordant pair, if otherwise.

So, given this data, we will get a Kendall's Tau and significance value. If the Kendall's Tau is very big with big enough significance, that would mean V* and V are tightly related or the diameter metric is tightly related to actual volume metric V. In that case, we can choose any. 

In opposite case, it would mean they are not at all related (probably in case of malignant tumours) and we would have to device some tricky math to decide which metric to use. A starting point, in that case would be to considere the abundance of data on each metric. 

Hope that helps!

It would be awesome if you shared these results :)

--Asheesh

https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient

Dear Asheesh,

Thanks you very much for your message. That is really a good idea. Actually, what you are suggesting is a kind of hypothesis testing for the spherical lesion assumption, right? Further, it would be possible to estimate the proportion of the population diverging from the spherical assumption. Eventually, a conclusion would be that the two biomarkers report different informations, even if one cannot say which one outperform the other in terms of the clinics.

That is just perfect.

For sure, I will forward you the result of this work. Please be patient as I am just drafting the protocol of this trial, supposed to end within, let say between six month and a year.

Thank you again for this very interesting discussion.

cordially,

Hubert