While correct in concluding that Bland-Altman is not appropriate, I'm still not sure your reasoning in strictly correct.  Even "theoretically", the 3 compartments will probably not be in equilibrium. You might want to review your respiratory physiology notes, particularly the notion of ventilation-perfusion (VQ) mismatch. Imagine a blocked alveolus, which sees no gaseous isolflourane (V = ventilation = 0). There is still arterial blood flow (Q = flow  > 0), but it will not equilibrate with the inspired isoflourane. This happens to lesser degrees even in normal lungs, where VQ mismatch results in blood that is not equilibrated with inspired O2  return to the left heart via the pulmonary veins, which will now have a lower concentration than the alveolar gas. Depending on the gas (oxygen, CO2) and the corresponding solubility of the alveolar-capillary barrier, it may also contribute to lack of equilibrium. But in any case, you can't test for equality between them, since they are fundamentally different quantities. If you only had two variables, you would use some form of correlation coefficient for continuous variables. In fact, with only 3 variables, you might simply report pair-wise correlations (arterial with inspired, venous with inspired, arterial with venous), as long as you adjust for multiple comparisons. 

But I suggest you pick up a good text on the biostatistics and read about inter-rater agreement when the > 2 outcomes are continuous numeric. The 2005 Encyclopedia of Biostatistics (Wiley) has a brief (14 pages) chapter by Shoukri on "Agreement, Measurement of" that states:

For interval or continuous scale measurements, we estimate interclinician reliability with the “intraclass correlation coefficient” (ICC)

The "class" in this case is the subject being studied and the "clinicians" or "raters" are the 3 different measurements (this is a bit confusing, since we're not really looking at agreement here, but the analogy should be clear :). In your reading, you will discover that there are actually 6 different types of ICC depending on your specific experimental design. I think it would be good for you to try to understand which best applies to your trial before asking this list. The distinctions are important, and you will be able to ask a better question if you first understand your experiment. 

Atul

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Atul Sharma MD, MSc, FRCPC, M.Stat

Senior Consultant | Biostatistics Group | Data Science Platform

George & Fay Yee Centre for Healthcare Innovation

University of Manitoba

@Atul, did you comment anything here? I cant see your comment here?

Many thanks.

There's a long comment in my browser. Try re-loading. 

Dear Atul, 

Many apologies for my simple questions. I just begin to involve in research and there is so many things that are new to me. I believe I have learn many valuable lessons from your comments. I appreciate that a lot. I admit that I am not very clear about the questions as I am only part of the members of the approved protocol. And I am trying my best to pick this specialist's knowledge during my undergraduate study as this is the field that I am interest.

My approved protocol questions as below:

1. The primary aim is to measure the isoflurane concentration in coronary sinus blood.

2. Secondary aims:

- to assess whether the myocardial concentration of isoflurane had equilibrated with its arterial concentration by before the aortic cross clamp had been applied by comparing the difference in concentrations between the coronary sinus (vein) and arterial blood concentrations.

- to assess whether coronary sinus venous blood concentrations (vein) may be estimated from oxygenator exhaust levels of isoflurane.

For the 9 samples (artery ug/ml, vein ug/ml, exhaust anaesthetic gas %), I run though the normality test (non parametric-Shapiro Wilk test) for all the three variables.

 (Shapiro-Wilk p>0.05): artery (p=0.228), vein (p=0.996), exhuast (p=0.035)

Artery and Vein variables are both normally distributed (despite my very very small samples (n=9), does that ethically right to use parametric tests to analyse the relationship between both variables)

Checking the correlation between both artery and vein variables, Pearson (parametric test) was 0.833 (p=0.039) whereas Spearman (non parametric test) was 0.829 (p=0.042). It indicates that the correlation between artery and vein is statistically significant. Please correct me if I was wrong. Should I use the value from Pearson coefficient given my normally distributed variables athough my sample size is 9 or otherwise? Please advise further. 

So far, I have completed part of my protocol's questions. However, I am still struggle for the last part of my research question (which is to check the agreement between the vein and exhuast variables, despite their different units ugml vs %)

Exhaust variable is not normally distributed. What can I do with this to check the correlation between vein and exhuast variable (taking Spearman reading?)? How can I check the agreement between these two different units variables? My lab cant do a calibration curve for to convert % to ug/ml. 

THank you very much for your attention and time solving my doubts about statistics.

Looking for correlations between arterial concentration and coronary sinus concentration will not answer your research question. You are asking if the cardiac isoflurane has equilibrated - by looking to see if the coronary sinus concentration has approached the arterial concentration. So you need to calculate the arterial to coronary sinus concentration difference for each subject. Do you need a p-value for this? Maybe not. You just need to report how close coronary sinus concentration is to the arterial. (Reporting a correlation would be meaningless. It is obvious that subjects with higher arterial concentration will also have higher higher coronary sinus concentration).

As for the other question, "Can the exhaust concentration be used as a surrogate for coronary sinus concentration?", I cannot see how this would be possible if you cannot convert the % gas measurement to a blood measurement.  You will need to know the blood gas partition coefficient (which might vary between subjects).  Once you make the conversion then Bland Altman would make sense.  But on first principles, the oxygenator exhaust will surely be representative of mixed venous isoflurane, and cannot be expected to be specific for the coronary sinus.

Dear Timothy, 

Many thanks for your comment. Would be great if I could exchange some opinions with you. My lab can only provide me with the concentration of isoflruane in ug/mL (number of mole/volume).

1. if the cardiac isoflurane has equilibrated between both artery and coronary sinus?

Applied the ideal gas law, both anaesthetic tension in artery and coronary sinus should be the same under equilibrium. In the literature, most papers reported as partial pressure to check the relationship of anaesthetic tension between arterial and vein. Also, to calculate partial pressure, we need to have blood: gas coefficient which varies in patients as what you proposed. This is well supported by various studies that the coefficient ranges from 1.1 to 1.55 among different patients' group as compared to textbook value 1.4. The reason being using ug/mL instead of partial pressure is lesser potential error because different hypothermic temperature (depend on surgeon preference) during CPB, hemodilution and gas flow during CPB have proved to affect the solubility of isoflurane.

PV=nRT

Given both anaesthetic tension in artery and coronary sinus have equilibrated, the concentration of isoflurane in ug/mL (number of mole/volume) would not be exactly the same as both artery and coronary sinus contain different volume of blood. Perhap, it is not possible for me to check whether both have reached equilibrium as partial pressure (blood:gas coeffcient) is not measured. 

Please advise further. Thank you for your time.