How is mass error (ppm) actually used?

[sbashkyrtsev]

I've seen a paper that reported the distance (in m/z) between the EMW and the measured masses. It's supposed to give a score that tells us how confident we are this is our mass. However I don't understand the essence of this metric.

The formula of the mass error (ppm) is:

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|theor_mz - measured_mz| / theor_mz x 1e6

But why do we divide by the theor_mz? Let's take a TOF for instance - it measures the time that it takes the ions to reach the detector. Smaller masses are affected by the electrical field stronger than larger masses. So the difference in speed (and time) between 100Da and 100.1Da will be greater than the difference between 1000Da and 1000.1Da. So if there's 0.1Da error for a large mass, then it's less likely that this happened due to a chance - it must be that the masses were actually different.

However, the formula says the opposite. What am I missing?

[bcd_GCLCMSMS]

This equation is similar to the standard relative percent difference equation, except that it is in parts-per-million instead of percent.

It is used to calculate the error between a measurement and the theoretical value. Since the theoretical value is taken as fact, it is the divisor and not the average of the two values.

We use it to find the error of an instrument's measurement or to rank order lists of possible molecular formulas based on the difference to their theoretical mass.

[sbashkyrtsev]

I understand why people usually use it, but I don't get why and how it works. My problem is with the formula itself - I didn't understand why you need to divide at all..

But I may have understood it eventually. But my understanding is that the formula still should be different - we need to divide by theor_mz raised to 3/2 power.

Why divide at all

Since TOF is probably the easiest to understand, let's take it as an example. My understanding is that it's harder to differentiate between 100.1 vs 100Da as opposed to 10 vs 10.1Da. TOF uses time-to-digital converters, so if the ions reach the detector closely spaced (time wise) it's easy to mistake it for some other mass.

As an example, suppose E=1 and the length of the tube is 1m, then the time difference between small molecules is (let's pretend `m` is measured in Da and not in `kg` for simplicity):

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v(10Da) = sqrt(1/10) = 0.3162278 m/s
v(10.1Da) = sqrt(1/10.1) = 0.3146584 m/s
t(10Da) - t(10.1Da) = 0.3162278 - 0.3146584 = 0.0015694 s

And the time difference between large molecules is:

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v(100Da) = sqrt(1/100) = 0.1 s
v(100.1Da) = sqrt(1/100.1) = 0.09995004 s
t(100Da) - t(100.1Da) = 0.1 - 0.09995004 = 0.00004996 s

So clearly it's harder to differentiate between 100 & 100.1Da vs 10 & 10.1Da.

Why we should not divide by just theor_mz

However, we know that E=mv^2, so the speed is v=sqrt(E/m). And if we take a derivative

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dv/dm= - sqrt(E) / (2 * m * sqrt(m))

By that logic the mass accuracy should be:

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|theor_mz - actual_mz| / (2*theor_mz * sqrt(theor_mz))

But even this.. It's only valid for TOF, can it be justified for other mass analyzers?

[Hollow]

I guess you're thinking too far and mix up some things.

Just look at it as the normalized error of a measurment. Similar to the relative standard deviation.

Then, as in your examples, as you said it's harder to distinguish between 1000.0 and 1000.1 than between 100.0 and 100.1. The first measurement has an error of 100 ppm while the second has 1000 ppm
Or just assume you have ruler with mm scale. Then you measure two distances 100 m and 1000 m. Each one will be accurate to +/- 1 mm. So how do you express which one is more accurate? -> by means of the relative error

Instrument wise, the time resolution(in ToF) has a certain minimum, but probably stays +/- the same for all masses (because it's more related to the electronics and quartz etc); like the scale of the ruler.
So then you need to calculate the minimal mass difference, that can be still differentiated for a set time difference.

And another aspect is the view as residuals from a calibration curve.
E.g for a ToF, the real measurement is the time. But what you want to know is the molecular mass. So you need to transfer the units by doing a calibration curve. You infuse multiple known asubstance and measure the time. Then model a function that best fits the measured times to known masses. And probably the correlation r will not be 1.000000 because of measurment uncertainty.
So you'll have some residuals and you want to know how big they are, related to the measured mass. And it's easier to say/write 1500 Da +/- 10 ppm than (1500 +/- 0.015) Da

BTW: your formulas for the time are erroneous faulty. The difference of velocity won't give time...
E=1/2mv^2 = 1/2mL^2/t^2 which then can be solved for m or t

BTW2: personally I like to get the difference by m(measured) - m(theor); so the sign of the difference will be correct if one does not take the absolute value

[Hollow]

Addendum
take some numbers

Let's assume your instrument is capable to measure time differences of at least 5 μs for all masses.

Then you could differentiate between
10 & 10.0000447 Da (+44.7 μDa)
100 & 100.000141 Da (+141 μDa)
1000 & 1000.000447 Da (+447.2 μDa)

like this it's hard to interpret the resolution
But as
10 Da +/- 4.47 ppm
100 Da +/- 1.41 ppm
1000 Da +/- 0.447 ppm

it's much easier to get the idea of the uncertainty of your mass

[sbashkyrtsev]

Just look at it as the normalized error of a measurment. Similar to the relative standard deviation.

RSD is used to compare sets that can't be compared in absolute values. Like comparing an effect of a weight control drug on elephants and mice. Comparing their absolute SD doesn't make sense, so you normalize by their mean value.

Or just assume you have ruler with mm scale. Then you measure two distances 100 m and 1000 m. Each one will be accurate to +/- 1 mm. So how do you express which one is more accurate? -> by means of the relative error

This one I don't get.. They have the same accuracy (or better call it precision). What makes you switch to relative units? In the example with elephants & mice I wanted to compare the effect of the drug on one animal vs the other. But in this example why would you want to compare?

So you'll have some residuals and you want to know how big they are, related to the measured mass. And it's easier to say/write 1500 Da +/- 10 ppm than (1500 +/- 0.015) Da

Here again - why is it relative to the measured mass? When I look at the mass spectra, I'd like to understand whether this particular measured value is my analyte or not. So I'll look how far the value from the theoretical value, and for this it's much easier to use +/- 0.015. I can see these values with my eyes on the chart, while ppm doesn't tell me anything..

To make it concrete, you have 2 analytes A (100Da) and B (1000Da), and the measurements find 101Da and 1001Da. Would you say that A wasn't found, while B might actually be there?

BTW: your formulas for the time are erroneous faulty. The difference of velocity won't give time...

I used length=1m in the calculations (though forgot about 1/2 coefficient).

[Hollow]

RSD is used to compare sets that can't be compared in absolute values. Like comparing an effect of a weight control drug on elephants and mice. Comparing their absolute SD doesn't make sense, so you normalize by their mean value.

Not only.
if you want to compare results of the same analysis with different method / on different systems /different labs etc.

Or your method gives you a default error of 0.1%w/w. Then it depends if you measure an assay >95%w/w or related compounds <1%w/w.
Of course you can always give the error in absolute measures but in relative terms the precision will be much easier to recognize.

You can also look at electronics components e.g. resistors and capacitors, they are usually specified by their relative error, not absolute ones.

This one I don't get.. They have the same accuracy (or better call it precision). What makes you switch to relative units? In the example with elephants & mice I wanted to compare the effect of the drug on one animal vs the other. But in this example why would you want to compare?

yeah maybe a missuse of the two terms..
Both measurements will have the absolute accuracy of 1 mm but their precisions are 10 and 1 ppm.
If you think of it as multiple measurments you will get results of 99.999 - 100.001 m or 999.999 - 10000.001 m.
With those multiple measurements you will get a (Gaussian) distribution with their means and SD.

Here again - why is it relative to the measured mass? When I look at the mass spectra, I'd like to understand whether this particular measured value is my analyte or not. So I'll look how far the value from the theoretical value, and for this it's much easier to use +/- 0.015. I can see these values with my eyes on the chart, while ppm doesn't tell me anything..

Why not?
It may also be a matter of taste how to describe the error. But the relative became more popular, because it expresses the precision of the measurement.
Just like the retention time window for peak recognition. If you know that your flow is stable (precise) and accurate to e.g. 1% then your retention times will also be in that order, so it's easier to define the RT windows as +/-1%. So late eluter will get a broader window than earlier ones.
of course you could also specify it in absolute time, but then later peaks may fall out of the window quite frequently. Or the window is to broad for earlier peaks and missidentification may occur.

To make it concrete, you have 2 analytes A (100Da) and B (1000Da), and the measurements find 101Da and 1001Da. Would you say that A wasn't found, while B might actually be there?

probably yes, depending on the accuracy. If both have an error of 1000 ppm, the A is significantly different but B is not.

[sbashkyrtsev]

You can also look at electronics components e.g. resistors and capacitors, they are usually specified by their relative error, not absolute ones.

Maybe, but they were not picked willy-nilly, there are some underlying physical reasons for that. It's not about mere (in)convenience. Even if the error is relative, the relationship isn't always linear! I'm trying to get to these underlying reasons in the Mass Spec (:

It may also be a matter of taste how to describe the error. But the relative became more popular, because it expresses the precision of the measurement.

You can convert absolute to relative and back, yes. But only as long as the values stay correct! It can't be a matter of taste if it's incorrect/misleading.

So if it's true that the absolute error of the Mass Spec increases with the absolute m/z being measured, then sure - relative units are the way to go. But if it doesn't depend (or depends non-linearly), then dividing by theoretical_mz seems wrong. And I'm trying to understand whether people just traditionally use incorrect values without questioning them or, more likely, I don't understand something about the instrument. As I mentioned with TOF, I do believe they should be relative, but to m/z value raised to 3/2. And even if this is right, I don't know if the same logic is applicable to other mass analyzers.

I've seen a paper, where the team calculated the error and made decisions based on that. If they used incorrect calculations, then the decisions might have been wrong too.

[Hollow]

You can convert absolute to relative and back, yes. But only as long as the values stay correct!

A conversion should never change the (base) values. Therefore they will stay correct.

ppm is just a form of expressing the relative error of a measurement.
Independent of the process how it was measured.

Just as the definitions by IUPAC:

measurement result: https://doi.org/10.1351/goldbook.M03796

error of measurement: https://doi.org/10.1351/goldbook.E02194

Result of a measurement minus the true value of the measurand. Since a true value cannot be determined, in practice the conventional true value is used.

relative error: https://doi.org/10.1351/goldbook.R05266

Error of measurement divided by the true value of the measurand. Since a true value cannot be determined, in practice the conventional true value is used.

percentage relative error: https://doi.org/10.1351/goldbook.P04486

The relative error expressed in percent. It can be calculated from the relative error by multiplying by 100. Comment: The term 'percentage relative error' should always be quoted in full, rather than 'error' or 'percentage error' to avoid confusion.

...just that it is not given in % but as ppm

None of those definitions are correlated to the measuring process itself.

[Hollow]

Some more references on MS

https://www.waters.com/nextgen/us/en/ed ... ution.html

https://fiehnlab.ucdavis.edu/projects/s ... urate-mass

https://fiehnlab.ucdavis.edu/projects/s ... resolution

Debating Resolution and Mass Accuracy
LCGC North America-02-01-2004, Volume 22, Issue 2
https://www.spectroscopyonline.com/view ... l-research

Accurate Mass Measurement: Terminology and Treatment of Data, JAmSocMassSpec
https://doi.org/10.1016/j.jasms.2010.06.006

[sbashkyrtsev]

So eventually I talked to an application scientist from Sciex, and he also doesn't sound very sure about these calculations of Mass Errors.

It seems like there are more factors that impact the error than just the flight path & velocity. Apparently, TOF surprisingly has higher Relative Mass Error (ppm) when masses are <100 Da. And after this cutoff the mass error scales with the molecule mass more or less linearly. But it's not clear what the additional factors are.

And.. This certainly depends on the instrument, as Orbi and trap instruments have higher Relative Mass Errors with larger masses.

So the overall conclusion is that the usual Mass Error formula is a hand-wavy math expression that sometimes works and sometimes it doesn't. But it mostly should overestimate the error (at least for TOF and when the masses aren't too small), so using the formula should be more or less safe. But there's certainly not enough rigor in these calculations.