I was reading a brief theory of measurement from ftp://ftp.sas.com/pub/neural/measurement.html when a passage got me stuck. It says that

Consider a rat in a Skinner box who pushes a lever to get food pellets. The number of pellets dispensed in the course of an experiment is obviously an absolute-level measurement of the number of pellets dispensed. If number of pellets is considered as a measure of some other attribute, the measurement level may differ. As a measure of amount of food dispensed, the number of pellets is at the ratio level under the assumption that the pellets are of equal size; if the pellets are not of equal size, a more elaborate measurement model is required, perhaps one involving random measurement error if the pellets are dispensed in random order. As a measure of duration during the experiment, the number of pellets is at an ordinal level. [...] In the example above with number of pellets as a measure of duration, the errors would be cumulative, not additive, and the error variance would increase over time.

From reading about the classical model of error, I gather that an additive error is one that can be factored out by averaging many measurements. How does the cumulative error work? I nearly thought they are the same, but this passage makes it clear they are not.

• This is more of an issue with the quote than your question, but I don't see how the error in that duration case would be cumulative. As I understand it the idea is that the longer a rat is in the box, the more opportunities they have to press the button, and so more time in box = more pellets dispensed. But unless the interval between button presses is always increasing, I don't see why the error is necessarily cumulative. Jul 19 '16 at 15:01

The cumulative error (also referred to as system error) - It's a single direction error. e.g, - If you are to measure 10 km run & your stopwatch is running 2 sec faster every minute. So at the end of the experiment to calculate error you will add 2 secs for each minute.

Additive errors (also referred as multiplicative) - Error term which can go either way based on the scenario. Good example is residuals in a linear regression model. The residuals can be both negative or positive based on your observation (unlike 1st scenario)

Hope this helps.

• This is correct, with the exception that additive errors and multiplicative errors are not synonymous. Can you correct that? Apr 6 at 0:49

Interestingly, this is a problem that psychologists and sociologists come across when considering theories of identity. Sam being black, male and disabled are all part of his identity and these are obviously not mutually exclusive. However the effect of them when considered together is cumulative not additive. So to check if you are using the right term see if an aspect of the 'thing' (in this case Sam's multiple identities) can be removed. If not, the correct term to describe the combined effect is cumulative. Only when non-identity concepts are added to Sam's profile (such as him being temporarily ill, on holiday with no money) may the term additive be legitimately used, in place of cumulative.

• This is not the sense of cumulative vs additive that is being asked about. Apr 6 at 0:48