Let's say there's a measurement hierarchy for discrete variables, that looks like nominal (grade pass/fail) < ordinal (grade A/B/C) < interval (grade 90-100; 80-89; 70-79, etc). Hierarchy is used to refer to the possiblity that we can assign order to the discrete values.

I read a line that methods applicable for one type of variable can be used for the variables at higher levels too, but not at lower levels. Can any one explain and exemplify why this is true (if it is)?

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    $\begingroup$ Although not an answer, it seems like all your examples are all ordinal. Pass and fail are not just names, but measures of ability on a scale (ordinal). Grade as A/B/C... is ordinal. Your example of interval (percent grades) is ordinal as well, as the difference in ability between a 90-80 range is probably not the same difference in ability between 40-30. It also has no meaningful zero point so it is not ratio. Percent grades or scores are really not ratio or interval, but ordinal. Although we treat them routinely as interval, we should feel awkward about it. $\endgroup$ Apr 25 '13 at 20:49
  • $\begingroup$ Pass/fail is not nominal, grade (at least as you have it) is not interval. Then there's ratio. $\endgroup$
    – Peter Flom
    Apr 25 '13 at 20:59
  • $\begingroup$ "Hierarchy is used to refer to the possiblity that we can assign order to the discrete values." --- Well, rather 'hierarchy' there refers to the relative 'strength' of the measurement classification (ratio>interval>ordinal>nominal). If data is interval it also satisfies the requirements to be ordinal data; if it's ordinal it also satisfies the requirements to be nominal. Just as an example, coefficients of variation, means, medians and modes are meaningful for ratio scale; means, medians and modes for interval; medians and modes for ordinal and only modes for nominal. $\endgroup$
    – Glen_b
    Apr 25 '13 at 22:46
  • $\begingroup$ Pass-fail is also binary. Allow, if only as a convention, coding as 0 for one state and 1 for another, and then you can feed it to mean and regression commands, etc. For example, proportion pass is just the mean of lots of 0s and 1s. Many techniques depend on this, but that is often forgotten in more dogmatic statements about what measurement level prohibits or inhibits. So binary scales are nominal if you focus on the names and ratio if you focus on (0, 1) codes. $\endgroup$
    – Nick Cox
    Apr 25 '13 at 23:34
  • $\begingroup$ Related: stats.stackexchange.com/questions/536040/… $\endgroup$
    – Galen
    Jul 27 at 19:06

Basically, measurement level describes certain assumption about the variables at hand. These assumptions imply additional information or structure of the data.

For example, take the interval level. Just as at the ordinal level, values are ordered in a certain progression. In programming terms one could say that the interval class inherits the properties of the ordinal class (not to worry if you are unfamiliar with the concept). In other words, the interval class has the same properties as the ordinal, but an additional property is added. Namely, the distance between the "ticks" on the scale is now equal.

So we know that just as on ordinal level 4 is above 2, but we also know that the distance between 0 and 4 is twice as large as the distance between 2 and 4.

Ordinal methods use the information about the order of values. The same information is contained on the interval level. However, by applying ordinal methods to the interval level you waste the information about the homogeneity of distance between values. Still, it is possible.

On the other hand, interval methods rely on the assumption of distance homogeneity. This is why they cannot be (correctly) applied to data which violate that assumption. Still, this is often done e.g. with Likert scales, but this is a whole other discussion.

Hope this helps.

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    $\begingroup$ +1, nicely done Maxim. Much has been previous discussed previous post We could add examination scores to routine violation as well. $\endgroup$ Apr 25 '13 at 20:58
  • $\begingroup$ A thread discussing inheritance of these types of variables appears at stats.stackexchange.com/questions/23200/…. $\endgroup$
    – whuber
    Apr 29 '13 at 19:50

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