Are ratio, interval, ordinal and nominal variables nested? It seems to me that ratio, interval, ordinal and nominal variables nested in the following sense:


*

*A ratio variable is also interval, ordinal and nominal.

*An interval variable is also ordinal and nominal.

*An ordinal variable is also nominal
Are the above statements accurate? If not, are there any counter-examples where one or more of the above statements will not hold?
 A: Leaping in here, I generally disagree with the proposition.
Let's look at how the categories are defined, by accepted convention.
There is the nominal category, which includes variables such as Sex, Marital Status, Religion, etc. Here we have discrete categories, where there is no ordering to them. If we gave them numbers, we could swap those numbers around however we liked because they make no difference to the underlying features which define those categories. This swapping is not a feature of the other 3 categories, therefore nominal data are quite special.
Next, we have the ordinal category. This is where order is important, for example the top 3 placings in a race, but the spacing between the categories is not consistent. The time to cross the finish line between 1st and 2nd could be 10ms, then 3rd might cross 500ms later. Or we might split income into quartiles, where the range of income in each quartile is not consistent.
Some people do not distinguish much between nominal and ordinal variables, it really depends on the analysis one wishes to do. Both can be analysed by nonparametric tests like chi-square, and the interpretation does not depend on category ordering. However, for analyses like ordinal logistic regression, it does matter that one has ordinal variables.
Coming to interval and ratio variables, I will treat these the same because my argument here does not depend on whether there is a true zero point or not (so yes, in practice there is no difference in statistical treatment of interval and ratio variables for at least the main tests). For both interval and ratio variables, the key feature is that their values have true mathematical meaning, so we can do various types of operations on them, e.g. take differences, linear or nonlinear transforms, and the results of these operations are meaningful. Whereas for nominal or ordinal variables, you can't halve "Sex" or square "Income quantile" - this would be a nonsensical action.
The key features of these two variables should not be confused with convenience of measurement or convenience of analysis: while we may reduce height to m/cm or ft/in with our rulers, height can keep being measured at finer and finer distinctions - it is only the limits of our measuring tool (and the issue of significant places) that makes some interval and ratio variables appear to be categorical. Because interval and ratio variables are essentially infinitely divisible, I disagree that they can be considered ordinal or nominal. Sometimes we choose to make them categorical, e.g. by using age bands in questionnaires, but the underlying variable (in this case, age) is not categorical.
A: I also think "no".
You can say these types are in a hierarchy of sorts because (for example)


*

*"an interval variable can be recoded as an ordinal variable if you are prepared to lose some of its information" and 

*"an ordinal variable can be treated as a nominal variable if you are prepared to lose some of its information"... 

*etc.  


However, my second dash point above (for example) is very different from "an ordinal variable is also a nominal variable" as per the original question.  A hierarchy of sorts is not the same as being "nested".
A: Yes, you are correct.  It's better to think of the levels of measurement as describing properties of data, rather than types of data.
All data that exhibit ratio properties also exhibit interval, ordinal, and nominal properties.  And so forth.
It makes no difference that (e.g.) interval or ratio data can take on an infinite number of possible values — any individual datum will have a specific value, which can be considered in virtue of its ordinal properties or nominal properties.
