Issue 17561: Scales of Measurement (smm-rtf) Source: CAST Software (Dr. Bill Curtis, b.curtis(at)castsoftware.com) Nature: Enhancement Severity: Significant Summary: A standard component of measurement theory is 'Scales of Measurement' because they constrain the types of mathematical operations that can be performed on a measure. The base set of scales are 'Nominal', 'Ordinal', 'Interval', and 'Ratio'. The representation of a measure in SMM should contain information about the scale of measurement to indicate how it can be used and the limits of the operations that can be performed. If there is a way to enforce these constraints that would be best, but lacking that it would be useful for the specification of a measure to contain information about its limits for those who would incorporate the measure in their applications. Resolution: Revised Text: Actions taken: August 22, 2012: received issue Discussion: End of Annotations:===== m: webmaster@omg.org Date: 22 Aug 2012 15:05:29 -0400 To: Subject: Issue/Bug Report ******************************************************************************* Name: Bill Curtis Employer: CAST mailFrom: curtis@acm.org Terms_Agreement: I agree Specification: Structured Metrics Meta-model (SMM) Section: 10 FormalNumber: 2012/01/05 Version: 1.0 Doc_Year: 2012 Doc_Month: Month Doc_Day: Day Page: 41-46 Title: Scales of Measurement Nature: Enhancement Severity: Significant CODE: 3TMw8 B1: Report Issue Description: A standard component of measurement theory is 'Scales of Measurement' because they constrain the types of mathematical operations that can be performed on a measure. The base set of scales are 'Nominal', 'Ordinal', 'Interval', and 'Ratio'. The representation of a measure in SMM should contain information about the scale of measurement to indicate how it can be used and the limits of the operations that can be performed. If there is a way to enforce these constraints that would be best, but lacking that it would be useful for the specification of a measure to contain information about its limits for those who would incorporate the measure in their applications. From: Bill Curtis To: Henk de Man , Larry Hines CC: "smm-rtf@omg.org" , Alain Picard , "William Ulrich (wmmulrich@baymoon.com)" , Arne Berre Subject: RE: On scales of measurements (ref Bill Curtis' email) Thread-Topic: On scales of measurements (ref Bill Curtis' email) Thread-Index: AQHNftrkgyL6+FXtdEeczaDM8uGfd5diulLw///m64CAADfgAIAA+sIAgABfgACAATnkAIAAyKSg Date: Wed, 22 Aug 2012 19:27:25 +0000 Accept-Language: en-US X-MS-Has-Attach: X-MS-TNEF-Correlator: x-originating-ip: [99.44.4.184] Henk, Larry, Here is what I submitted as an issue. >>A standard component of measurement theory is 'Scales of Measurement' because they constrain the types of mathematical operations that can be performed on a measure. The base set of scales are 'Nominal', 'Ordinal', 'Interval', and 'Ratio' each of which has a different set of allowable operations. The representation of a measure in SMM should contain information about the scale of measurement to indicate how it can be used and the limits of the operations that can be performed. It would be best if there is a way to enforce these constraints, but lacking that it would be useful for the specification of a measure to contain information about its limits for those who would incorporate the measure in their applications.<< If SMM is primarily a specification language to help programmers formalize their implementation so it can be shared, then the significance of the issue may be less. However, if we intend for SMM to be used by non-programmers who are defining measures to be implemented, then it should have a home as an attribute of the measure even if we can.t enforce the constraints electronically. If we expect measurement experts and statisticians to use the standard we will need to provide a good tutorial to help them understand the worldview behind the SMM, since it will not be natural for some of them. In trying to specify the Automated Function Point calculation process in SMM, I found it challenging and difficult to see mistakes, compared to representing the calculation process in algebraic notation. I think the Architecture Board is hoping I got it right since they are struggling to read the graphical representation of the measure in SMM which Alain helped me clarify. - Bill Dr. Bill Curtis SVP & Chief Scientist CAST PO Box 126079 Fort Worth, Texas 76126-0079 USA +1-817-228-2994 b.curtis@castsoftware.com www.castsoftware.com From: Henk de Man [mailto:hdman@cordys.com] Sent: Wednesday, August 22, 2012 4:12 AM To: Larry Hines Cc: Bill Curtis; smm-rtf@omg.org; Alain Picard; William Ulrich (wmmulrich@baymoon.com); Arne Berre; Henk de Man Subject: Re: On scales of measurements (ref Bill Curtis' email) Bill (Curtis), I think you feel with us where the tricky part is regarding S-o-M: constraining whether or when a measure, dependent on what it's S-o-M is, can serve as "base" for rescaled, binary or collective measure. And constraining is difficult, given the lack of grip on type-correctness of formula's, functors and accumulators. But it is likely that you might not want to even go that far. Maybe you only wanted a property on a measure, to serve as a piece of information to the human user of SMM, and without any further consistency checking and validation by the system? Maybe you explain your need in more detail (your use case): you want S-o-M in SMM, but for what purpose, to do what ?. What would be your ambition level in SMM, with this? That would help. Otherwise we might get into very complicated anticipated designs of something that you yourself did not even dream of ? But it is important that you then also indicate whether just having the property will make SMM more useful ? And if we can agree on the usefulness of having it (depending on your further explanation), and we see there might be way, can you then submit the issue, so that the RTF team can consider it further. ((Note that the issues that Larry and I were discussing, and still are discussing, are all formally submitted issues.)) Regards, Henk de Man Cordys On Tue, Aug 21, 2012 at 4:29 PM, Larry Hines wrote: Henk, Bill, Concerning TempProduct, I agree that we rarely and maybe never conceive of areas of temperature or square temperatures. Multiplying something else by a temperature is, nonetheless, valid. Let.s say I want the cost of lowering my thermostat to 68F. I might multiply the temperature delta below 72F (in degrees) by the cost per degree (of variance below 72F). I do agree with your larger point, Henk. At the minimum we would need to say that the calculations (via formula or functor) of collective measures are type correct. We would also need to specify the return type of each Dimensional measure beyond .Number. of unit of measure. Possible restrictions would include minimal and maximum. Also, whether .Number. was from a field such as the rationals and reals or whether .Number. was just from a total order. Doing so would, nevertheless, still not answer the question of whether a measure was meaningful. That is definitely beyond the scope of SMM. For example, I know a personality psychologist who doubts the value of the MNPI because of item overlap. Larry From: Henk de Man [mailto:hdman@cordys.com] Sent: Tuesday, August 21, 2012 3:47 AM To: Larry Hines Cc: Bill Curtis; smm-rtf@omg.org; Alain Picard; William Ulrich (wmmulrich@baymoon.com); Arne Berre; Henk de Man Subject: Re: On scales of measurements (ref Bill Curtis' email) Larry, Bill, On one hand I believe Bill that scale of measurement (S-o-M) is an important piece of information. But... On the other hand, it is not easy to see how SMM can take advantage of it. Let me explain as follows (... you can correct me if I get derailed here ...): Note that TemperateMeasure is interval measure. Not ratio. Now consider the following two examples, both fitting perfectly in SMM: Remember P*V/T = C (the empirical law on pressure, volume and temperature. ). So, T = (1/C ) * P * V. (EXAMPLE ONE) So, T can be binary measure, or rescaled measure (in SMM sense). It can be determined based on formula(e), functor(s) and operation(s). But TempQuotient = Tcalif / Teurope, or TempProduct = Tcalif * Teurope aren.t meaningful measures. (EXAMPLE TWO) But SMM would allow to express both as binary measure and TempQuotient even as ratio measure. Using functor, or, as discussed with Larry in other e-mail, formula. This says: in SMM you can use Temperature as part of functors, operations and formulas. The fact that temperature is of interval scale of measure, not ratio scale of measure, cannot easily be applied as constraint in SMM. It does not constrain temperature for being used in functors, operations and formulas. Because we must not exclude something like EXAMPLE ONE.. More detailed .parsing. of formula or functors would required, to see whether the measure is doing something of the sort of EXAMPLE ONE or of the sort of EXAMPLE TWO. Which gets complicated, the more so, as "content" of formulae is not normatively structured. And let alone operations. As implementations of operations aren.t structured in SMM, and can be of any nature (even Java programs), .parsing. of code would be required, which is hard. So, possibly, S-o-M would only serve as piece of info to the human user, with the discipline only between his/her ears ? That would imply that SMM cannot take real advantage of it, but can just store the property, without any constraint on it, or it by itself constraining nothing ? Regards, Henk de Man On Mon, Aug 20, 2012 at 7:49 PM, Larry Hines wrote: Bill, Henk, Here are some rambling thoughts. Bottom line: fully embracing scales of measurement terminology seems to be a slippery slope. >> Nominal scales are categorical with no implied ordering. This is precisely the Grade Measure. The symbols of grade measurements have no implied ordering. That does not mean that users of such measurements may not impose their own notions of order. Measures can be defined upon the all grade measurement to derive modes. SMM currently has no mechanism to restrict defining a measure upon set of grade measurement to derive the set.s median. But, the definition of the median deriving measure certainly implies some order. >> Ordinal scales have an implied order but with no requirement to have equal intervals between the points. The intervals behind a Grade Measure can vary in size. To impose a total order requirement on the symbols of the grade would require an extension or subclass of the Grade Measure. Imposing an implied order seems odd to me. Either the order should be explicit or left the users of the measurements. Measures can be defined upon the such grade measurement to derive modes and medians. Alternatively, one can use a Ranking Measure. The intervals can still vary in size, but there is an explicitly named unit of measure. That is, ranking indicates a point of a dimension. The order is the standard inequality of Numbers. Ordinal rankings would need an extension which restricted the possible values to the positive integers. One important aspect of ordinal scales is that averaging is meaningless. This suggests that we somehow restrict how such measurements can be used in further derivations which implies that ordinal scales are not ranking measures. >> Interval scales have equal intervals between points, but have an arbitrary point for zero. (Farenheit, years in BC/AD) This is precisely Dimensional Measure. Dimensional Measures are based upon its specified unit of measure. That unit is the interval. >> Ratio scales have equal intervals between points and an absolute zero representing the absence of the attribute measured. Yes, I got this one wrong. Ratio scales of measurement are a subclass of Dimensional Measures which have the minimum possible value of 0. To impose that requirement would require an extension of the Dimensional Measure. >> There are other types of scales such as Guttman (if you answered this question correctly, then you probably answered all the easier questions correctly), but they would probably be treated as specialized restrictions or attributes of a measure. Guttman imposes additional semantics on the symbols of grade measurements. To be clear, let.s consider the following Guttman style question. To what degree does SMM minimally describe the aspects of Metrology? None A little Some Half Most Almost all Completely The answers that we collect are grade measurements. We may want to analyze the responses, but we cannot do calculations on them until we turn them into numbers which we could do in many different ways. Let.s choose a simple method. .None. maps to 0, .A little. maps to 1, .Some. maps to 3, .Half. maps to 5, .Most. maps to 7, .Almost all. maps to 8, and .Completely. maps to 10. You might prefer starting at 1, using even increments of 1, or even a completely different mapping. We are, nonetheless, applying some dimensional measure to the responses (which are grade measurements). From the dimensional measurements I can measure their total, average, etc. The mapping from symbols to numbers happens to be consistent with Guttman. .A little. implies that SMM minimally describes at least .None.. The mapping of .A little. is greater than the mapping of .None.. By consistent with Guttman we are saying that if response R1 subsumes response R2 then mapping(R1) > mapping(R2). To me the operative word here is .subsumes.. That is, Guttman is an ordinal scale of measurement where the implied ordering is containment. I could ramble on and on, but time is short. The real questions are how can we tiptoe into statistics without biting off more than we can chew? Which part of statistics? Probability? Scale Analysis? Should we tiptoe into statistics before we add logarithms? Is this a SMM RTF issue or is it the need for a new OMG task force? From: Henk de Man [mailto:hdman@cordys.com] Sent: Monday, August 20, 2012 9:30 AM To: Bill Curtis Cc: Larry Hines; smm-rtf@omg.org; Alain Picard; William Ulrich (wmmulrich@baymoon.com); Arne Berre; Henk de Man Subject: Re: On scales of measurements (ref Bill Curtis' email) Bill, Larry, So, in SMM terms, what Bill suggests, is, probably, about the following: The S-o-M of a measure, determines whether a measure of that sort-of-scale, can serve as: A grade measure on its own A "base measure" to a grade measure A "base measure" to a ranking measure (see issue discussion with Larry on one of the issues) A "base measure" to a rescaled measure A "base measure" to a binary measure of some sort, including base to ratio measure A "based measure" to a collective measure of some sort .. (note that when I say "a base measure to", I am not following precise SMM wording per se here ...) If this is true, and there's consensus on this, there would be some more analysis required to fine out what the SMM contraints and implications are in more detail. The basis for this would be that Bill Curtis first brings it up as SMM RTF issue via Juergen Bold juergen@omg.org . And then this one can be discussed further between you, Larry, Alain, me, and whoever else is interested. Is that ok Bill (( note that I am a recent new-comer, but I feel that issue discussion works very well )). But maybe Larry should responds first, before you actually submit the issue. I am just trying to interprete here, in advance of the actual issue to come. Regards, Henk de Man On Mon, Aug 20, 2012 at 4:15 PM, Bill Curtis wrote: Henk.s table gets to the heart of the issue. The type of scale places constraints on the types of mathematical/statistical operations that can be applied to the data. Nominal scales are categorical with no implied ordering. Ordinal scales have an implied order but with no requirement to have equal intervals between the points. Interval scales have equal intervals between points, but have an arbitrary point for zero. (Farenheit, years in BC/AD) Ratio scales have equal intervals between points and an absolute zero representing the absence of the attribute measured. The scale type would be an attribute of the measure that limits its allowed operations and should be declared as an aspect of the measure. There are other types of scales such as Guttman (if you answered this question correctly, then you probably answered all the easier questions correctly), but they would probably be treated as specialized restrictions or attributes of a measure. - Bill Dr. Bill Curtis SVP & Chief Scientist CAST PO Box 126079 Fort Worth, Texas 76126-0079 USA +1-817-228-2994 b.curtis@castsoftware.com www.castsoftware.com From: Henk de Man [mailto:hdman@cordys.com] Sent: Monday, August 20, 2012 8:51 AM To: Larry Hines Cc: Bill Curtis; smm-rtf@omg.org; Alain Picard; William Ulrich (wmmulrich@baymoon.com); Henk de Man; Arne Berre Subject: On scales of measurements (ref Bill Curtis' email) Bill (Curtis), Larry, ((Larry, note my response, with updated attachment, to our issues-under-discussion, per separate/previous e-mail )) This here is just in response to Bill Curtis' measurement scale issue. Note, Larry, that you interpreted ratio scope as relating to ration measure. I think that is not the same. Ratio measure means (in SMM) a measure that is based on devision of two other measures. Ratio scale (Bill Curtis ...) means: 2 measurements of same measure, regardless of whether it is Ratio measure or not, can be e.g. multiplied by each other, divided by each other, etc. I think you are right that grades are like nominal. (though some might also be ordinal ..) But there might be some more in it to understand how the measurement scales should map to the various types of measures in SMM. That might sometimes be many-many. And, if it is useful to add these to SMM, what they will do, e.g. are there any constraints, or semantics coming with them, etc. See e.g. this table of what measurement scales imply (... taken from the web somewhere ...): OK to compute.... Nominal Ordinal Interval Ratio frequency distribution. Yes Yes Yes Yes median and percentiles. No Yes Yes Yes add or subtract. No No Yes Yes mean, standard deviation, standard error of the mean. No No Yes Yes ratio, or coefficient of variation. No No No Yes If you also think that more discussion is required on this, Bill Curtis should probably go the regular issue route, submitting an SMM RTF issue to Juergen Bold, so that it will appear on the formal issue list of the OMG. Larry, do you agree ? Bill, if so, can you then do that ? Regards, Henk de Man Cordys On Sun, Aug 19, 2012 at 7:48 PM, Larry Hines wrote: Interval scales of measurement are SMM Dimensional Measures. The unit of measure is a core aspect of Dimensional Measures. Ratio scales of measurement are SMM Ratio Measures. Ratios are differ from dimensional because ratios are dimensional-less (i.e., dimensions of the dividend and divisor cancel). Nominal scales of measurement are both Ranking measures in SMM (soon to be renamed to be Grade Measure). The symbols mapped to by the Grade Measure are arbitrary names for nominal scales. Ordinal scales of measurement are likewise Ranking (or Grade) Measures. The symbols of the Grade Measure Interval map to the ordinal numbers. As with Ratio Measures, Grade Measures have no unit of measure. From: Bill Curtis [mailto:b.curtis@castsoftware.com] Sent: Saturday, August 18, 2012 12:29 PM To: Larry Hines; Henk de Man; smm-rtf@omg.org; Alain Picard; William Ulrich (wmmulrich@baymoon.com) Subject: Quick question Has the concept of scales of measurement been added to the spec (Nominal, Ordinal, Interval, Ratio)? Excuse my lateness to the discussion if it has already been discussed or added. - Bill Dr. Bill Curtis SVP & Chief Scientist CAST PO Box 126079 Fort Worth, Texas 76126-0079 USA +1-817-228-2994 b.curtis@castsoftware.com www.castsoftware.com From: Larry Hines [mailto:Larry.Hines@microfocus.com] Sent: Saturday, August 18, 2012 12:16 PM To: Henk de Man Cc: Juergen Boldt; issues@omg.org; smm-rtf@omg.org; Alain Picard; Pete Rivett; Arne Berre; fred.a.cummins Subject: RE: issues 17473 -- SMM RTF issues -- Add attribute .type. (String) to Observation As I see it, we are down to two issues, 17472 and 17483. See attached for my comments (mostly just agreed) about the other issues. 17483 comes down to mapping the variable names of the formula/operation to the base measures. I don.t see how that is done. Until we have such a mapping we can.t move forward on resolving this. 17472 is not just a technical issue. SMM is about measures and measurements. 17472 steps into probability and statistics. That might be a good follow up standard for SMM. I.ll try to make time on Monday for a more extended note. Thanks, Larry From: Henk de Man [mailto:hdman@cordys.com] Sent: Saturday, August 18, 2012 8:09 AM To: Larry Hines Cc: Juergen Boldt; issues@omg.org; smm-rtf@omg.org; Alain Picard; Pete Rivett; Arne Berre; fred.a.cummins; Henk de Man Subject: Re: issues 17473 -- SMM RTF issues -- Add attribute .type. (String) to Observation Larry, 17473 can indeed be closed, with no action. I concluded on that one also, earlier this week. See my red comment to that issue in the attachment. Can you check status and my last feedback on all issues in that same attachment. We need some more discussion on the ones that we did not yet resolve. And maybe confirmation on the rest. Regards, Henk de Man On Fri, Aug 17, 2012 at 10:34 PM, Larry Hines wrote: Given the resolution of 17474, is there a reason for 17473 anymore? Larry Hines, PhD Software Systems Developer, Sr. Principal Micro Focus larry.hines@microfocus.com 8310 Capital of Texas Highway, Suite 100 Austin, Texas, 78731, USA Telephone : 512-340-4740 This message has been scanned by MailController. -- Henk de Man Research Director hdman@cordys.com www.cordys.com T +31 (0)341 37 5541 . M +31 (0)6 51 43 09 45 CORDYS . Improving Business Operations -- Henk de Man Research Director hdman@cordys.com www.cordys.com T +31 (0)341 37 5541 . M +31 (0)6 51 43 09 45 CORDYS . Improving Business Operations -- Henk de Man Research Director hdman@cordys.com www.cordys.com T +31 (0)341 37 5541 . M +31 (0)6 51 43 09 45 CORDYS . Improving Business Operations -- Henk de Man Research Director hdman@cordys.com www.cordys.com T +31 (0)341 37 5541 . M +31 (0)6 51 43 09 45 CORDYS . Improving Business Operations -- Henk de Man Research Director hdman@cordys.com www.cordys.com T +31 (0)341 37 5541 . M +31 (0)6 51 43 09 45 CORDYS . Improving Business Operations