# Is the sum of a number of ordinal variables still ordinal?

I have performed a survey where I have a number of questions which can be answered Strongly Agree, Agree etc. to Strongly Disagree. Some of the questions have been designed to measure the same thing.

I have summed these variables to get a 'score' for this underlying trait, and I wish to predict this score in a regression

My question is then, can I still considered this summed variable an ordinal variable? And therefore should be using an ordinal regression. Or should I consider this now a continuous variable and use a simple regression?

There have been a few questions I have seen that discuss whether agree / disagree variables can be considered ordinal at all, and others which question whether it is ok to sum these kinds of questions, but I have not found one that discusses the properties of the new variable.

• Well, you say they were asked if they strongly agree, agree, some neutral answer, disagree, strongly disagree. I assume you assigned them 5, 4, 3, 2, 1, right? Now think about this: if the respondents weren't told you were using that numerical scale, why aren't you using a different one? 5:1:1 is as good as 500,120,9,4,1, but surely this is not going to give you the same estimates, right? – pkofod Sep 25 '14 at 6:00
• If you are convinced by the answers already given to you then you might arrive to a reply to your exect question can I consider the summed variable an ordinal variable? Or should I consider this now a continuous variable?: that variable is generally nominal, neither ordinal or interval. It may prove to be ordinal occasionally, under some circumstances, - but who can say under which? – ttnphns Sep 25 '14 at 6:45
• Thankyou all for the excellent response. I was also drawn to this question by the 1+4 = 2+3 = 5 idea, but was confused because I had been assured it was common practice to sum agree/disagree variables to capture an intrinsic trait. I had though that because these questions SHOULD be measureing the same thing, most people should be answering one way or the other. But I fear this is too big of an assumption, because you could never KNOW this. The responses here lead me to another question - how do people measure underlying traits with agree/disagree questions? Because I am sure it is being done – SamPassmore Sep 25 '14 at 23:02

When you say things like 4+1 = 3+2 = 5, -- which you must do when you sum the components -- you (pretty much unavoidably) assumed they were interval at that time.

[If the components weren't interval, in general 4+1 $\neq$ 3+2 ... so you'd certainly have no business calling both of them "5".]

If the components were interval when you summed them, their sum is certainly interval.

[People may well disagree with me on this, but I can't see any basis for saying things like 4+1 = 3+2 = 5 -- along with all the similar statements that must be made -- unless you have assumed an interval scale. What basis would there be for thinking the summed-category-labels are equivalent outside the assumption that all gaps between adjacent values are equi-distant?]

Don't take this as an assertion that people should not add scale-items; in general I think it's a pretty reasonable thing to do. But in any case, once you do it, you shouldn't be uncomfortable about calling the sum interval-scale; you already went there.

• ZOMG we both used the same numeric example codes. :D – Alexis Sep 25 '14 at 3:29
• @Alexis That we might both do so surprises me not at all. I actually recycled the example (4+1 vs 3+2) from a comment under an answer by Nick Cox that I made earlier today. – Glen_b -Reinstate Monica Sep 25 '14 at 3:43
• +1. @SamPassmore: Think of a ordinal scale levels as twisted. So you cannot reliably sum them up. The only thing that you know is that ordinal scale is related to the unobserved "underlying" and "true" interval scale is some unknown and monotonic way. – ttnphns Sep 25 '14 at 6:26

(What a fun question! My answer is a reflection, and others might have interesting opposing viewpoints.)

No. The pairs of ordinal numbers 1 & 4 and 2 & 3 both sum to 5. But suppose such variables were coding, for example, functional mobility in an individual, where 1 indicated no impairment, 2 impairment not requiring assistance, 3 impairment requiring assistance, and 4 immobility. The codes meaningfully represent a progression along a continuum of impairment. But problems obtain:

• The idea of summing 'no impairment' and 'immobility' does not necessarily have conceptual validity (beyond two individuals with different experiences).

• The substantive meaning of a sum=5 is unclear.

• The meaning of an average of these numbers (i.e. a normalized sum), for example 2.5 (the average of 1 and 4, or 2 and 3) means what? Requiring or not requiring assistance in this example?

• One of the individuals measured by the 1 and 4 pair will ever complete any kind of race, but both of the 2 and 3 pair will eventually traverse a distance.

• Recall also that the many different quantities in ordinal codes will be ordinally identical (i.e. preserve rank), but will be radically different in quantity. For example the same four values above might be coded 0, 1.5, 7 and 92, both the sums and the averages change then, and ordering is not preserved in the resulting sums compared to the 1, 2, 3, 4 codes.

• At the last point I prefer the coding "Jack, Queen, King, As" in order to prevent summation. How to sum them up clearly depends on the rules of the game, i.e. the unknown monotone transformation @ttnphns mentioned. ;) – Horst Grünbusch Sep 25 '14 at 17:57

In order to add the numbers of the codification of two ordinal variables you have to make two assumptions:

1- The distance between any two adjacent categories is the same in each variable. That is, if you measure the distance between categories as "the effort it takes to change a customer's mind", then you must consider that changing somebody's mind from "strongly disagree" to "disagree" takes the same effort as changing it from "disagree" to "neutral", etc.

2- The impact of scoring minimum and maximum is the same for each variable. One example in which this would not apply would be these two variables: "Do you like the service given by this company?" and "Do you like the candies on the front desk?". Of course, the impact of each variable on the overall evaluation is much different.

All this problems could be avoided if you tune the score that each category adds to the global punctuation, so for these two variables you could use:

"Do you like the service given by this company?" 100, 90, 30, 5, 0 (You consider "Agree" and "Strongly agree" as nearly the same answer, "Neutral" as a bad outcome, and "Disagree" and "Strongly disagree" as VERY BAD)

"Do you like the candies on the front desk?" 5, 4, 3, 2, 1 (In this case you may consider the first assumption as real, but of course this variable contributes much less than the previous one to the global image of your company)

Another BIG problem is how to tune the category scores properly. In this case I can only say... Good luck, and don't forget to tell me how to do it!

I guess there are enough answers that state that sum of ordinal variables is itself not a well defined concept. But the OP mention Likert scales and indeed there are arguments that it is OK to sum up the Likert items (each answer) which results in what can be considered an interval number! See the discussion on scoring and analysis on the wikipedia page of Likert scales