<p>Likert scales are ordinal. But for so many statistical analyses, they are treated like interval scale variables.</p>
<p>For example, in developing scales using factor analysis, you find the mean of the Likert scores, demean them, then find correlations (usually not rank correlations) etc. All of this requires your variable to be interval scale, as you cannot add, subtract, multiply or divide ordinal variables.</p>
<p>I am a student of Economics who has an interest in Behavioural Economics. I wanted to work with some psychometric scales and see their relationship with economic variables, and therefore I have read up quite a bit on factor analysis and the development of scales.</p>
<p>But this treating of ordinal scales as interval scales has not been convincing for me.</p>
<p>If I draw an analogy: in Economics there used to be a great debate between whether utility is ordinal or cardinal. Turns out, if you only have data on choice behaviour under deterministic conditions, the best you can do is infer ordinal utility.</p>
<p>But then, von Neumann and Morgenstern published their theorem proving that by empirically observing choice behaviour, not over the options themselves, but over lotteries that give those options as prizes, it is possible to pin down a cardinal value for utility of any option, unique upto an affine transformation: basically, we get a unique interval scale measure for utility. (This is extended in Savage which still has some problems, but there has been substantial theoretical effort put into it).</p>
<p>In the Psychology literature, I found claims that if the Likert scale has 5 or more options to choose from, then you can treat it like an interval scale variable. But I could not find any solid justification for this.</p>
<p>Is there any solid statistical or mathematical justification (like the von Neumann Morgenstern theorem of utility) for treating Likert scales with 5 or more options as interval scales? Where can I read them?</p>

CC BY-SA 4.0
2024-09-24T15:54:43.473Z
There is: https://en.wikipedia.org/wiki/All_models_are_wrong I'm also certain that there are papers where people have made these arguments quantitatively, but those arguments are always going to be based on particular assumptions as well as qualitative decisions about what is "good enough". From my perspective, it's usually those assumptions that fail before the "treat as interval" fudge does.
Bryan Krause
CC BY-SA 4.0
2024-09-24T16:32:16.253Z
So, if I try to apply the logic of "all models are wrong some are useful" here, this is how I would interpret it: there is no a priori justification for treating Likert scores as interval variables, but if we take a leap of faith and do that anyway, the results that we get are quite robust (I know that many such scales like OCEAN are quite robust) across time and populations. Therefore, we do not theoretically understand why this works, it is an empirical fact that this works well. This is actually a good enough justification to satisfy me, but I want to make sure that is what you meant.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2024-09-24T16:36:17.737Z
Because that is how I understand this adage. This is somewhat similar to Friedman's idea: An experienced pool player isn't actually an expert Physicist, but if we make a mathematical model assuming she is an expert Physicist, and predict her moves in the game under that assumption, we may be remarkably successful. So, that model is good to go. Is that what you mean?
Ishan Kashyap Hazarika
CC BY-SA 4.0
2024-09-24T16:42:33.317Z
I don't know that I would take it quite so far as "if we take a leap of faith and do that anyway, the results that we get are quite robust", more like "if we take a leap of faith, we get results that might be useful in answering our research questions". A bit like we *know* that Newtonian mechanics are wrong, but are happy with using that wrong model in certain cases where it's useful (like when speeds are relatively slow within a reference frame).
Bryan Krause
CC BY-SA 4.0
2024-09-24T16:45:28.620Z
I think it's questionable to treat Likert or other ordinal scales as if they're interval, but if it's intractable to treat them as ordinal in some context, and aggregating them and treating them as interval is useful in making predictions about behavior, that might be okay. Specifically in the case that you are aggregating, you might use https://en.wikipedia.org/wiki/Central_limit_theorem as a theoretical basis, though you'll run into arguments with people about how many observations you actually need for CLT to "apply" (there is of course no way to agree on what that number is).
Bryan Krause
CC BY-SA 4.0
2024-09-24T19:34:32.673Z
Regarding the first point, I guess we are agreeing with the same thing, just that I was a bit too generous in the first comment. The pool player example fits perfectly with your "useful for answering our research questions" point: we _know_ that the pool player isn't a perfect physicist, but assuming she is solving differential equations in her head to decide her moves may still allow us to predict her moves reasonably well _if she is experienced enough_. Similar to the Newtonian mechanics thing
Ishan Kashyap Hazarika
CC BY-SA 4.0
2024-09-24T19:35:45.340Z
Regarding the CLT though I am not sure. We can apply the CLT only after we have already assumed the variable is cardinal, not vice versa, as far as I know. The CLT cannot directly be applied to ordinal data, can it?
Ishan Kashyap Hazarika
CC BY-SA 4.0
2024-09-24T19:51:16.653Z
Not directly, no, but it comes into play when you decide to treat your ordinal scales like they are interval by doing things like giving ordinal responses a numerical value and summing them up. The resulting variable will behave more like a continuous, normally-distributed variable than the original scores do (in most cases). It's controversial whether this is "okay" or not for the most common reason anything in statistics is controversial: because (going back to Box) we all know that it's all "wrong" and are arguing about which levels of "wrong" are okay and which are not.
Bryan Krause
CC BY-SA 4.0
2024-09-24T19:54:26.880Z
https://stats.stackexchange.com/questions/10/under-what-conditions-should-likert-scales-be-used-as-ordinal-or-interval-data is relevant, the 10th ever question asked on Cross Validated.
Bryan Krause
CC BY-SA 4.0
2024-09-26T14:41:14.597Z
Although there are practical justifications (especially when averaging multiple items), I think the answer to "Is there any solid statistical or mathematical justification?" is "No"
pep
<measurement><methodology>
Justification for treating Likert scales as interval
Ishan Kashyap Hazarika
# Justification for treating Likert scales as interval

(Asked by: username_0 on 2024-09-24)

Likert scales are ordinal. But for so many statistical analyses, they are treated like interval scale variables.

For example, in developing scales using factor analysis, you find the mean of the Likert scores, demean them, then find correlations (usually not rank correlations) etc. All of this requires your variable to be interval scale, as you cannot add, subtract, multiply or divide ordinal variables.

I am a student of Economics who has an interest in Behavioural Economics. I wanted to work with some psychometric scales and see their relationship with economic variables, and therefore I have read up quite a bit on factor analysis and the development of scales.

But this treating of ordinal scales as interval scales has not been convincing for me.

If I draw an analogy: in Economics there used to be a great debate between whether utility is ordinal or cardinal. Turns out, if you only have data on choice behaviour under deterministic conditions, the best you can do is infer ordinal utility.

But then, von Neumann and Morgenstern published their theorem proving that by empirically observing choice behaviour, not over the options themselves, but over lotteries that give those options as prizes, it is possible to pin down a cardinal value for utility of any option, unique upto an affine transformation: basically, we get a unique interval scale measure for utility. (This is extended in Savage which still has some problems, but there has been substantial theoretical effort put into it).

In the Psychology literature, I found claims that if the Likert scale has 5 or more options to choose from, then you can treat it like an interval scale variable. But I could not find any solid justification for this.

Is there any solid statistical or mathematical justification (like the von Neumann Morgenstern theorem of utility) for treating Likert scales with 5 or more options as interval scales? Where can I read them?

**Tags:** measurement, methodology

> **Comments:**
> **username_1**: There is: https://en.wikipedia.org/wiki/All_models_are_wrong I'm also certain that there are papers where people have made these arguments quantitatively, but those arguments are always going to be based on particular assumptions as well as qualitative decisions about what is "good enough". From my perspective, it's usually those assumptions that fail before the "treat as interval" fudge does.  
> **username_0**: So, if I try to apply the logic of "all models are wrong some are useful" here, this is how I would interpret it: there is no a priori justification for treating Likert scores as interval variables, but if we take a leap of faith and do that anyway, the results that we get are quite robust (I know that many such scales like OCEAN are quite robust) across time and populations. Therefore, we do not theoretically understand why this works, it is an empirical fact that this works well. This is actually a good enough justification to satisfy me, but I want to make sure that is what you meant.  
> **username_0**: Because that is how I understand this adage. This is somewhat similar to Friedman's idea: An experienced pool player isn't actually an expert Physicist, but if we make a mathematical model assuming she is an expert Physicist, and predict her moves in the game under that assumption, we may be remarkably successful. So, that model is good to go. Is that what you mean?  
> **username_1**: I don't know that I would take it quite so far as "if we take a leap of faith and do that anyway, the results that we get are quite robust", more like "if we take a leap of faith, we get results that might be useful in answering our research questions". A bit like we *know* that Newtonian mechanics are wrong, but are happy with using that wrong model in certain cases where it's useful (like when speeds are relatively slow within a reference frame).  
> **username_1**: I think it's questionable to treat Likert or other ordinal scales as if they're interval, but if it's intractable to treat them as ordinal in some context, and aggregating them and treating them as interval is useful in making predictions about behavior, that might be okay. Specifically in the case that you are aggregating, you might use https://en.wikipedia.org/wiki/Central_limit_theorem as a theoretical basis, though you'll run into arguments with people about how many observations you actually need for CLT to "apply" (there is of course no way to agree on what that number is).  
> **username_0**: Regarding the first point, I guess we are agreeing with the same thing, just that I was a bit too generous in the first comment. The pool player example fits perfectly with your "useful for answering our research questions" point: we _know_ that the pool player isn't a perfect physicist, but assuming she is solving differential equations in her head to decide her moves may still allow us to predict her moves reasonably well _if she is experienced enough_. Similar to the Newtonian mechanics thing  
> **username_0**: Regarding the CLT though I am not sure. We can apply the CLT only after we have already assumed the variable is cardinal, not vice versa, as far as I know. The CLT cannot directly be applied to ordinal data, can it?  
> **username_1**: Not directly, no, but it comes into play when you decide to treat your ordinal scales like they are interval by doing things like giving ordinal responses a numerical value and summing them up. The resulting variable will behave more like a continuous, normally-distributed variable than the original scores do (in most cases). It's controversial whether this is "okay" or not for the most common reason anything in statistics is controversial: because (going back to Box) we all know that it's all "wrong" and are arguing about which levels of "wrong" are okay and which are not.  
> **username_1**: https://stats.stackexchange.com/questions/10/under-what-conditions-should-likert-scales-be-used-as-ordinal-or-interval-data is relevant, the 10th ever question asked on Cross Validated.  
> **username_2**: Although there are practical justifications (especially when averaging multiple items), I think the answer to "Is there any solid statistical or mathematical justification?" is "No"  

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## 0 answers

