There is an important difference between individual Likert-scored items (which are ordinal and thus do not have a mean) versus multi-item scales (which can be treated as interval and thus have a mean).

The mean of Likert scale statements is statistically invalid. The score can be computed of Likert scale based statements using mean or other appropriate method then mean of the score can be calculated.

This is a question that has troubled me for some time. Researchers seem split on whether the mean of Likert-scored items are inherently meaningful. I'll be following this question for sure.

David L Morgan I recently had a discussion with an editor, and I made the exact points you mentioned - explaining that the mean of multiple Likert items are traditionally treated as interval granted the tau equivalence assumption is not violated. They would have none of it, insisting that the only appropriate treatment is to use dummy coding. If you don't mind, how would you respond?

First, there is almost never any point in arguing with editors. Second, if you had a scale composed of multiple-likest scored variables, that is one thing, but tau-equivalence does not applied to a single variable.

Finally, if you had a multiple-category independent variable, then converting it to a series of dummies generates the maximum possible explained variance because it allows for non-linearities.

Thank you for the reply. In this instance, I had multiple, ordinal Likert-scored items, and I derived the mean from these items. The order of response options mattered, so I couldn't use dummy coding. Perhaps it's an assumption that some choose not to grant?

Likert-scored items, if looking at them as a summative scale, then the mean is computed, but there are issues that are resolved by analyzing Likert-type items using modern test theory (Item Response Theory-IRT). I hope this is helpful

If you have multiple Likert-scored items, then you should consider converting them to a single interval-scored scale. ARe you familiar with this approach, via coefficient alpha and Exploratory Factor Analysis?

Also, the reason why a set of dummies maximizes R-sq for an item is because it considers all possible orders, rather than assuming a single ordering. This makes it possible to include non-linear equivalents in the scoring of an item.

The bad news: Don't use AM for ordinal data! It is not well-founded, it is even meaningless. The good news: there is a fairly simple way out. Pool all your data in a single discrete frequency distribution ("histogram") on the 5-point scale (or any other point scale) and calculate the distribution's RANK (between 0 and a maximum which only depends on the total number of observations and the number of scale points (5)). Once you have this RANK and its maximum, you may convert it into a percentage, for convenience and standardization. This procedure only assumes that the Likert points represent ordered categories, nothing more. See my RG project "Educational Assessment Engineering" and/or contact me directly.

Paul Hubert Vossen I'm not sure I follow your advice. If the items are scored 1 to 5, won't the ranks also be scored 1 to 5?

David L Morgan No, scores are associated with single items (here, scored on a scale consisting of 5 ordered categories, aka "5-point scale"). Ranks are associated with a distribution or profile of multiple scores (on multiple items). Thus, a possible distribution would be: (0, 0, 2, 1, 0), i.e. 2 items scored 2 and 1 item scored 4. With n = 5 and F = 3, there are in total 35 different profiles ranked 0 to 34. The profile (0, 0, 2, 1, 0) has rank 15 according to the co-lexicographic ordering principle applied to n-point profiles. See attachment. P.S.: There is of course a not-to-difficult formula to compute the rank of any profile without looking it up in a table.

David L Morgan Yes, this is how I typically treat multiple Likert-scored items (i.e. coefficient alpha, EFA). However, there are apparently some individuals who do not agree with this conversion. That is, the assumption that multiple ordinal items can be treated as a single interval scale.

I understand their concern. I'm just curious what the alternative approach(es) are. Dummy coding seems to be one option.

Blaine Tomkins The problematic issue is: "... the assumption that multiple ordinal items can be treated as a single interval scale." Is it *just* an assumption or is it a mathematical fact? And what conditions (on the data) should the satisfied in order that the assumption may be safely said to be corroborated (in practice)?

I have often read about this assumption but never seen a strict mathematical proof of it (probably some sort of asymptotic law, i.e. having enough data lowers the error implicit in the assumption to almost 0%). Can you provide us with a reference to the relevant literature? Would be nice. Thank you in advance.

Paul Hubert Vossen Your questions are precisely those that I have as well. It seems this question has been addressed at length already in an earlier thread:

https://www.researchgate.net/post/Is_a_Likert-type_scale_ordinal_or_interval_data

I don't believe there is mathematical proof for the conversion. At least, I've never encountered one. But then, nearly all statistical tests include assumptions about data that may not be true - or the data may (somewhat) conform to the assumptions of the test.

As a non-expert, my understanding is that data from multiple ordinal items will tend to approximate an interval scale assuming the items all measure the same latent variable (i.e., the data are unidimensional), the data display symmetry, and of course, one has a sufficient amount of data.

Please find below four references pro and contra the metric analysis of Likert scale data as if they were uni-dimensional quantitative data obeying the axioms of interval measurement:

Paul Hubert Vossen Many thanks for sharing the above references. Any good resources on this topic are greatly appreciated. The Carifio & Perla (2007) paper is compelling and well-articulated. The Heene (2013) article is a bit dense, so I'll have to re-read it a few times.

One of the most radical alternatives to the classical metric analysis of so-called Likert "scale" type data has recently been proposed by

• James Camparo www.researchgate.net/profile/James-Camparo
• Lorinda Camparo www.researchgate.net/profile/Lorinda-Camparo.

Their basic idea is astonishingly simple: don't treat the respective responses 1 ... 5 on the 5-point Likert items as 5 distinctive ordered points on a single uni-dimensional continuum, but as the 5 axes of a 5-dimensional space in which each Likert item will find its own place, or position.

Each time a respondent emits a response on an item, e.g. response category 4, he/she reveals that the 4th dimension is most closely related with that item, according to his/her mental state at the moment.

Now the Camparo's claim that this situation can be quite well likened to the measuring of e.g. a particle's position as modelled in quantum mechanics, including the standard interpretation of such measurement and the formal methods of dealing with such data and calculating interesting parameters and variables over many repeated measurements with the same device, i.e. Likert questionnaire.

They are now trying to find empirical evidence for their suggestion/proposal. Here are some references to their work:

• Article The Analysis of Likert Scales Using State Multipoles An Appl...

• Article Being of “two minds”: Assessing vacillating and simultaneous...