Well, one advantage you have is that your ordinal categories are tied to numeric values, so you could consider each ordinal observation as approximately equal to the midpoint of the numeric range. That is, "0" is 0; "1" is 50; "2" is 250. You could convert the ordinal observations to these numeric values, and then, for each field, use the measure of central tendency you feel is most appropriate (mean, median, trimmed mean, Huber M-estimator...).... I would convert your final observations to some more common units, like plants / m2, or whatever is used.

A difficult topic. These ordinal scales are so difficult to model. Perhaps starting with a jitter or boxplot to see if there is a pattern along the different gradients.

For monitoring purposes of the aquatic ecosystems with aquatic plants they often use the Kohler scale, which consist of the ordinal scales 1-5. In boxplots I do not observe a clear tendency of this scale to any factor (see the second tab https://snwikaij.shinyapps.io/Debunked/). This often results in me switching to a binary scale present/absent (1/0) as it is easier to model and more accurate. I am not sure if this issue is similar in terrestrial ecosystems.

As Sal Mangiafico suggested you can convert the values. However, you can also treat the values as integers and plug them in an MVA. If the specific goal is to explore variability between different species you can repeat the same for present/absent to see if there are "large" differences. However, with increase in sample size it seems that the "explained" variation of an MVA often decreases (for aquatic plants at least).

Good luck!

Why not try the more reasonable approach of using ordinal logistic regression. This was designed exactly for your situation. Most major statistical packages containe this procedure. If you google ordinal logistic regression and a major statistical package you will get full instructions. Remember that an ordinal variable is ordinal not something else. Best wishes, David Booth

Thank you all. David, before proceeding to analysis we struggle to generate a response variable. We have around 100 points in each field, in each we have an estimate of weeds density using an ordinal variable. We wish to somehow generate a value that describe weeds density in each field in order to compare 50 fields in total.

This is a typical situation where a novel approach I have worked out recently (with the help of a researcher from France) might be applicable.

It's all about distributions of "points" or "units" over a discrete scale of ordered categories (also sometimes called points, confusingly).

Let's say we have 5 ordered categories from lowest to highest, e.g. 0, 1, 2, 3, 4 (5-point-scale). Let's assume there is some procedure thru which a fixed number of points, e.g. 20, are distributed over this scale.

We want to know the RANK of this distribution among all possible ranks of distributions with a given number of categories (here:5) and a given number of observations (here:20).

There is a rather simple formula by which this RANK can be calculated. There is an even simpler (but similar) formula by which the total number of such distributions can be computed.

Given both RANK and MAX RANK a percentage can be calculated which is a good representative for this profile of data.

See attachment for an example calculation scheme.

Thanks Paul Hubert Vossen. Very interesting. We don't fully understand how you calculate the rank. Any recommended paper we can read to get more details? Is there a code or UI we can use to input our data and get the COLEX score? And just to get a bit more sense of the method, the rank will be more influence (and get larger COLEX) by larger categories, that is 001 will be ranked higher than 010 (in case of three categories)?

Lior Blank Over the last five years or so I have developed (with the help of colleagues from South Africa and the United Kingdom) a novel approach to the problem of conjoint group-peer assessment. We have published several papers on it and my colleagues from the UK have already started developing a web-based implementation of it (the project is not yet fully completed , though). See my project www.researchgate.net/project/Educational-Assessment-Engineering-2021.

Although that system might also be used for analysing the data collected by means of one or more Likert scales, it is not meant for it and it may be too sophisticated for that purpose. Therefore I started a new project as of the beginning of this year to solve this ever-lasting problem of dealing with ordered category scales (instead of the standard percentage scale, which was the focus of the earlier project).

It's only a couple of weeks/days ago that I did indeed solve the mathematical challenges of ranking and unranking user response or judgement profiles, i.e. discrete frequency distributions, and implemented those solutions in Excel VBA.

Therefore, alas, there is not yet any paper that I could offer you to read the full story about it. However, I plan to publish about this novel approach to ordered category data analysis (which includes of course "Likert scale" data), and yes, there will also be some sort of software tool in the form of a simple Excel VBA template (or a standalone application or website, if I happen to find some software developer who is eager to co-operate with me in this project).

For those who can't wait so long, I offer to give you any support you need to experiment with the novel approach with your own data. And in the attachment, I can already give you an idea of how simple this approach may work for putting together the responses or judgments of any number of subjects on any number of questions or items. It may even be used to quickly convert between n-point and m-point scales, for arbitrary m and n, although I don't yet know if people would be convinced of the (mathematically correct) result of the conversion.

Here is a quick explanation of the template. Instructors and peers of students in a workgroup give their judgment of the products and processes of the workgroup. They will do that by means of any sort of well-defined quality criteria (the items). A judgement will be in the form of an integer number (e.g., 1 or 3, up to .... ) placed into one or more of the ordered categories (here: 7). That's all they have to do. The code will calculate the judgment counts per category, so we will get a 7-point response profile. This profile has a rank, depending upon the number of categories (here: 7) and the count of responses given (here: 38). Finally, a percentage score will be computed, which is simply the rank divided by the maximal attainable rank. Because the ranking is based on the co-lexicographical ordering principle, each ordered category starts a series of ranks that is definitely larger than any preceding rank of the same profile. This has a high plausibility and guarantees a high spread of the students' scores.

... and here is an example of the ranking table in case of 5 ordered categories x1 ... x5 (what are called "points" in the misnomer "n-point Likert scale") and just 3 observations, which can be judgements, opinions etc.

For more technical details, including the ranking and unranking formulae and code, please stay tuned on my current project here on RG:

COLEX ranking, unranking and scoring ~ simplified diagram

Many thanks Paul Hubert Vossen . A method like this seems promising and could help us quantify a response variable. Further, it is likely to be relevant in other similar studies. We will start working on implementing it using our data. I wonder if we can email you personally to consult you on the matter in the future. We will appreciate it a lot.

Lior Blank Of course, see my notification to you here on RG. Thanks for your interest.