Dear Sirous Yassery.

I agree with you on your wise considerations on Likert-type scales. Suffice it to say that in Likert-type scales a given score (50, for example) can be obtained through different choices and hence have different psychological meanings. So, to apply parametric statistics to these scores do not go without problems. As you certainly know, the neo-Popperian Paul Meelh (1978), in his seminal article, Theoretical risks and tabular asterisks: Sir Karl, Sir Ronald and the slow progress of soft psychology, Journal of Consulting and Clinical Psuchology, 46, 806-834, cogently argued that social sciences, namely psychology, uses and overuses of statistical analyses (i.e., presents tabular asterisks) at the cost of conceptual clarification or grammatical investigations a la Wittgenstein (i.e., theoretical risks). This procedure may give us an impression or illusion of rigor, when this is not the case. I think that results coming from Likert-type scales cannot be treated, say, as interval results.

Of course, we can perform non-parametric, but not parametric, operations or analyses on ordinal and nominal data. To be frank, I must confess that, to my undertanding as a Ph D developmental psychologist, it is often the case that statistical analyses in psychological papers are there more to hide (e.g., lack of conceptual clarity) than show significant psychological findings. It is not a mere coincidence that famous psychologists (e.g., Freud, Piaget, Skinner) never appealed to statistical analyses, and yet they offered us worthwhile findings and results.

I hope that I has got your challenging post.

Best wishes,

Orlando

Measurements in nominal and ordinal scales are best analyzed using nonparametric methods. These do not call for assumptions of normality and independence of data because the measurements do not have a true zero value. These nonparametric techniques have been observed to be reliable. They are comparable to their parametric counterparts. They should be used.

Dear Professor Etuk,

Thank you for your contribution. A concrete example would help me a lot. I took a big bottle of cola to the market place and offered a small sample to passerby's and asked them to mark their liking using the Linkert scale of 1 to 5. Ones mean the least liking and 5 the most liking. Now I have 100 numbers. What conclusion can I reach using this data according to you.

Thanks

Dear Professor Lourenco

Thank you for your contribution.

You are welcome, Professor Yassery,

I really like your post and it was a pleasure to deliver my contribution to it.

Best regards,

Orlando

Sirous Yasseri,

There are statistical analyses appropriate for ordinal data.

One is ordinal regression, which is very flexible. In R, the most common package for ordinal regression uses cumulative link models. In the ordinal package, the spacing between ordinal categories can assumed to be unknown, or symmetric, or equidistant.

Another approach are tests for contingency tables of ordinal data. This includes the Cochran-Armitage test and the linear-by-linear test. It's not clear to me if these tests make assumptions about the spacing between the ordinal categories.

I assume Ette Etuk was referring to traditional nonparametric tests like Wilcoxon-Mann-Whitney and Kruskal-Wallis. There is some controversy with using these tests with purely ordinal data, but their use is common, and supported by some authors.

As to your example, it is not clear what you would want to know this data. I think with ordinal data, you can clearly calculate the medians and other quantiles. You could calculate the proportions of responses that are say, 3 or higher. Since you have a single sample, you could run a one-sample sign test or one-sample Wilcoxon signed-rank test to determine, for example if the data (or median, depending) is greater than, say, 3.

Hi Salvatore,

My question is it meaningful to do arithmetic on ordinal numbers. The distance between the ordinal number is not constant. That is in a scale of 1 to 5, giving a score of 5 doesn't mean that it is five time better than giving a score 1.

My trouble is everybody is calculating the average of the ordinal number there is little to show that why this is meaningful.

To be clear, I'm not in favor of using means, standard deviation, or traditional parametric statistical tests (e.g. t-test) for Likert item data.

There are sufficient summary statistics, statistical tests, and plots to use for Likert item data without introducing additional assumptions. And practically speaking, Likert item data is often not symmetric, and so the mean may not be the best summary statistic.

But to make the case for treating Likert item data as interval data...

First: to use means or t-test, the data needs to be interval in nature. It doesn't need to be ratio. That is, a score of 4 doesn't need to be twice a score of 2. It only needs to be the case that the space between a 3 and a 4 is equal to the space between 4 and a 5, and so on.

A second assumption is that there is a continuous variable underlying the ordinal variable.

Of course there are ordinal variables that wouldn't meet these assumptions. But what about typical Likert items?

I think the assumption of an underlying continuous variable makes sense. We assume that below a 5-point scale ranging from "Strongly disagree" to "Strongly agree" that there's a continuum.

I also think that assuming that the categories are equally spaced makes sense in a lot of cases. To see what I mean, take a 5-point scale --- "Strongly disagree", "Disagree", "Neutral", "Agree", "Strongly agree" --- and try to give a range for each category, say on a 1 to 100 scale, like percent. When I do this, I seem to naturally equally space the categories. Neutral is 40-60%, Agree is 60-80%, Strongly agree is 80-100%. This probably doesn't work for all item scales, or for all people, but I see how people would be okay making this assumption in some cases.

Dear Salvatore ,

Thank you for your comprehensive answer.

I agree with you if the spacing is equal and 5 means five time of 1, then I have no difficulty to perform arithmetic on them.

I am working on maturation of a technology to be used in a system. To measure the maturation, NASA's Technology Readiness Levels (TRL) are used. TRL has 9 level, and it is obvious Level 6 does not mean twice as mature as level 2.

TLR becomes so popular that most organisation such as the US department of defense and UK ministry of defense use a variation if it. In last 10 years many papers were published in them authors do various calculations on TRL as if they are cardinal numbers.

Sirous,

The one point I'll disagree with you: It is not necessary that 5 means five times 1. Consider temperature measured in Celsius. Obviously 20°C is not twice 10°C, as absolute zero is – 273°C. Yet, it still makes sense to take the mean of temperature in °C because the values are equally spaced (interval data).

As to the proper statistics to use with TRL, I will be happy to instruct the US DOD and UK MoD if they provide sufficient funding. :)

Scales made up of 5-point items can often end up with pretty like a normal distribution. You can realise why: the inequalities of spacing of the five categories vary between items, leading to a more equal average spacing over the sum of the items.

The advantage of giving average scores is that these can be mapped back onto the original scale. So if you have a 1-5 scale ranging from disagree strongly to agree strongly, a mean value of 3.8 is up near 4, which would be 'agree'.

While the mathematical purists may come out in goosebumps, I do find that this is a useful way of communicating results. Of course, it doesn't take into account the variation in answers, but then that's not what a measure of central tendency does.

I think nonparametric statistical tests would be better.