# Testing associations between nominal and ordinal variables

I know there have been many questions on this topic. Sorry for another one.

I have done a survey in which both questions using Likert scale (strongly disagree-strongly agree, so 5 point) have been asked and yes/no (binary) questions. I want to explore whether one of the yes/no questions (if a certain quality team exist) is associated with one of the likert scale questions (an ordinal question: if a leader helps to facilitate quality). What would be the best test to explore this? Is a Pearson chi-squared test sufficient for this?

Yes and No on each of the dichotic questions divide the dataset into two groups. In each of these groups we find independent ordinal values to compare. That is a classical setting for Wilcoxon rank sum test (aka Mann-Whitney U). If you plan to calculate a lot of them, think about how you want to deal with alpha error inflation.

One single answer on a scale of 5 possible answers is hardly a Likert scale. A Likert scale is the sum of a number of such answers and is then often treated as quasi-metric. With only one answer on a 5-step scale, your answers are not metric, which is why a t-test is not recommended.

I suppose, yuo want to know, whether Yes or No leads to higher or lower scores on the 5-point scale. A $\chi^2$ test is not the optimal solution. Consider yes-Answers to be all "1" and "5" on the 5-point-scale and no-answers to be all "3". The $\chi^2$ will tell you, they are different, even though none of them has generally "higher" or "lower" answers.

• Thank you! I do indeed want to see whether having a quality team makes people give higher scores (more towards the agree answers) on the likert question whether a leader helps to facilitate quality (because that is what we assume). I doubted between Mann Whitney U and chi squared so thank you very much for you reply. You can also use Mann Whitney to compare for example gender and one of the likert questions. Would you be able to explain maybe when I would use a chi squared test for a likert question? Thank you. – Leanne Nov 15 '16 at 13:25
• See my edit. I was wondering when I would use a chi squared test for a likert question? As for example gender is also dichotomous. – Leanne Nov 15 '16 at 13:29
• $\chi^2$ testing drops the ordinal information in your 5-point answer. It treads your data like nominal data. Whenever you think, that you are interested in the answer like you are interested in nominal data, use a $\chi^2$ test. If the ordinal nature of the data matters, use a test that acknowledges ordinal information, like the Mann-Whitney test or rank sum test or signed rank test do. – Bernhard Nov 15 '16 at 13:33
• Thanks. So if I understand it correctly, if I would want to analyse the possible differences between gender/educational level/occupation with a likert question I use Mann Whintey U? If I want to compare gender/educational level/occupation with one of the binary questions I use chi squared? – Leanne Nov 15 '16 at 13:41
• I cannot answer that as you have not stated, which of the variables are of which scale niveau. To compare a dichotic with an ordinal, use Mann-Whitney, to compare to dichotic variables use the chi-square independence test. – Bernhard Nov 15 '16 at 14:21

As far as I know, a chi-squared test is best for comparing two categorical variables because it only tells you if the distribution of your likert variable is different for the different groups. It doesn't tell you which group has a higher score than the other. If the scale is (sort of) normally distributed, you can do a t-test to test for differences between groups. if it's highly skewed, I would suggest an ordinal regression. You can check the distribution with a histogram.

• +1 for consideration of scale niveau and explanation, that $\chi^2$ does not tell if one group has higher scores than the other. – Bernhard Nov 15 '16 at 13:37

You need to conduct some univariate analyses to investigate if your data qualifies the assumptions for parametric testing. If it does, I would suggest to compare the means among/between your groups using T-test or ANOVA. If the assumptions are violated, you need to look at non-parametric tests.

• I think you need to expand on this more if it is to be helpful. Which preliminary tests are appropriate for the OPs scientific hypothesis? – mdewey Nov 15 '16 at 9:40
• No preliminary tests on normality should be done, as we know perfectly well, these data are not normally distributed. If any test is not significant, then n is to small. In case of a small n, there is a risk of parametric tests being all wrong. In case of large n, many parametric tests become quite robust, but it is then, that the normality tests excludes them. Don't do a normality test in this situation, as it will lead you in the wrong direction, no matter how it turns out. – Bernhard Nov 15 '16 at 13:30
• Thanks for the feedback, mdewey. I will expand and structure my answers in a more helpful manner in future. – jeyss Nov 21 '16 at 6:26

If you believe that what is being measured by the Likert scale and the binary questions are a continuous latent variable (e.g., the amount of agreement with the statement, or the extent to which the quality exists) then you might want to consider the use of polychoric correlation. This would not only tell you if there was an association, but you would get a correlation coefficient which provides information on its strength and direction.

You can use polychoric correlation with any combination discrete variables (i.e., with 2 or more levels), as long as you assume that each variable is representing a continuous latent variable.