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i would go for a simple "kendall tau" correlation coefficient (aka rank agreement), accounting for ties (they could be many because of the "categorical" variable which is indeed treated as ordinal)

https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient

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Try Kendall's W, also.

https://www.statstodo.com/Concordance_Exp.php

Regards!

@ Fabrice Clerot

For appropriate answer the following circumstances shoul be explained first: Can you measure and/or classify both variables for every observed item, or differently: for a portion of the sample you can observe only the continuous variable, and for the remaining portion - only the second feature.  The adviced Kendall's tests are applicable in the first case, only. The second case is almost hopeless.  Probably some additional information is required. The monotonicity mentioned in the question seemingly is unsufficient for suitable test construction. Small light may be seen if there are known some (may be fuzzy) limits of the continuous variable determining the classification for the second categorial one. E.g.  in case of two categories - it might be helpful to know, that the first class consists of elements for which the continuous variable lies on say  below  3  and that the second class consist of elements with values not less than -3 (minus 3). And if the full range of possible values of the continuous variable contains the (-3,3) as a proper subinterval! The last requirement is necessary, since otherwise ther is no relation between the two classes.

Regards, Joachim

The continuous data I am using is facial electromyography data (mean response amplitude in microVolts). I have made video recordings of the face as well, using a new software package that uses a digital neural network to assess movement of specific facial landmarks (i.e. action units). The software package unfortunately assesses activation of these action units at 5 levels: from trace to maximum activity. I want to see whether the software is as accurate in establishing an emotion expression as EMG activation is.

I am really sorry for directing the last answer to Fabrice, only. Actually, it  directed first to Nina Kupper, and then to ALL Followers of this interesting question.

And now according to the last explanation by Nina:  It seems that, fortunately, the data are good for the Kendall tests, since every measurement returns two data: the voltage  and one of the five levels of the reaction taken from the software.

Another problem is, how to restore from the (probably very expensive device) full info about the movement of the landmarks.  Good luck with respect to this problem!

Regards, Joachim

@Joachim

no worries and, moreover, you are right : i  had silently assumed a paired experiment scenario

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regarding kendall's tau versus kendall's W, see

https://www.researchgate.net/post/When_to_use_Kendalls_Tau_correlation_against_Kendalls_W_concordance_A_bit_confused_on_both_of_their_distinctiveness_on_usage_and_requirement

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Nina,  Nina. th

e appropriate statistical technique for your analysis is Analysis of Variance which usually requires  continuous  dependent variable and categorical independent variables.

@ Nina & All Other Followers,

Let me share a remark, which can help in simplifying the test of agreement between  the continuous variable  (call it  X in  R  )   and the discrete one (call it  J ,  with values in {1,2,3,4,5}). The simplification is possible due to the claimed by Nina  monotone dependence (between the variables).  This mean, that (as it is implied by Harold's answer) we can apply just the  covariance/correlation, in the same way as it is preceeded for regression. Note, that the Kendall tests work without any knowledge about the monotonicity.

The simplest algorithm of best  fitting  by the least square method (LSM) will return the averages  m1, m2,...,m5  in the classes  1,2,...,5, respectively. The expected order  is then  m1< m2 < m3

@Nina, It is interesting how many answers can you get from different point of view ;)

Respect to ALL followers to this sophisticated issue you have. Anyway, it can help all of us to choose a proper statistical approach sometimes. 

@Fabrice, @Joachim, @Harold have their arguments and they are absolutely right.

Initially, my first answer was Kendall's W (before the explanation Nina did). The arguments of @Joachim and @Fabrice are correct. I did some simulations on that (random numbers vs random Likert scale numbers).

Very impressive results... Regards, Anton

Thanks very much everyone for all your suggestions. It will take the whole of Summer to process the electromyography data (needs to be done by hand mostly). I will definitely try your suggestions. I suppose R would be preferred when doing @Joachim 's suggestions?

It would be interesting to see whether the various statistical methods come to the same result as well :)

Dear Nina, your issue is actually not as difficult as it appears.  One way analysis of variance (ANOVA)  will give you all the necessary output for your analysis,   ANOVA  deals with one continuous data dependent variable and categorical or nominal independent variables.