# Correlation with ordinal outcome variable and four ordinal (?) independent variables

I have scored student's solutions according to a rubric which is scaled from 1-5. This accounts for the dependent variable. After videoing student's interactions in groups, I have coded the number of times they said four specific words I was interested in (it was a little more complicated than this, but for the sake of understanding). These four factors account for the independent variables. There are 27 samples in the data which include the student's scored argument and the number of times they said each of the four specific words. I am hoping to discover if there is a significant positive or negative correlation between their scored argument and the number of times they said each of the four words. Is there a statistical test for this? I have tried Spearman's rank correlation, but by my understanding I can only use one independent variable for this test. I have included my data below.

• If this is the complete data, it is maybe few cases to investigate four independent variables. But I would see it as more of a regression problem, but start out with some tables and visualizations. Look into ordinal regression, one post is: stats.stackexchange.com/questions/302254/… – kjetil b halvorsen Jun 15 at 17:49
• You are correct that Spearman correlation looks at the relationship between exactly two variables, but you can just run a Spearman test 4 times, once for each independent variable comparing against the same dependent variable. To be complete, you'll then have to adjust the significance of your results due to multiple hypothesis testing. – Nuclear Wang Jun 17 at 17:43

This appears to be too small a sample to obtain anything strongly significant.

Performing simple linear regression gives the following p-values for the slopes on the explanatory variables:

     0.298377646979805
0.242069284775947
0.542518498332371
0.0765452504092274


and the following coefficients:

     0.150821087611921
-0.205764501620133
-0.22273623562583
-0.449611423943887


Of if you run the regression with the logarithm of the "scored argument" then you get the following p-values:

     0.293969089451282
0.290364255905711
0.499341526214123
0.0555191236104219


and the following coefficients:

    0.0454807348691385
-0.0553966456864823
-0.0738823488768918
-0.146133621656525


(Reasonable as taking logarithms of variables which are always positive often improves inference.)

However, as a commentator pointed out, it's not really appropriate to do simple linear regression when the dependent variable is ordinal / categorical. In this case, an ordinal multinomial model should be used. See here for further details (Matlab link since I was working in Matlab).

With an ordinal multinomial model we end up with the following p-values on the explanatory variables:

     0.316637233111995
0.312612749358216
0.436799749905584
0.082584197771639


and the following coefficients:

    -0.238697920928262
0.293986734960427
0.0516709598386895
2.39433385577948


(Signs have the opposite interpretation to above.)

• I disagree with your point about statistical significance. At a traditional 0.05 significance threshold and 80% power, 27 samples is sufficient to identify a correlation coefficient as low as 0.23, which isn't a particularly strong effect. – Nuclear Wang Jun 17 at 17:48
• @NuclearWang How did you get 0.23? Using pwr in R I fond 0.51. – Ous Jun 17 at 23:19
• @NuclearWang Don't forget that when I estimate the multinomial model, in total I'm estimating 8 parameters simultaneously (4 offsets for all but one of the categories, plus the 4 slope coefficients). – cfp Jun 18 at 11:36