I am not sure what you exactly mean by "scale variable"; I assume that you are talking about an at least interval scaled (i,.e. "quantitative") variable.

In contrast to Emilio I think that any normality test is not appropriate. Two reasons: 1) there is no normal model applicable for ordinal data. Think of expressing the "values" of the ordinal variable as letters ("a" < "b" < "c"...), what is absolutely admissible (a numbering like 1,2,3... is also just used to indicate the order; these are also just *names*, not *numbers*!). You immadiately see that the calculation of averages, variances and all that is impossible and pointless. 2) Normality tests are of little value generally. A "significant" result (decide to call your data "non-normally distributed") does not help (or is conter-productive) if the deviation (kind and strength) is not relevant. This is especially and likely the case for large data sets. For small sets you will likely have no power to detect relevant deviations from normality. The decision if your data (more precisely: your model residuals!!) are normal distributed should be based on theoretical considerations, not on looking at the data. If you decide that a normal error model is appropriate and you have enough data, a residual analysis may be performed to find out if your model is ok or considerably suboptimal. In how far the model can be improved and how relevant the changes in your conclusions will be has to be decided in the light of all your expert understanding of the topic - and not soley by the fact that some p-value of some normality test is "significant" at some arbitrary level.

Now to my answer: One of your variables is ordinal, so you cannot use more than ordinal information. This means that the information from the "scale variable" has to be reduced to its ordinal information, too. Spearman's rho is the rank-correlation, so this uses the rank information of the variables, and this is the correct choice for your problem.

PS: The linear regression supposed by Emilio is also not proper IMHO, because one of your variables is ordinal. Recall the coding as a,b,c...: how will you do a linear regression with this? But: there are ways to quantify expectations about one variable ("response") given another ("predictor") also when one of these variables if ordinal. If the predictor is ordinal, I don't know how to use the order-information, but is can be used as a nominally scaled predictor (i.e., the model yould essentially be an Anova). An ordinal response can be modelled by ordered logistic regression.

Hello, Cherry.

I am not an statistician, but I´d do as follows:

1. Normality of both distributions with K-S test? r-Pearson.

2. Non-normality? In this case:

- Rho-Spearman or Tau-Kendall.

- A linear regression in which your dependent variable is the quantitative one. Although both variates are not normally distributed, regression model may fulfill the assumption of linearity. Hence, check standardized residuals: histogram with normality curve, in which mean~0 and SD~1, test K-S again... If so, R or Nagelkerke´s-R² may be your measures of correlation.

- Another option are log-transformations, repeating the process again. Verify you are not biasing the data with this procedure.

Regards.

I am not sure what you exactly mean by "scale variable"; I assume that you are talking about an at least interval scaled (i,.e. "quantitative") variable.

In contrast to Emilio I think that any normality test is not appropriate. Two reasons: 1) there is no normal model applicable for ordinal data. Think of expressing the "values" of the ordinal variable as letters ("a" < "b" < "c"...), what is absolutely admissible (a numbering like 1,2,3... is also just used to indicate the order; these are also just *names*, not *numbers*!). You immadiately see that the calculation of averages, variances and all that is impossible and pointless. 2) Normality tests are of little value generally. A "significant" result (decide to call your data "non-normally distributed") does not help (or is conter-productive) if the deviation (kind and strength) is not relevant. This is especially and likely the case for large data sets. For small sets you will likely have no power to detect relevant deviations from normality. The decision if your data (more precisely: your model residuals!!) are normal distributed should be based on theoretical considerations, not on looking at the data. If you decide that a normal error model is appropriate and you have enough data, a residual analysis may be performed to find out if your model is ok or considerably suboptimal. In how far the model can be improved and how relevant the changes in your conclusions will be has to be decided in the light of all your expert understanding of the topic - and not soley by the fact that some p-value of some normality test is "significant" at some arbitrary level.

Now to my answer: One of your variables is ordinal, so you cannot use more than ordinal information. This means that the information from the "scale variable" has to be reduced to its ordinal information, too. Spearman's rho is the rank-correlation, so this uses the rank information of the variables, and this is the correct choice for your problem.

PS: The linear regression supposed by Emilio is also not proper IMHO, because one of your variables is ordinal. Recall the coding as a,b,c...: how will you do a linear regression with this? But: there are ways to quantify expectations about one variable ("response") given another ("predictor") also when one of these variables if ordinal. If the predictor is ordinal, I don't know how to use the order-information, but is can be used as a nominally scaled predictor (i.e., the model yould essentially be an Anova). An ordinal response can be modelled by ordered logistic regression.