I am wanting to run correlations on a number of measurements where Likert scales were used. Looking at the scatterplots it appears the assumptions of linearity and homoscedasticity may have been violated.
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From my experience, there is a difference between running analyses on a likert item as opposed to a likert scale. A likert scale is the sum of multiple items. After summing multiple items, likert scales obtain more possible values, the resulting scale is less lumpy. Such scales often have a sufficient number of points that many researchers are prepared to treat them as continuous. Of course, some would argue that this is a bit cavalier, and much has been written in psychometrics about how best to measure psychological and related constructs.
From my casual observations from reading journal articles in psychology, the majority of bivariate relationships between multiple-item likert scales are analysed using Pearson's correlation coefficient. Here, I'm thinking about scales like personality, intelligence, attitudes, well-being, and so forth. If you have scales like this, it is worth considering that your results will be compared to previous results where Pearson may have been the dominant choice.
It is an interesting exercise to compare Pearson's with Spearman's (and perhaps even Kendall's tau). However, you are still left with the decision of which statistic to use, and this ultimately depends on what definition you have of bivariate association.
A correlation coefficient is an accurate summary of the linear relationship between two variables even in the absence of Homoscedasticity (or perhaps we should say bivariate normality given that neither variable is a dependent variable).
If there is a non-linear relationship between your two variables, this is interesting. However, both variables could still be treated as continuous variables, and thus, you could still use Pearson's. For example, age often has an inverted-U relationship with other variables such as income, yet age is still a continuous variable.
I suggest that you produce a scatter plot and fit some smoothed fits (such as a spline or LOESS) to explore any non-linear relationships. If the relationship is truly non-linear then linear correlation is not the best choice for describing such a relationship. You might then want to explore polynomial or nonlinear regression.
You should almost certainly go for Spearman's rho or Kendall's tau. Often, if the data is non-normal but variances are equal, you can go for Pearson's r as it doesnt make a huge amount of difference. If the variances are significantly different, then you need a non parametric method.
You could probably cite almost any introductory statistics textbook to support your use of Spearman's Rho.
Update: if the assumption of linearity is violated, then you should not be using the Pearson correlation coefficient on your data, as it assumes a linear relationship. Spearman's Rho is acceptable without linearity and is meant for more general monotonic relationships between the variables. If you want to use Pearson's correlation coefficient, you could look at log transforming your data as this might deal with the non-linearity.
one thing is pretty sure that correlation requires linearity in relationship in general. now you say your data somewhat curve shaped,so nonlinear regression seems to be the left choice
