Is Pearson's correlation the best method to compare strength of relationship between two Likert items across groups? My dataset comprises of 400 respondents. They are shoppers from different sociodemographic backgrounds. 
I asked each of them (among other things) how likely or unlikely they are to purchased ice-cream. Similarly, I asked them how likely or unlikely they are to purchase yogurt. They responded on a five point Likert scale.
I am doing a correlation between ice cream purchase and yogurt purchase by gender. So I have r=.8296 for females and r=.7784 for males. Both are significant at p<0.05.
I have then compared the two corrleation coefficients to determine if there is a difference between purchase by male vs. female. I have used one of the online calculators that can compare the coefficients (The calculator is from a reliable source.). The result is not significant.
Is this the right way to test the three relationships? (i.e. male, female, and male-female)
I have asked this question because someone pointed out that using correlation coefficients to assess relationships in this way is at best a weak approach and at worst it is wrong and misleading. I was told to consider more conventional analyses.
I am not statistically trained (I know that is no excuse) and I have to present my findings to an audience who is not very statistically orientated, so I used Pearsons corrleation (which appears quite straight forward to me).
 A: At the very least, you would have wanted to consider polychoric correlations, which are specifically formulated to provide estimates between correlations of the underlying traits rather than their crude categorizations (as is done by Pearson correlation). Depending on the purposes of your analysis, you may or may not be OK with the simple correlation. If you are thinking whether to place yogurt and ice cream on the same shelf in the supermarket, without knowing who will be buying it, then this information may suffice. If you are doing a more in-depth analysis of who are the buyers of the two products, you would have to consider other demographic characteristics, like age, marital status, number of children, whether the person is the one who does the regular grocery shopping, and probably a ton of other things. Then you could put an ordinal logistic regression for each Likert scale to explain the purchasing behavior. It may or may not make sense to try to combine them, as the products are rather different.
A: I am still not sure what exactly you are trying to find out.
People who buy frozen yogurt are likely to buy ice cream, and vice versa. This is true for both men and women, and the correlations are about the same.
Fine.  If that is what you wanted to know, you know it. 
You could look at a scatterplot of the two variables, with dots coded for sex (red for women, blue for men, or whatever).  You would probably want to jitter the points a bit to avoid overlap.
If you are trying to predict the purchase of one of the products, then you want ordinal logistic regression, not correlation.  If you want to know how much yogurt and ice cream people buy, you want univariate statistics.  If you want to know if men buy a different amount of these products than women do, you want a t-test or a nonparmetric version.
