How to interpret and test for a significant difference between two correlations drawn from independent groups (e.g., males and females)?

Question

I read a paper recently where the author compared the correlation coefficients (Pearson) between household budget and the type of meat purchased by male and female shoppers.

So the correlation is between household budget and meat type purchase for males versus the same for females (i.e. comparison by gender).

The author concluded that the "two correlation coefficients differ significantly".

I have difficulty understanding what this mean. Can someone explain this in simple English?

(Sometimes in the complexity of statistical analysis, we tend to get confused on the basics!)

Answer 1

The relation between meat type and household budget is of significantly different size.

It could interpreted such that males and females significantly differ in their behaviour considering what kind of meat they purchase depending on their available household budgets.

It is a bit difficult to construct an example, as I do not know how meat type is measured. But, one example supposing that meat type is an ordinal measure of meat quality: A significantly higher correlation coefficient for females could mean that when deciding what quality of meat to purchase, females are more likely to take the houshold budget into consideration, i.e. they buy cheaper meat if the budget is low, whereas for males, if the correlation coefficient is significantly smaller, they are less likely to choose poor meat if their budget is low, or they switch to not that much worse meat if their budget is low.

Answer 2

Data tends to have a certain amount of random variability. For example, one could observe that a particular woman purchased primarily beef while a particular man purchased mostly chicken. However, that might just be random chance so we'd sample several men and women and note what the buying preference was on average. A larger sample (more men and women) would give us a more accurate measure of this phenomenon, just as one sample is easily very inaccurate. If we sampled all of them then we'd just know the answer. However, it's not easy to sample everyone so we randomly sample a subset, a group of sufficient size that we can infer what the true values for the population are likely to be with a degree of accuracy that we're satisfied with.

The significantly different part of the statement refers to a belief that whatever differences you've observed were unlikely to have occurred by random chance so we believe that the differences between the two groups are real and representative of what men and women purchase in general.

More specifically, if the typical 0.05 cutoff were used then it is tantamount to saying that, "if there really was no difference between the groups then the probability of this data occurring is quite small, probability <0.05, and that is so small that we don't believe that there really is no difference." That's typically what saying "significantly different" really means... even though it's not what's usually intended.

PS: which is rather convoluted method at determining that the groups are different since you need to hypothesize there really is no difference in the first place... but that's the way it's usually done