# Can a set of means be used as the dependent variable in a correlation?

The following is a purely hypothetical scenario. It is not based on any study I am conducting or plan to conduct, it is only me trying to understand what kind of statistical test I would need to run on a hypothetical dataset. Let's say I collected a dataset which uses weight as an independent variable. I have 5 weight categories: 0–50 kg, 50–70 kg, 70–90 kg, 90–110 kg, and 110+ kg.

For each of the individuals who fit into each weight category, they are asked to state their agreement on a scale of 1 to 5 with a question. Counts are taken for each response. So hypothetically, the 0-50 kg weight category has 5 people respond with a level of agreement of 1 to the question, 4 respond say 2, 4 people respond 3, 3 people respond 4, and 1 person responds 5. Now let's say I take the mean of these counts: (5+4+4+3+1) / 5 = 3.4.

I now repeat this for the individuals of the other weight categories. I find a mean of 3.6, 3.6, 4.14, and 4.2.

Now let's say I want to run a correlation test, to see if there is a statistically significant positive correlation between the weight category and the mean response to the survey question. Is this possible, and if so, what test would I use? I notice I can't use Pearson's correlation coefficient, as the independent variable is an ordinal category. But am I allowed to use Spearman's correlation with the set of means as the dependent variable? I'm unsure if this is valid, and I feel like it entails a problem: namely, if I just use my 5 means, then my "sample size" is 5 (because I have 5 means) and there's a much stronger possibility of getting a non-significant result.

If this is not a valid way to do a correlation test, is there a better way on this dataset (with or without using the mean of counts) but without dividing it into like 5 different correlation tests for each weight category? Can a correlation coefficient be found, or do I need to use a statistical procedure which does not calculate a correlation coefficient?

# Spearman correlation

I will assume that you want to treat your intervals as ordered categories (think as 0, 1, 2, 3, 4) and check if this order coincides with the order of the average answers. Then, yes, you should use Spearman correlation. However, note that Spearman correlation will only change values if, for instance, the average answer of group 1 surpases that of group 2. If the order does not change, the Spearman correlation won't change either. Thus, it will not be very sensitive. Spearman correlation is just the correlation of the rankings.

# Other ideas

## Forget groups

Why don't you do the scatter plot of answer vs weight and visually check if you have something? A simple measure could be the correlation between weight and answer.

## Make groups scalar

If you only have the groups, you can assume a single weight for each group and then compute the correlation again (mean or max). I would also recommend visualizing this.

## Use unordered groups

If you assume the groups unordered, you could have the data to estimate the probabilities $$P(answer | weight\; group)$$, in particular estimating mean and variance and assuming a Gaussian for each group. Maybe that is useful for further analysis, for instance computing the expected answer of a new patient.