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I want to understand the relationship between a region's health and wellness demographics (% obese, % smokers etc.) and their injury rates, for ex. how does change in a metric say '% obese people' affect my injury rate (# of injuries for 100 people)

I don't want to do this for the purpose of prediction, I just want to test my hypothesis which in this case says that regions with high percentage of obese people have significantly higher injury rates than regions with low obesity rates.

I started with checking the correlation of each of my demographic metrics with the injury rates and the correlation values were extremely small (less than 0.2) for all my independent variables, denoting a week linear relationship.

I then proceeded with checking the scatterplots to identify if there is any curve but I am failing to identify the underlying relationship through the plots. Here are the scatterplots:

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Is there a relationship evident from the scatterplots? How should I proceed knowing the correlations are bad?

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There's no clear bivariate relationship that's obvious in your plots. A next step you can consider is constructing a multiple regression model that attempts to predict injury_rate using ALL the predictors you are hypothesizing contribute to it, simultaneously. You can then investigate this more complex model to see if the variables jointly explain/predict injury_rate well, using whatever metrics you are happy with (p-values are common, though they have their drawbacks)

In other words, just because there is no clear relationship in a set of 2-dimensional plots does not mean that there is no relationship in $n$ dimensions.

But it's also worth preparing yourself for the possibility that your hypothesis is unsupported - there may just be no relationship, or too weak of a relationship to identify.

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