# Between two variables with weak correlations and no significant prediction rate from simple regression, what are the next research steps?

I am working to determine the association between crime rates and economic inequality using income brackets. I have found that some crime rates are weakly, or almost moderately, correlated with rates of households in various income brackets. I have also found, using simple linear regression to predict crime rates from rates of households in various income brackets (where the correlation values are largest) do not significantly predict crime rates. For example, the percentage of households in the $0-10k income range have appx. a .32 correlation value with burglaries, but the predictive value of that household percentage on the burglary rate isn't significant. What might be the next statistical research steps? • This is a very general and perhaps vague question, but sharing more details about your initial analysis would be certainly useful. You could explain what is your sample size, share a snippet of what the data look like, what is the output of your model, etc. For instance, I'd be a bit wary of using rates with linear regression (e.g. if you just use rates, you lose information about the number of crimes). You should also examine if there could be some confounding variables at play (but this is a question that can be addressed by economic or sociological theory, not really by statistics alone). Commented Jul 31 at 17:34 • n = 1,441 U.S. counties. I have crime rates for several crime types and percentages of county populations in various income brackets. Data is annual from 2018-2022. I determined correlation values between each income bracket and each crime type, producing correlational values. I then examined which crime type values exhibited the greatest range of correlation values. Burglary had the largest: .32 with respect to the$0-$10k population, and -.15 with respect to the$200 and up population (all years were remarkably similar), suggesting it was most associated with income inequality. Commented Jul 31 at 20:27

You could look into statistical interactions. A community with many low-income households may have higher burglary rates when there is a substantial number of high-income households as well. If you conceptualize and measure wealth and poverty as two variables rather than two sides of the same coin, you find some interesting effects. See this short piece.

You might also investigate spatially. High-income neighborhoods that have a certain proximity to low-income neighborhoods may show particular patterns that differ from what you find for more insulated wealthy enclaves.

• I did perform multiple linear regression attempting to fit both the lowest income bracket and highest income bracket. The results were similarly insignificant. For many reasons, often having to do with the inherent problems with the crime data I'm using, I'm not necessarily surprised. That said, I've thought about your proposed spatial investigation. My data is at the county level, so I can't get to neighborhoods, but exploring any potential spillover effect county-to-county might be interesting. Commented Jul 31 at 20:37

I'm not sure a typical regression works here. Since you are dealing with rates (and to my knowledge crime rates are discrete), this may be better modeled using a discrete response approach, perhaps with something like a negative binomial (NB) regression with week used as an offset. I would also see if you can find out if there is a version of the income variable that isn't an interval. A more continuous version that includes all of the data would give you a lot more information versus binned data. Some of the approaches from Rolando may also work.

Setting aside these concerns, the following statement:

What might be the next statistical research steps?

Is problematic given it follows this statement:

I have also found, using simple linear regression to predict crime rates from rates of households in various income brackets (where the correlation values are largest) do not significantly predict crime rates. For example, the percentage of households in the \$0-10k income range have appx. a .32 correlation value with burglaries, but the predictive value of that household percentage on the burglary rate isn't significant.

The next steps should be, if your model is not misfit in some way, to simply report the results as is. Whether or not a predictor is statistically significant is not as important as determining why the association is weak or why one would want to hold their judgement about a particular null effect.

In any case, I would see if you can describe, model, and plot your data using a an NB regression and see what the results say.

• It may be surprising, but Census data, structured as "__% are in __ bracket, __% are in __ bracket, etc." provide more information than a single, continuous, mean-income-per-community variable. Commented Jul 31 at 17:48
• Interesting. I admit that this field is not my own, so I will defer to your wisdom on that point, but not that in other contexts this is usually not ideal. Commented Jul 31 at 18:02
• @rolando2 I’m most interested in the rationale behind that. Would you be willing to post a question with a self-answer?
– Dave
Commented Jul 31 at 20:11
• @Dave Suppose we discuss it out loud instead? You can reach me at [email protected] Commented Jul 31 at 20:21
• @Dave My own thoughts: Mean income often says little, since incomes in the U.S. are highly right-skewed. A median is better at saying something about incomes in general. Income brackets break geographic areas down into subgroups, and so can say something about economic inequality. Commented Jul 31 at 20:32

You could consider using an income index like the Gini coefficient, instead of individual income brackets. I haven't tried it, but the R package binequality seems to be able to help you with this.

It is well known from other research that income inequality and crime rates can be positively correlated. See e.g. US homicide rates increase when resources are scarce and unequally distributed, Dynamic linkages between poverty, inequality, crime, and social expenditures in a panel of 16 countries: two-step GMM estimates and Revisiting the Income Inequality-Crime Puzzle to pick three freely available examples, the last one being a meta-study. You can also take a leaf out of their books and consider covariates/control variables like demographics and economic growth.

If you cannot dismiss the null hypothesis, and if the null hypothesis represents a surprising or controversial result, then the next step is to support that result with a power analysis.