Relevant reading for this problem set: ModernDive Chapter 5: Basic Regression.
Please indicate who, if anyone, you collaborated with on this problem set:
In this problem set we will work with a data set concerning hate crimes occurring in the US. For more context about these data, you can read a FiveThirtyEight article about these data that appeared in January of 2017: “Higher Rates Of Hate Crimes Are Tied To Income Inequality”
In this problem set, we will model these data using regression models with a single categorical predictor variable (i.e., explanatory variable).
First load the necessary packages
library(ggplot2)
library(dplyr)
library(moderndive)
library(readr)Next, load the data set from where it is stored on the web:
hate_crimes <- read_csv("http://bit.ly/2ItxYg3")You can take a glimpse at the data like so:
glimpse(hate_crimes)Be sure to also examine the data in the RStudio data viewer.
Each case/row in these data represents a state in the US. The response variable we will consider is hate_crimes, which is the number of hate crimes per 100k individuals in the 10 days after the 2016 US election as measured by the Southern Poverty Law Center (SPLC).
This week we will examine the explanatory strength of three categorical variables in the data set:
trump_support: level of Trump support in 2016 election (low, medium or high; roughly equal number of observation at each category level)unemployment: level of unemployment in a state (low or high; split below or above mean)median_house_inc: median household income in the state (low or high; split below or above mean)Let’s start by modeling the relationship between:
hate_crimes per 100K individualstrump_support in the state: low, medium, or highCreate a visualization that will allow you to conduct an “eyeball test” of the relationship between hate crimes per 100K and level of Trump support. Include appropriate axes labels and a title. Please note that because of alphanumeric ordering, the levels of trump_support are alphabetically ordered high, low, medium, and hence the baseline group is high. Although this default ordering is counter-intuitive, we will use the ordering as-is for this problem set.
Comment on the relationship between these two variables. Is this what you would’ve expected?
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Now run a model that examines the relationship between hate crime rates and the level of Trump support. Generate a regression table.
What does the intercept mean in this regression table?
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What value does the model estimate for the number of hate crimes per 100,000 people in states with “low” Trump support?
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Does the model estimate that hate crimes are more frequent in states that show “low” or “high” support for Trump?
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How much greater were hate crimes in “medium” trump-support states compared to “high” trump-support states?
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What are the three possible fitted values \(\widehat{y}\) for this model? Hint: use the get_regression_points() function to explore this if you are not sure!
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Below we calculate the group means of hate crimes for the high, medium and low levels of Trump support. How do these numbers compare to the three possible fitted values \(\widehat{y}\) for this model?
hate_crimes %>%
group_by(trump_support) %>%
summarize(mean_hate_crimes = mean(hate_crimes, na.rm = T))| trump_support | mean_hate_crimes |
|---|---|
| high | 0.1910000 |
| low | 0.4601111 |
| medium | 0.2224000 |
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The regression equation for this model is the following:
\[ \widehat{y} = 0.191 + 0.269 \times 1_{\mbox{low support}}(x) + 0.031 \times 1_{\mbox{med support}}(x) \]
So for instance, in a state in which trump_support is “low” you would plug in 1 for \(1_{\mbox{low support}}(x)\), and 0 in for \(1_{\mbox{med support}}(x)\) and solve like so:
\[ \begin{aligned} \widehat{y} &= 0.191 + 0.269 \times 1 + 0.031 \times 0 \\ \widehat{y} &= 0.191 + 0.269 + 0\\ \widehat{y} &= 0.460 \end{aligned} \]
Solve the regression equation for a state in which trump_support is “high”
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Which 5 states had the highest rate of hate crimes? What was the level of Trump support in these 5 states? You can solve this any way you choose, using code or not…
Do these results surprise you? (There is no right answer to this question.)
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We will next model the relationship between:
hate_crimes per 100K individuals after the 2016 US electionlow, or high)Create a visualization that will allow you to conduct an “eyeball test” of the relationship between hate crimes per 100K and unemployment level. Include appropriate axes labels and a title.
Now run a model that examines the relationship between hate crime rates and the unemployment level. Generate a regression table.
Write out the regression equation for the above model. Try writing out the equation using the same LaTeX formatting used to write the equation for the first hate crime ~ Trump support model. You can copy the example LaTeX code that appears just after “The regression equation for this model is the following:” in this document, and edit it to describe the hate crime ~ unemployment model you just fit. You don’t have to write the equation using LaTeX, but this is a great statistical communication skill to practice!
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What does the intercept mean in this regression table?
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What does the model estimate as the number of hate crimes per 100,000 people in states with “low” unemployment?
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What are the two possible fitted values \(\widehat{y}\) for this model? Why are there only two this time, instead of the three like the previous model?
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Use the get_regression_points() function to generate a table showing the fitted values and the residuals for the model relating hate_crimes to unemployment. Examine the first row: How are the residuals calculated here?
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Now run a model that examines the relationship between hate_crimes and median household income in the state median_house_inc.
Were there more hate crimes in areas with high or low median household incomes? How large was the difference between states with “low” and “high” levels of household income?
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Run the get_regression_points() function for the hate_crimes and median_house_inc model, and examine the data for Maine (row 2). Did the model overpredict or underpredict the hate_crimes level, compared to what was observed in the data?
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