Relationship between Crime and Punishment: Regression Analysis

Analysis of Crime vs ProbPrison

1. This question  considers the relationship between Crime and Punishment.  The data are from 47 different states of the U.S.A.   

The response variable Crime is the   crime rate i.e.  number of offenses per 1,000,000 population in 1960.

And the explanatory variable is   ProbPrison,  the probability of imprisonment, i.e. the  ratio of number of people committed to prison  to the number of offenses.

(a) Fit a simple linear model of Crime (y)  vs ProbPrison (x). Show output.

(i) Is there a statistically significant linear  relationship? Quote evidence.

(ii) Does the output suggest  this a useful model for predicting the crime rate?   Explain why or why not?  

(iii) What is the value of the correlation between Crime and ProbPrison? Is this  statistically significant? Quote evidence.

(b) (i) Show a scatterplot of Crime vs ProbPrison, with individual 95% Confidence Intervals.

(ii) The graph indicates some problems with the linear model. Comment what these problems are.  

(c) Calculate new variables  lnCrime = ln(Crime)   and lnProbPrison = ln(ProbPrison).

(i) Regress  lnCrime on lnProbPrison.   Is this a better model than in part (a)?

(ii) Produce a scatterplot of lnCrime vs lnProbPrison, again with line and prediction interval (like in graph (b))   Have the problems identified in part (b) been fixed?

(iii) Estimate ln(Crime) when ProbPrison = 0.1  (hint convert to ln).   Then back-transform to estimate the Crime rate using this model.

2. (a) Fit a regression of  Crime on Police60pc (the  per capita expenditure on Police in 1960). Save the studentised residuals and unstandardized fitted values.

(i)  Show the regression output.

(ii)   Interpret the regression coefficients.

(iii) Find a 95% confidence interval for the slope.

(iv) Plot the studentised residuals against the fitted values. Does the plot show any problems with the assumptions of linear regression?

(v) Produce a normal probability plot and normality tests for the standardized residuals. (Hint: Analyze> Descriptive Statistics > Explore).  What do you conclude about normality?

(b) Re-fit the regression of Crime on Police60pc without an intercept.  Show output.   Does this model fit better than the one with an intercept?  Quote evidence.

(c) For the rest of this assignment include an intercept.  Fit a regression model for Crime including both Police60pc  and ProbPrison as predictors.   Show the regression output.  Is this a better model than the one with just Police60pc?  

(d) Fit a regression model for Crime including both Police60pc  and Inequality  as predictors.   Include partial residual plots.    Save the Centered Leverages,  Cook’s Distances and standardized DDFITs    

(i) Is this a better-fitting model? Show output as evidence (comment what part of the output you are referring to).  

(ii)  Edit your Partial Residual Plots to include a linear regression line  for each one.

What is the purpose of these graphs? (i.e. What are they supposed to reveal?)  

How do you interpret the slopes of these lines, in terms of the overall model?

How do you interpret the R²s for these lines, in terms of the contribution of each variable to the fit?

(iii)    Plot the Centered Leverages against Police60pc.   What is the criterion for high Centered Leverage?  Include a horizontal line on the Centered Leverages plot at this value. Are there any points with high Centered Leverages?  If so, what is unusual about them?     

(iv)   Plot Cook’s Distance against Police60pc.  How do you interpret what this graph tells you.