Statistical analysis of crime rate

Summary of Variables

Total violent crime over time

Discussion

dd

Median income

Graduation Rate by age 25

Results

• Total rate per 100,000 inhabitants plotted against a 45 year index (from 1967 to 2012)
• Crime has been dropping since the apex in year 1991.

Possible improvements to model

Unemployment rate

Median age

• The commonly believed factors of crime yielded surprising results.

• Models show that graduation rate and median age is positively correlated with violent crime.

• Income and unemployment rate, were negatively correlated.

ddd

D

• Consider more variables, find data that would improve fit.

• Change the scope by increasing timeframe or analysing other countries.

• Analyse non-violent crime as well.

M2: Transformmed model

Correlation between variables

Limitations

Transform graduation variable using log()

fitTrans1 = lm(total~I(log(graduation))+income+unemployment.rate+median.age,data = crime)

summary(fitTrans1)

• The final best fitting model only had two variables, there are many more factors of crime not considered.

• Data may have been incomplete: Census only offers US population every 10 years.

• Very small sample size, only 40 inputs for each variable

• High corrleation between median age, income, and graduation rate

• Multicollinearity will have to be dealt with.

vif decreases

especially for graduation

Compared to the original model, graduation and unemployment rate are more strongly correlated, while income and median age are weaker.

Problem Description

Exclude unemployment rate

Transform income variable using X^2 based on the last model

fitTrans3 = lm(total~I(log(graduation))+I((income)^2)+median.age,data = crime)

fitTrans2 = lm(total~I(log(graduation))+I((income)^2)+unemployment.rate+median.age,data = crime)

vif decreases a lot

but...

• We can improve accuracy by dropping the unemployment rate as a variable.

p-value for unemplyment rate = 0.93

we will exclude unemployment rate in next model

• However, the p value for the unemployment rate is too high.

Conclusion

M3:

• The media is reporting an increasing amount of crime

• We seek to analyse the correlation from commonly believed factors.

• Establish a regression model to explain the changes in crime rates, identify important causes

• Discover if crime is really increasing like the media portrays

Fitted Original Model (M1)

Crime rates has been steadily decreasing since 1991. As of 2007, we are safer now than 20 years ago.

Final Model

Data collection

Because there are still strong correlation bewteen our three predictor varables, one of the variable is considered to be dropped.

All the p-values are approximately equal to zero.

All predictor variables are significant.

Drop median.age

fit = lm(total~graduation+income+unemployment.rate+median.age,data = crime)

Our model becomes:

fitR = lm(total~I(log(graduation))+I((income)^2),data = crime)

Gathered from online datases and the United States census From 1967 to 2012

Figure 2: qqnorm and qqline

Figure 1: fitted value vs. residual

Function for crime prediction is 90.5 (B1) - 14.95 (B2) - 2.461 (B3) + 235.8 (B4)

Our model shows that graduation rate has the highest positive impact on violent crime, while median income had negative.

Response variable:

Model selection :

vif drops to 4.87

However.....

• crime rate

stepwise method

Questions?

vif is pretty high, which is greater than 100 , serious multicollinearity exists

AIC=398.66 (ALL variables included)

Predictor variable:

The new predictor function is -1417 + 730.1 (B1) - 8.540E-7 (B2). Graduation rate is positively correlated with crime rates, while income is negative.

• income

• unemployment rate

• median age

• graduation rate

Statistical analysis of crime rates

Group member:

Yehao Zhang

Weiwei He

Xinze Li