Dr. Tom O’Grady - Sex, Education and Age - Criminal Justice - Recognition of LGBT Relationships - Statistics Assignment Help

Assignment Task:

QUESTION A. Many organizations working on criminal justice argue that longer prison sentences do not cut crime. However, many voters disagree and express strong support for increasing the length of prison sentences. For this question, suppose that a campaign group advocating shorter prison sentences asks for your help. They plan to run a campaign targeted at groups that most strongly support longer prison sentences for convicted criminals, in the hope that they can change their minds. Your job is to tell them which types of people are most supportive of longer prison sentences. To help measure the likely effectiveness of their campaign, they also want to know how much each characteristic matters in explaining support. 

QUESTION B. This question uses data from the USA in the mid-2000s, a time when the gay marriage had not been legalized nationally and individual states differed a lot in their recognition of LGBT relationships. Some, like Massachusetts, were in the process of fully legalizing gay marriage or introducing civil unions. Others were in the process of passing constitutional amendments defining marriage as only between a man and a woman. In this question, we want to find out how much public opinion toward marriage equality also differed across the states. 
One previous, authoritative, estimate of these figures has been produced in the past. Your tasks in this question are (i) to produce estimates of the percentage of people that supported marriage equality in every US state using multilevel modeling and post-stratification that are as close as possible to this existing set of estimates, as measured by the Mean Absolute Error (MAE), and (ii) to use your results to explain why people supported or opposed marriage equality. You need to: 
i) Estimate an appropriate logistic multilevel model explaining support for marriage equality, using the predictors in the dataset.1 
ii) Present the multilevel model results and interpret how the variables affect support for marriage equality (Note: you do not need to discuss statistical significance). 
iii) Produce post-stratified estimates of the percentage of people who supported marriage equality in all 51 US states2 
iv) Compare your results to the existing estimates using the Mean Absolute Error 
You should present and explain your approach and results in a brief report, explaining why your estimates do or not perform well compared to the existing estimates. Note: if you cannot get very close to the existing results, do not worry. Your grade depends on the quality of your analysis, presentation, and interpretation, not how close your results are to the existing estimates. 

Question 1: Sex, Education, and Earnings 
In this question, we will investigate whether women are discriminated against in pay, and whether the marginal effect of education differs between men and women (i.e. whether the monetary returns to education differ by sex). If the marginal effect differs, there is an interaction between sex and educ; we may also say that the effect of educ is “moderated” by sex. 
a) Fit a regression model (without interaction terms) to estimate the effect of education and sex on hourly income 

b) Carefully interpret the meaning of the coefficients on education and sex from your previous answer. Can we reject the null hypotheses that each coefficient is equal to zero at the 5% significance level? 

c) According to the estimated model from (a), how many more years of education would women need on average in order to earn the same as men? Code Hint: To calculate this directly in R, you can extract the coefficients from a model called mod1 using mod1$coef[]. You need to fill in the square brackets with the correct number. 
mod1$coef[2]/mod1$coef[3] 

d) Do these results suggest that there is pay discrimination against the women in this sample? 
Explain your answer 

e) The model from (a) assumes that women at all levels of education have the same wage disadvantage. Stated differently, the model assumes that the returns on education do not differ between the sexes. To relax this assumption, we’ll now study a regression of the form: 
hrinc = β0 + β1educ + β2sex + β3sex : educ 
In the above equation, what are the estimated marginal effects of education for men and for women (stated in terms of the βs only)? 
Hint: Use differentiation! 

f) In the equation in (e), explain what it would mean if: 
i) β3 < 0 
ii) β3 = 0 
iii) β3 > 0 

g) Now estimate the equation from (e). What do you conclude about the relationship between sex and the return to education? Code Hint: See what happens when you include only the code sex*educ as an explanatory variable in the lm function 

h) From your estimated equation in (g), what is: 
i) The predicted hourly wage for a man with 12 years of education? 
ii) The predicted hourly wage for a woman with 15 years of education? 

i) Now, we’ll create a 95% confidence interval for our prediction from (h) (ii) using simulation, with the following steps: 
1. Install the arm package, which contains the sim() function 
2. Take the regression from (g) and create 1000 simulations of it, using the sim() function. 
Store this as an object called sims 
3. Create a vector named values for the prediction from (h) (ii), containing four numbers 
in total 
4. Create a matrix of regression coefficients from your simulations using the code: 
coef <- coef(sims) 
5. Create the sampling distribution of predicted outcomes using matrix multiplication: 
values %*% t(coef) 
6. Use the quantile() function to find the 95% confidence interval for your prediction 

Question 2: Sex, Education and Age 
We will now test whether the effect of education on earnings changes as a function of age. One argument for this could be that the skills acquired during formal schooling are gradually forgotten or become irrelevant due to technological innovations. 
a) Estimate a model for the effects of sex, age and education on earnings, allowing for the effect of education to differ as a function of age 

b) In your model from (a), what is the estimated marginal effect of education on earnings for: 
i) a 50-year-old male? 
ii) a 30-year-old female? 
What do your answers suggest about the hypothesis posed at the start of this question? Hint: Use differentiation! 

c) In your model from (a), what are the predicted hourly earnings for: 
i) a 50-year-old male with 12 years of education? 
ii) a 30-year-old female with 15 years of education? 
iii) a female of average age and education? 
d) Create a 95% confidence interval for the prediction from (iii), using the same steps as Q1 (i) 

POLS0010 Data Analysis Term 2: Practical Session 2 
Dr. Tom O’Grady 
This set of exercises will introduce linear probability and logit models using two different datasets. You should complete the exercises in your own time. 
Load the data for this tutorial using the command: 
setwd(...) load("practical2.Rda") 
where you will need to adjust the working directory to your own directory (use the command getwd() if you don’t know the path). 
The dataset is called u. It consists of data on men in their forties and fifties who were unemployed at the start of the study period. We can use the data to predict the probability that an individual finds a new job. It contains the following variables: 
• studytim: Time it took to become employed again, weeks 
• age: age at beginning of study, years 
• health: index of health at beginning of study (1=high, 4=low) 
• edu: education level at beginning of study (1=high, 5=low) 
• exp: Number of previous spells of unemployment (1=min, 4=max) 
• soccap: index of expert assessment of individual’s social capital at beginning of study (1=low, 4=high) 
• msrch: index of expert assessment of individual’s effort at job-seeking (0=low, 6=high) 

Question 1 
a) Using the studytim variable, add a new binary outcome variable called work to your dataset 
equalling 1 if the respondent found a job within one year and 0 otherwise. Code Hint: Use the ifelse() command: see the help file. The syntax is as follows: var <- ifelse(condition, value if condition is met, value if condition not met) 

b) Estimate a linear probability model relating work to all of the other variables except studytim. 
Give a precise interpretation of the coefficients on age and exp 

c) Use the model to predict the probability of finding a job for a relatively “high-risk” person: a man aged 55 in the worst possible health with the lowest possible education, one previous spell of unemployment, and the lowest social capital and effort at job-seeking Code Hint: You can extract the coefficients using model name$coef[] 

d) Produce a plot of the predicted probabilities (on the y-axis) against age (on the x-axis). Using your plot and your answer from (c), how successfully do you think this model predicts the probability of finding a job? Code Hints: Extract the fitted values from a linear regression using fitted.values(model name). Set the limits of the y-axis to above 1 and below 0 with the ylim() argument. Include horizontal lines at 0 and 1 using abline(h=...) 

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