Exam projects - Bayesian Statistics

Exam’s rules

• Prepare your final presentation by using RMarkdown. Any template (pdf, html, word, slides, etc.) is admissible.

• Send the professors via email your presentation the day before the exam.

• Your project’s discussion should not last longer than 25 minutes! Respect for time will be evaluated.

• A list of possible datasets for the final exam are provided below. They are ordered in somehow increasing difficulty. You are free to choose any of them: though, one project cannot be selected by more than four students in a year. You can book your project’s choice here. The link is also available in the moodle page.

List of possible projects

This is the list of possible projects for the final exam. Choose one among them. Alternatively, you may propose your own dataset by contacting the professors of the course via email. You have to bring your printed final work when you deal the oral examination.

Problem A: Wages and Education of Young Males

The R package Ecdat contains the Males dataset about wages and several other variables for a sample of 545 individuals observed over a longitudinal study of 8 years (1980-1987), for a total sample size of 4360 observations. The data have been used by several scholars, notably

Vella, F. and M. Verbeek (1998). Whose wages do unions raise? A dynamic model of unionism and wage. Journal of Applied Econometrics, 13, 163-183.

The help file description of the 12 variables in the dataset is as follows:

• nr subject identifier
• year year (1980-1987)
• school years of schooling
• exper year of experience (=age-6-school)
• union factor; whether wage was set by collective bargaining
• ethn factor with 3 levels
• maried factor marital status
• health factor for presence of health problems
• wage log of hourly wage
• industry factor with 12 levels
• occupation factor with 9 levels
• residence factor with 4 levels

After performing some explanatory analyses:

1. Build a model for studying the effects of covariates on wage with rstan package, taking into account the hierarchical structure of the data. Pay some particular attention to the relation between unionism and wage.
2. [optional] Model the effect of time.
3. Comment the final model and check the model fit by using the proper tools of posterior predictive checking.

Problem B: Short-term effect of air pollution on mortality

The R package SemiPar contains the milan.mort dataset on short-term effect of air pollution on mortality. The data comprise 3652 observations on 9 variables, whose description can be found in the help file. The data are also analysed in the book by Ruppert, Wand and Carroll (2003). The original reference is

Vigotti, M.A., Rossi, G., Bisanti, L., Zanobetti, A. and Schwartz, J. (1996). Short term effect of urban air pollution on respiratory health in Milan, Italy, 1980-1989. Journal of Epidemiology and Community Health, 50, 71-75.

After performing some explorative analyses:

1. Taking total.mort (or a suitable transformation of it) as a normally distributed response variable, build a model for the average number of deaths, checking if some of the covariates may have a nonlinear effect (do not consider the resp.mort variable). Follow a Bayesian approach for the task and check the model fit via pp checks.
2. [optional] Model the nonlinear effects of some covariates.
3. Now consider a GLM with a Poisson distributed response for total.mort, comparing the fitted response values with those obtained previously.

Problem C: Scores attained by students in Scotland

The R package mlmRev contains the ScotsSec dataset on scores attained by Scottish secondary school students on a standardized test taken at age 16. The data include 3435 observations on 6 variables. The help file description is as follows:

• verbal The verbal reasoning score on a test taken by the students on entry to secondary school
• attain The score attained on the standardized test taken at age 16
• primary A factor indicating the primary school that the student attended
• sex A factor with levels M and F
• social The student’s social class on a numeric scale from low to high social class
• second A factor indicating the secondary school that the student attended

After performing some explorative analyses:

1. Consider the binary variable attain01 which takes values 1 if attain is greater than 5 and 0 otherwise. Build a model for studying the effects of covariates on attain01 with rstan, taking into account the hierarchical structure of the data.
2. Check the model fit and comment the results.
3. Draw inference on school random effects. Does the primary school matter?
4. [optional] Propose an alternative model for the variable attain (stan fit is not required).

Problem D: Italian football data

The R package engsoccerdata contains many datasets with results for National Leagues, European Cup and Champions League matches (including qualifiers) from 1871 to 2016. The dataset italy consists of 25404 match-observations on 8 variables. The help file description is as follows:

• Date Date of match
• Season Season of match - refers to starting year
• home Home team
• visitor Visiting team
• FT Full-time result at 90 mins
• hgoal home goals at FT 90mins
• vgoal visitor goals at FT 90mins
• tier tier of football pyramid: 1

The reference

Baio, G., Blangiardo, M. (2010). Bayesian hierarchical model for the prediction of football results. Journal of Applied Statistics, 37, 253–264

explains how to model two independent Poisson distributions for the number of the goals scored by two teams in a Bayesian framework.

1. Build a model with the rstan package, trying to replicate the model of the paper.
2. Interpret the results and check the fit of your selected model with posterior predictive checking tools.
3. Using the same data of Baio and Blangiardo (2010), try to compare your results with those reported in the paper.
4. [optional] Compare different models using predictive information criteria, such as LOOIC. What can you conclude about the model?
5. [optional] Propose a different model for the number of the goals (stan fit is not required).

Problem E: Positive patients due to Covid-19

Download the data for the Covid-19 spreading outbreak from the official website of Protezione Civile, by using the following command:

read.csv("https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-regioni/dpc-covid19-ita-regioni.csv")

The dataset contains the following variables:

• data Date of notification
• stato Country of reference
• codice_regione Code of the Region (ISTAT 2019)
• denominazione_regione Name of the Region
• lat Latitude
• long Longitude
• ricoverati_con_sintomi Hospitalised patients with symptoms
• terapia_intensiva Intensive Care
• totale_ospedalizzati Total hospitalised patients
• isolamento_domiciliare Home confinement
• totale_positivi Total amount of current positive cases (Hospitalised patients + Home confinement)
• variazione_totale_positivi New amount of current positive cases (totale_positivi current day - totale_positivi previous day)
• nuovi_positivi New amount of current positive cases (totale_casi current day - totale_casi previous day)
• dimessi_guariti Recovered
• deceduti Death
• totale_casi Total amount of positive cases
• tamponi Tests performed
• casi_testati Total number of people tested

Consider your dataset until 1 April 2020. After performing some explanatory analysis:

1. Build a model for nuovi_positivi with the rstan package. Poisson distribution is ok, but you can explore other ones.

2. Evaluate the inclusion of the following covariates:

• time
• lockdown measures adopted by the Italian Government (see here for more details https://en.wikipedia.org/wiki/2020_Italy_coronavirus_lockdown)
• number of medical swabs
• regional membership.

3. Study the temporal trend of your selected response variable.

4. Check the fit of your final model by using posterior predictive checking tools and comment.

5. [optional] Provide 3/4 days-forward predictions.

6. [optional] Compare alternative models in terms of predictive information criteria and comment.