Bayesian Linear Regression

In regression analysis, we look at the conditional distribution of the response variable at different levels of a predictor variable

Response variable

- Also called “dependent” or “outcome” variable- What we want to explain or predict- In simple linear regression, response variable is continuous

Predictor variables

-Also called “independent” variables or “covariates”-In simple linear regression, predictor variable usually is also continuous-How we define which variable is response and which is predictor depends on our research question

Bayesian Linear Regression

Quick review of linear functions

Y is a response variable that is a linear function of the predictor variable

β0: intercept; the value of Y when X=0

β1: slope; how much Y changes when X increases by 1 unit

XY 10

Intro to Bayesian simple linear regression

Likelihood

Prior distribution

The posterior distribution is not straighforward

We have to implement MCMC techniques with WinBUGS

2102

10 ,~,,,| iii xNxy

vuIG

JjhN jjj

,~

,...,1,~2

00

Examples

Willingness to Pay for Environmental Improvements in a Large City. For example, we can study the social benefits of a set of environmental and urban improvements planned for the waterfront of the City of Valencia (Spain):

Response variable:

How much are you willing to pay for this policy?

Covariates: Sex, Age, Income

Data: 80 individuals

for(i IN 1 : n)

for(j IN 1 : 4)

v

u

h0[j]

beta0[j]

mean[i]

sigma

sigma2

h

beta[j]

income[i]

age[i]

sex[i]

WTP[i]

h0[j]

name: h0[j] type: constant

Examples

Random Utility Model

Probit Model

Logit Model

Discrete choice experiment

Objectives:

1. Revealed preference models use random utilities

2. Probit models assume that utilities are multivariate normal

3. Probit MCMC generates latent, random utilities

4. Logit models assume that the random utilities have extreme value distributions.

5. Logit MCMC uses the Hasting-Metropolis algorith

Random Utility Model

Utility for alternative m is:

where there are n subjects (sample),

M+1 alternatives in the choice set, and

Ji choice ocassiones for subject i

1,...,1

,...,1

,...,1

,,,,,,

Mm

Jj

ni

xY

i

mjiimjimji

Subject picks alternative k if

for all m

The probability of selecting k is

for all m

Statistical Models:

{εi,j,m} are Normal Probit Model

{εi,j,m} are Extreme Value Logit Model

mjikji YY ,,,,

mjikji YYP ,,,,

Logit model in WinBUGS

Likelihood:

Prior distributions:

kki

ii

XXp

pBernouilliy

...logit

p-1p

ln

;~

110i

i

JjhN jjj ,...,1,~ 00

Logit model in WinBUGS

Example: Discrete choice experiment

To study the value that car consumers place upon environmental concerns when purchasing a car

Response variable: Yes/No

Attributes: safety (Yes/No), carbon dioxide emissions, acceleration from 0 to 100 km/h(<10sec. And < 7.5 sec)2.5 sec), second hand, and annual cost (900€, 1400€, 2000€).

Sample size: 150

Logit model in WinBUGS

Probit model in WinBUGS

Likelihood:

Prior distributions:

kki

ii

XXp

pBernouilliy

...

;~

1101

vuIG

JjhN jjj

,~

,...,1,~2

00

Probit model in WinBUGS

Hierarchical Logit

The hierarchical logistic regression model is a very easy extension of standard logit.

Likelihoodyij ~ Bernoulli(pij),

logit(pij) <- b1j + b2jx2ij + … bkjxkij

Priorsbjk ~ N(Bjk, Tk) for all j,k Bjk <- k1 + k2 zj2 + … + km zjm

qr ~ N(0, .001) for all q,rTk ~ Gamma(.01, .01)