Ray poynter advanced quant - 2011 - 3

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Advanced Quant Techniques July 14, 2011

An Introduc4on to Hierarchical Bayes

Ray Poynter, The Future Place

Speaker Ray Poynter, The Future Place, UK NewMR Advanced Quant Techniques, July 14, 2011

An Introduction to Hierarchical Bayes

Ray Poynter The Future Place


Why HB?


Traditional View To solve a linear equation

y = mx + c 1 unknown, 2 observations

To solve a quadratic y = ax2 + bx + c 2 unknowns, 3 observations

To solve a 2 variable linear equation Y=b0+b1x1 + b2x2

2 unknowns, 3 observations


HB?

≈ HB ≈ Magic

e.g. 40 unknowns 12 observations


A company has three drivers, Andy, Bob, and Chris. I tell you one of them had a crash today If you had to guess who, you’d have a ⅓ chance

But, if I say Andy and Bob have never had a crash And, Chris has had fiQeen crashes already this year You might start to think it was Chris

With thanks to Rich Johnson at Sawtooth SoQware for the example


Introducing Bayes

Reverend Thomas Bayes 1702 -‐ 1761


Bayes Theorem

)()(*)|()|(

YPXPXYPYXP =

This is much clearer with an example, based on a nasty disease Bayesi=s, which is fortunately very rare, only 1 in 1000 people have it.


Bayesi=s Example

•  1-in-1000 have Bayesitis - chance is just 0.1%

•  The test is very reliable: –  If you have it the test will be positive in 99% of cases –  If you don’t have the it, chance of a negative result is 95%

•  You are tested and get a positive result, what is the chance that you have the disease?

Just under 2%


Alterna=ve View of Bayesi=s Imagine we have a popula=on of 1 million

Have the disease? P=0.1%

Posi=ve Test? P=99%

Yes, n=990

No, n=10

Posi=ve Test? P=5%

Yes, n=49,5

00

No, n=940,500

50,490 receive a posi=ve result. 990 are real posi=ves, 1.96% of all posi=ve results.


Hierarchical Bayes

In the context of choice data


Conceptual Guide to HB 2 level model is assumed – the Hierarchy in HB

–  Upper level - the part worths for a respondent are drawn from a multivariate normal distribution

–  Lower level - the logit assumption that the probability of choosing x is its transformed utility divided by sum of all the utilities

Markov Chain Monte Carlo Simulations –  Very computer intensive

Great ability to produce individual level results from choice data

–  even with many parameters and relatively few choices


Comparison of Hit Rates with Disaggregation

Summary of Six Commercial Studies

Aggregate Logit

Hierarchical Bayes Improvement

Study One 75.8 99.5 23.8Study Two 24.8 79.5 54.7Study Three 60.5 62.6 2.1Study Four 61.2 79.3 18.1Study Five 59.2 78.8 19.6Study Six 71.9 68.1 -3.8

Note, the last study was par=al profile . . . Source: Pinnell (2000)

Does HB Work?


Why Did We Need HB?

Allows us to estimate individual utilities for DCM studies

– The missing piece

Opens up other cases where sparcity of data meant individual results not available:

– MaxDiff scaling – Bundled choice modeling


HB


The Impact of HB on DCM Data If data for a respondent is ‘typical’, the HB contribution to their results is small

–  If the respondent was an outlier, the contribution of HB to their solution is more

Data from groups of people who’ve been through the HB are normally good predictors of the group solution

– An individual’s post HB data is not necessarily a good predictor of that individual’s values


HB is very useful

But it’s not magic


Thank you



Q & A


Andrew Jeavons Survey Analy=cs


Ray Poynter

Read Ray’s blog @ h>p://thefutureplace.typepad.com/ Follow Ray’s tweets at @ h>p://twi>er.com/raypoynter Connect with Ray on LinkedIn @ h>p://uk.linkedin.com/in/raypoynter Find out about Ray’s book at @ h>p://bit.ly/cmFnbo

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Ray poynter advanced quant - 2011 - 3