“Survival” Analysis of Web Users

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Dell Zhang

DCSIS, Birkbeck, University of London

Outline

• What Is It

• Why Is It Useful

• Case Study

– The Departure Dynamics of Wikipedia Editors

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What Is It

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Time-To-Event Data

• Survival Analysis is a branch of statistics which deals with the modelling of time-to-event data

– The outcome variable of interest is time until an event occurs.

• death, disease, failure

• recovery, marriage

– It is called reliability theory/analysis in engineering, and duration analysis/modelling in economics or sociology.

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Y X

How to build a probabilistic model of Y ?

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Y X

How to build a probabilistic model of Y ?

How to build a probabilistic model of Y given X ?

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Y X

How to build a probabilistic model of Y ?

How to build a probabilistic model of Y given X ?

Censoring

• A key problem in survival analysis

– It occurs when we have some information about individual survival time, but we don’t know the survival time exactly.

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Y X

Options: 1) Wait for those patients to die?

2) Discard the censored data?

3) Use the censored data as if they were

not censored?

4) ……

Goals

• Survival Analysis attempts to answer questions such as

– What is the fraction of a population which will survive past a certain time? Of those that survive, at what rate will they die?

– Can multiple causes of death be taken into account?

– How do particular circumstances or characteristics increase or decrease the odds of survival?

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• Censoring of data

• Comparing groups

– (1 treatment vs. 2 placebo)

• Confounding or Interaction factors

– Log WBC

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Why Is It Useful

for Online Marketing etc.

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The Data Are There

• Events meaningful to online marketing

– Time to Clicking the Ad

– Informational: Time to Finding the Wanted Info

– Transactional: Time to Buying the Product

– Social: Time to Joining/Leaving the Community

– ……

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Time Matters!

Evidence-Based Marketing

• Let’s work as (real) doctors

– Users = Patients

– Advertisement (Marketing) = Treatment

Survival Analysis brings the time dimension

back to the centre stage.

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Predict whether a new question asked on Stack Overflow will be closed when

Case Study

The Departure Dynamics of Wikipedia Editors

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About 90,000 regularly active volunteer editors around the world 21

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Departure Dynamics

• Who are likely to “die”?

• How soon will they “die”?

• Why do they “die”?

“live” = stay in the editors’ community = keep editing “die” = leave the editors’ community = stop editing (for 5 months)

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Who are likely to “die”?

(WikiChallenge)

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2010-09-01

2010-09-01

2011-02-01

2010-04-01 2001-01-01

2001-06-01

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Behavioural Dynamics Features

months

Exponential Steps

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Web Search (SIGIR-2009), Social Tagging (WWW-2009), Language Modelling (ICTIR-2009)

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© 2008-2012 ~maniraptora 32

Gradient Boosted Trees (GBT)

Gradient Boosted Trees (GBT)

• The success of GBT in our task is probably attributable to – its ability to capture the complex nonlinear

relationship between the target variable and the features,

– its insensitivity to different feature value ranges as well as outliers, and

– its resistance to overfitting via regularisation mechanisms such as shrinkage and subsampling (Friedman 1999a; 1999b).

• GBT vs RF

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Final Result

• The 2nd best valid algorithm in the WikiChallenge – RMSLE = 0.862582: 41.7% improvement over

WMF’s in-house solution

– Much simpler model than the top performing system : 21 behavioural dynamics features vs. 206 features

– WMF is now implementing this algorithm permanently and looks forward to using it in the production environment.

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How soon will they “die”?

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birth & death

The evolution of Wikipedia editors' community. 40

110,000 random samples

January 2001

active editors

The evolution of Wikipedia editors' community. 41

January 2001

110,000 random samples

Survival Function

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What is the fraction of a population which will survive past a certain time?

The histogram of Wikipedia editors' lifetime.

Customary Editors Occasional Editors

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Kaplan-Meier Estimator

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The empirical survival function. 46

Normal Distribution

Probability Plot 47

Extreme Value Distribution

Probability Plot 48

Rayleigh Distribution

Probability Plot 49

Exponential Distribution

Probability Plot 50

Lognormal Distribution

Probability Plot 51

Weibull Distribution

Probability Plot 52

The survival function. 53

Weibull distribution

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Expected Future Lifetime

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median lifetime: 53 days

Hazard Function

The instantaneous potential per unit time for the event to occur, given that the individual has survived t.

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Of those that survive, at what rate will they die?

Bathtub Curve

57 http://en.wikipedia.org/wiki/Bathtub_curve

The hazard function. 58

59 The hazard function.

Conclusions

• For customary Wikipedia editors,

– the survival function can be well described by a Weibull distribution (with the median lifetime of about 53 days);

– there are two critical phases (0-2 weeks and 8-20 weeks) when the hazard rate of becoming inactive increases;

– more active editors tend to keep active in editing for longer time.

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Why do they “die”?

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Covariates

Last Edit

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Cox Proportional Hazards Model

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Semi-Parametric

• The semi-parametric property of the Cox model => its popularity

– The baseline hazard is unspecified

– Robust: it will closely approximate the correct parametric model

– Using a minimum of assumptions

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Cox PH vs. Logistic

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Maximum Likelihood Estimation

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Cox Proportional Hazards Model

β se z p

X1: namespace==Main

-0.1095 0.0172 -6.3664 0.1935e-9

X2: log(1+cur_size)

-0.0688 0.0036 -19.2474 0.0000e-9

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Hazard Ratio

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Adjusted Survival Curves

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Next Step

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Cartoon: Ron Hipschman Data: David Hand 74

Lightning Does Strike Twice!

• Roy Sullivan, a former park ranger from Virginia

– He was struck by lightning 7 times

• 1942 (lost big-toe nail)

• 1969 (lost eyebrows)

• 1970 (left shoulder seared)

• 1972 (hair set on fire)

• 1973 (hair set on fire & legs seared)

• 1976 (ankle injured)

• 1977 (chest & stomach burned)

– He committed suicide in September 1983.

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A Lot More To Do

• Multiple Occurrences of “Death”

– Recurrent Event Survival Analysis (e.g., based on Counting Process)

• Multiple Types of “Death”

– Competing Risks Survival Analysis

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Software Tools

• R

– The ‘survival’ package

• Matlab

– The ‘statistics’ toolbox

• Python

– The ‘statsmodels’ module?

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References

• David G. Kleinbaum and Mitchel Klein. Survival Analysis: A Self-Learning Text. Springer, 3rd edition, 2011. http://goo.gl/wFtta

• John Wallace. How Big Data is Changing Retail Marketing Analytics. Webinar, Apr 2005. http://goo.gl/OlMmi

• Dell Zhang, Karl Prior, and Mark Levene. How Long Do Wikipedia Editors Keep Active? In Proceedings of the 8th International Symposium on Wikis and Open Collaboration (WikiSym), Linz, Austria, Aug 2012. http://goo.gl/On3qr

• Dell Zhang. Wikipedia Edit Number Prediction based on Temporal Dynamics. The Computing Research Repository (CoRR) abs/1110.5051. Oct 2011. http://goo.gl/s2Dex

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