# Model for probability of song reaching top 10 ranking, over time?

I'm trying to model the probability of a song reaching Billboards top 10 over time.

My data has the columns "Day since release", "If reached top 10". For example, [12,1] means the song hit top 10 on the 12th day since release, and [350,0] means the song has been out for 350 days and has not become a top 10 hit yet.

Any suggestions for the best type of technique to approach this problem?

Thanks for any help!

• What predictors do you have? Commented Aug 5, 2014 at 22:50

## 1 Answer

You ask a general question, so I will give a general answer. This sounds like a case for event history models (aka survival models aka failure time models), which allow one to answer the dual question of whether and when an event occurs. In your case the event would be "reaching Billboard's Top Ten™." There are a variety of event history models, including:

These models allow one to estimate several quantities, including:

• the hazard function, which gives the instantaneous rate of event occurrence for a given point in time for continuous time models, or the probability of event occurrence in a specific time period conditional on it not yet having occurred in discrete time models

• the survival function, which gives the probability of an event not having occurred by a specific point in time since the start of study time; the converse of the survival function (i.e. 1 - survival) is sometimes called cumulative incidence and sometimes called uptake (and quite possibly called other things depending on discipline).

• median survival time and mean survival time

I like Singer and Willett's text, which details the Cox proportional hazards model (continuous time), and discrete time hazard models.

Depending on the precise nature of your research questions and data, these models may not adequately account for competition between songs in the rankings.

References

Singer, J. D. and Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. Oxford University Press, New York, NY.

• This approach does not seem incorporate the fact that rankings are rival. Commented Aug 5, 2014 at 20:18
• This is a good point. And I suppose it depends on the precise nature of the question being asked. I have edited to add this point. Commented Aug 5, 2014 at 20:21
• Thanks, Alexis. Do those event history models take into account the fact that some songs may never reach top 10? Commented Aug 5, 2014 at 21:14
• Yes. In the event history analysis lit, this is called "right censoring" meaning that you observe your units of analysis for some period of time, but the event has not occurred by the time you stop observing them. Commented Aug 5, 2014 at 21:16
• Ok, I understand. Does survival analysis assume that all cases will eventually "die", given enough time? Commented Aug 5, 2014 at 21:42