# Poisson or Linear Regression for Time Data

I am trying to model time-elapsed data (time from event A to event B) and am stuck on deciding between standard multiple linear regression vs. Poisson regression. A number of papers published on a similar topic seem to use Poisson regression, but in my head I've always associated the Poisson distribution with "count" data. What are your thoughts?

Thanks!

Unfortunately, linear regression is focused on cross-sectional data -- and Poisson regression usually collapses follow-up data over time and calendar period into cells of a multiway table -- thus each record in Poisson regression may represent age or calendar period, but the ensuing results (coefficients) are commonly interpreted as absolute or relative risk as a function of age group or calendar period (group).

Back to your question: if you have the difference in time between event A to event B for each object, i.e., $y =time_B-time_A$, then you could likely use $y$ as a dependent variable in linear regression, providing the residual values $e_i=y_i-\hat{y}_i$ derived from results are normally-distributed, where $\hat{y}_i$ are the predicted values of $y$.

There happens to be a niche in survival data analysis where a series of regression models allows one to regress a dependent time variable (not delta-time) directly on input predictor variables. The simplest example is called accelerated failure time (AFT) regression models, for which a host of time-based functions can be assumed (exponential, Weibull, Gompertz, logistic, etc).

If you have failed/survived (0,1) data for each object, and the time at which each object failed, you could also employ Cox proportional hazards regression -- which is a survival data analytic approach.

• Thank you for your help! I do, indeed, have the time difference between A and B (the value you labeled y). It sounds like the best approach for the specific problem I have is going to be using linear regression (as long as the residuals are normally-distributed. Thanks again! – Ben May 29 '16 at 1:01
• That will work, it's simply called using "delta time" as the dependent variable. Generate a histogram of the delta-time variable to see if it's normally-distributed -- if there are large left or right tails you'll violate regression assumptions. In this case, run a Shapiro-Wilk test or Lilliefors normality test on the delta-time variable. If any of these tests have a p-value<0.05, then the delta time variable is not normally distributed -- meaning you'd have to convert the delta time variable to ranks or van der Waerden scores before using in regression. – JoleT May 29 '16 at 1:05

If you are modelling the time between events, this can be thought of as being exponentially distributed, i.e a Poisson distribution can be thought of as being the number of events, while the exponential as the time between these events, https://en.wikipedia.org/wiki/Exponential_distribution. Based on this, $y=time_B−time_A$ probably follows an Exponential distribution.