External time-varying covariates

[Andrea Mangani]

Hi, I am using the Cox regression with censored data for analyzing firm survival; one firm=one observation. I have 500 firms and the time span is 20 years. Each firm has its own time-independent characteristics that I use as main explanatory variables. In addition, I want to include time-varying covariates. To include “internal” time-varying covariates (for example, firm size, measured with sales of each year), I guess that the unique methodology is to split the sample. However, first I want to include an “external” time-varying covariate, such as the yearly change of GDP or, as an alternative, a dummy variable equal to 1 if the GDP is increasing and equal to 0 if the GDP is decreasing. In other words, the time-varying covariate is the same for all firms every year. Is there a simple method to do it instead of splitting each observation in 20 observations? Thank you,
Andrea

[Steve Samuels]

A couple of questions:

1. Do all firms enter followup in the same calendar year?
2. What is the time unit for events: days? (i.e. you know the calendar dates of entry, failure, end of observation) or year?

[Andrea Mangani]

Hi, thanks for the message. In the time span of twenty years, some firms are already in, others enter during that period. I know the exact date (day/month/year) of entry/failure/end of observation.

[Steve Samuels]

Thanks for the information. I'm afraid that early change in GDP has a fatal defect as a time-varying covariate. The basic requirement for a time-varying covariate, internal or external, is that it's value x(t) at time t must be based on history up to time t. The yearly change in GDP is measurable only after the year is over. Therefore, the change cannot serve as a risk predictor for events during the year.

I'll note as a matter of interest that the categorization of time-varying covariates as internal or external first appeared in Kalbfleisch and Prentice (1980).

Reference
Kalbfleisch, J. D., & Prentice, R. L. (1980). The statistical analysis of failure time data. New York: Wiley.

[Steve Samuels]

I should add that GDP change in the prior calendar year is a possible time-varying covariate for events in the current year. Whether it is a useful one, I annot judge.

[Andrea Mangani]

Thank you Steve. Regarding your point, I agree with your suggestion that it would be reasonable using a variable in t to predict the risk of failure events in t+1. But there are situations in which the survival analysis must tkat into account the impact of time t on failures in t. For example, if you're studying the risk of suicide, it is logical to use the wheather conditions as a "current" time varying covariate: if today it is raining, suicides will probably increase (today). My point is: how to include this effect in the Cox regression?

[Steve Samuels]

I agree that current and recent weather are ideal external covariates for many purposes. My point is that year-to-year change in GDP is a statistic that by definition depends on history over the whole year. For an economic indicator at time t , you need a statistic based only on the history of conditions up to t. That could possibly be a forecast published by t. However something like that might risk reverse causation, if the forecast was based on early internal indicators that firms were in trouble. See Goodliffe's article "The Hazards of Time-Varying Covariates" here.