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I am using a Cox PH model to study at what point in their lives people decide to move outside of the United States. My sample is 500 people who live in the United States, and the unit of analysis is the person-year. Using the traditional survival model language, "death" here refers to moving abroad and leaving the US.

I anticipate that existing international ties (e.g., they have a friend who lives in France) will affect a person's decision to move abroad. I want to include this as a covariate in my Cox PH model. However, because my unit of analysis is the person-year, I am worried that in years where moving abroad occurs, I won't be able to distinguish between the effect of existing international ties before the move and new international ties created immediately after the move.

In a linear model, I would usually solve this problem by lagging the independent variable (e.g., international ties in t-1). But I have never seen lagging in a survival model before. Can I lag a covariate in a Cox PH model?

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    $\begingroup$ Lags are used in state transition models which can often be more versatile and interpretable than time-to-event models. See this. $\endgroup$ Feb 25 at 12:53

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A Cox model bases its estimates on the covariate values that are in place for all those at risk at each event time. The definitions of covariate values thus should represent the values that are most directly associated with the risk of an event at that time. In your situation, that would at least mean making sure that you don't use international ties developed after the emigration event as a predictor of the hazard of the event. The lag you propose is a simple way to deal with that.

You might need to consider other ways to incorporate the history of predictors like the number of international ties into a time-to-event model. Is it just the current number of those ties that helps determine the hazard of the emigration event, or something in the history of developing those ties? You need to apply your understanding of the subject matter to make that type of determination and, if warranted, to formulate a predictor whose current value depends in some way on the history of some variable and contributes to the current hazard of an event.

That said, I'm not sure that a Cox model is the best way to approach this particular problem. With data only available annually, a discrete-time model might be preferable to a standard Cox model with its assumption of continuous time. A discrete-time model then might best be handled as a state-transition model, as Frank Harrell recommends in a comment.

However you approach it, I fear that your sample size of 500 individuals might not be adequate to get a reliable model. The number of events is what primarily determines the power of a time-to-event model. Only a few percent of US citizens lives abroad. Unless your sample is already over-represented by individuals likely to emigrate, your sample of 500 might only include a handful who experience the event.

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