I want to implement an AFT model to study the failure time of a product given it's width.
$$S(t|width)=S_0((\beta_0+\beta_1 \times width)t)$$
However the prodction has varied during the time (less production at the beginning of the commercialisation of the product). Is it problematic ? And is there anyway to fix this problem by adding a variable in the model ?
Only someone with knowledge of the subject matter can say for sure whether it might be problematic that "production has varied during the time." I assume that survival is estimated for each individual item from its production or first-service date.
If you think that reliability has systematically varied as a function of calendar time or production rate at the time of each item's production, you could include either or both as covariates in your model. Again, the choice would be based on your understanding of the underlying processes.
Continuous predictors like those can be modeled flexibly with splines so that you don't have to hypothesize a particular form of the relationship(s) between calendar time (or production rate) and reliability; the data can speak for themselves to determine the form(s). Standard tests on the coefficients determine whether the hypothesized extra effects of calendar time or production rate on reliability are statistically "significant"; the magnitudes of those coefficients would illustrate their practical significance.
