I understand that an accelerated failure time model can be conceptualized as a cox model which includes covariates whose effects depend on actual time, so the convenient expression of the partial likelihood due to Cox is simply not possible when the data arise from this type of model. If this is true, i.e. if $\log \lambda(t; \mathbf{X}) = \log \lambda_0(t) + \mathbf{X}^T\beta $ how exactly does one simulate data from such a model? And if that's not exactly the case, how can one with only a classic understanding of Cox models express the AFT model?
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For simulation it's simplest to work directly with the linear log-time formulation of an accelerated failure time (AFT) model instead of trying to work with hazards. That form, adapted to your notation, is (e.g., Chapter 12 of Klein and Moeschberger):
$$\log T = \mathbf{X}^T \beta + \sigma W ,$$
where $T$ is the failure time, $\mathbf{X}$ includes the intercept, $\sigma$ is a scale factor, and $W$ has a probability distribution corresponding to the baseline survival curve. This has the conceptual advantage of similarity to a standard linear model of $\log T$, with linear predictor $\mathbf{X}^T \beta$ and error term $\sigma W$, and thus should be accessible to anyone with a basic understanding of linear modeling.
Choose $\sigma$ to describe the desired width of the distribution, sample independently from $W$ and your $\mathbf{X}$ values, and you're done (except for modeling censoring). For example, if $W$ is standard normal you have a log-normal survival model, if $W$ is logistic you have a log-logistic model, etc. For an arbitrary baseline survival curve, work with the corresponding probability distribution that has that survival function in the $\log T$ scale.
