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I was watching this video (https://www.youtube.com/watch?v=Wy-WmY6x4tg) and the presenter mentions (@ 8:10) that in a Multi-State Model, the user is required to specify number of "States" within the model and which "Transitions" between these States are "Valid":

enter image description here

Based on this diagram, I can understand that for the Multi-State Model to work, (obviously) the user must specify how many States are in the model, but I am not sure why it is necessary for the user to specify which Transitions are "Valid". In the above example, there are 3 States - The First State can transition to the Second State or the Third State, the Second State can only transition to the Third State, and the Third State is an "absorbing" State (i.e. no transitions are allowed).

This transition structure has been codified into the generator matrix "Q" - but I still do not understand why this transition structure is necessary. For example, suppose we assume a model in which "any state can be reached from any state" - but based on the observed data, there is only evidence of certain transitions happening (e.g. like the above diagram): When it came time to estimate the entries of the Q Matrix, wouldn't the entries corresponding to transitions between states that was never observed in the data be estimated as 0 (regardless if they were codified as "valid transitions" into the transition structure) ? Why can't we just say that all transitions are valid and then all transitions that never happen in the data would be estimated 0?

Thus, if the Q matrix is specified as in the diagram - or if the Q matrix is specified in such a way, such that there are no 0's : For the same observed data, wouldn't the estimated values of the entries in both matrix specifications be identical?

Thanks!

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    $\begingroup$ What happens when, as is usually the case in survival models, many of your observations are censored, so that you just don't know whether subsequent transitions occurred? $\endgroup$
    – whuber
    May 30, 2022 at 19:52

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I haven't watched the video in question, but I think there are two relevant points here. While you're absolutely right that these transitions could be estimated from the data, for exactly the reasons you say, there are good reasons to not do this, and to specify a priori which transitions are "valid".

First, there are two ways of thinking about the behaviour of this model. One is to say that once state 3 is reached, the model stops - there are no valid transitions once that state has been reached, so those entries are fixed at zero. The other is to say that once state 3 is reached, the only valid transition is the one that keeps you in state 3. This would be encoded as $q_{31} = 0, q_{32} = 0, q_{33} = 1 - (q_31 + q_{32}) = 1$ (I'm assuming these are probabilities, so each row should sum to $1$). Both representations work, but the author prefers to go with the former.

Second, it's conceptually useful to place constraints on Markov models like this, because these constraints usually correspond to theories about the system being studied. In this case, removing the zero constraints on the matrix would correspond to the graphical model below, which is clearly a different theory.

enter image description here

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