Explain the difference between probability and HR

[yinglittxs]

I've noticed an issue: in a model with states 1-2 and 2-3, the difference in transition probabilities between low and high levels of an exposure factor is smaller for state 1-2 than for state 2-3. However, the hazard ratio (HR) is higher for state 1-2. I am confused how can this strange phenomenon be explained?

[chjackson]

I don't fully understand what you write here, but I suspect the confusion is because the models in msm are continuous-time, not discrete-time. To answer the question in the subject line, the "hazard" (i.e. intensity, or rate) is the rate of instantaneously moving from one state to another. The "transition probability" in a continuous-time model depends on time $t$, and is the probability of being in a particular state s at a time $t$ in the future, given state $r$ now.

The transition probability from state r to state s depends not only on the transition intensity from state r to state s, but all the other intensities relating to states r and s. So if a covariate effects the r-s intensity in a particular way, the effect on the r-s probability is not easy to predict. Multiple transitions between different pairs of states could take place between time 0 and time t. Particularly if we are forecasting over a longer time t.

[yinglittxs]

Thank you~, I get it.

[yinglittxs]

Is the transition probability the absolute risk (probability) after the accumulation of these instantaneous rates over time, and is it affected by the baseline intensity and competing transitions and other related factors?

[chjackson]

Yes, I think that describes it roughly.