See jackson_flexsurv_2016 for available baseline functions. Proportional baseline: \[ \lambda_{ml}(dt_{k,n}) = \lambda_{m0}(t) \lambda_{l} \tau_{k} \text{ for } l \in S_{m} \text{and} \lambda_{s_{m,1}}=1. \] \(dt_{k,n}\) denotes tk,nstop - tk,nstart

Proportional hazard term: \[ e^{(\theta_{k} + \beta) D_{ml} } \]

  • add covariate later.

out-of-state, item, person parameters.

  • no incercept term in prop. hazard if baseline contains constant in the same level.
  • action m leads to more/less coherent action
  • \(D_{ml}\) is bi-directional similarity mapping.
  • including \(\beta_m\) doesn’t make it directional.
  • is \(\beta_k\) meaningful for item-specific action space? certainly not! this opens up the question about how actions should be defined. loosely defined without event_desciption or not.

option1: similar items share the same action space

no event description should be used.

option2: each item has its own action space (item-specific action space)

  • use multi-state modeling framework to explain?
  • target journal:
  • grant application (check deadline)
  • meeting at 4pm (CST)
  • online learning platform: interaction with online resources, with instructors, with other people (communication length, contents) - team collaboration.
    • data will be available on Aug.
    • team science program (NIH)

The intensity function \(q_{ml}(\cdot)\) represents the instantaneous risk of moving from action \(m\) to \(l\).

\begin{align*} q_{ml} (t ; \boldsymbol{\lambda}, \boldsymbol{\beta}, \mathbf{z}(t)) = & \lambda_{ml}(t) e^{\beta_j + (\beta_m + \theta_{\beta}) D_{ml}}, \end{align*}

where \(\boldsymbol{\alpha}\) is a vector of intercepts, and \(\boldsymbol{\beta}\) is coefficients associated with \(\mathbf{z}(t)\), \(\lambda_{k,m\rightarrow l}(t)\) is a baseline intensity function. For each state \(l\), there are competing transitions \(m_1, \ldots, m_{n_l}\). This mean there are \(n_{l}\) corresponding survival models for state \(l\), and overall \(K=\sum_l n_l\) models. Models with no shared parameters can be estimated Common out of state transition: \(\beta_{ml}=\beta_{m}\).

Baseline hazard: \[ \lambda_{ml}(t) = \alpha_{m1}(t) \alpha_{l} + \theta_{\lambda} \text{ for } l \neq 1. \] Proportional hazard term: \[ e^{\beta_j + (\beta_m + \theta_{\beta}) D_{ml}} \]

  • \(D_{ml}\) is bi-directional similarity embedding between actions \(m\) and \(l\).

The piecewise-constant baseline hazard is used:

\begin{equation} \label{eq:1} \lambda(t) = \lambda_j \text{ if } s_{j-1} \le t < s_{j}, \end{equation}

for \(j = 1,\ldots,J\). \(\lambda_{j}\) could be a function of the similarity. This would be similar to have a piecewise constant transition matrix (time-inhomogeneous Markov chain), but much simpler as you have a parametric model for constants. The cosine similiarity should be normalized before used.

\begin{align*} q_{ml} (t ; \boldsymbol{\alpha}, \boldsymbol{\beta}, \mathbf{z}(t)) = & \lambda_{ml}(t) \exp( \boldsymbol{\beta}_{m,l}’ \mathbf{z}_{i,m,l}(t) ), \end{align*}

\begin{align*} q_{ml} (t ; \boldsymbol{\alpha}, \boldsymbol{\beta}, \mathbf{z}(t)) = & \lambda_{k,m \rightarrow l}(t) \exp( \alpha_m + \alpha_l + \boldsymbol{\beta} d_{i,m,l} ), \end{align*}

where \(\boldsymbol{\alpha}\) is a vector of intercepts, and \(\boldsymbol{\beta}\) is coefficients associated with \(\mathbf{z}(t)\), \(\lambda_{k,m\rightarrow l}(t)\) is a baseline intensity function. For each state \(l\), there are competing transitions \(m_1, \ldots, m_{n_l}\). This mean there are \(n_{l}\) corresponding survival models for state \(l\), and overall \(K=\sum_l n_l\) models. Models with no shared parameters can be estimated separately.