LOCF (Last Observation Carried Forward)
The simplest possible longitudinal model. If {\(y_{it}\)} is the set of observed responses from early visits, and \(y_{i t_m}\) is the last observed value of \(y_{it}\), then the LOCF model for the final endpoint \(Y_i\) is
\[Y_i \mid \{y_{it}\} = y_{i t_m}\]
Beta Binomial
The Beta Binomial longitudinal model imputes a patient’s final endpoint given their most recent observed response.
The final endpoint response \(Y_i\) is modeled as:
\[Y_i \sim \text{Bernoulli}(\pi_{t y_{it}})\]
where \(\pi_{t y_{it}}\) is the probability that a patient is a response at the final endpoint given its early observed endpoint at time \(t\) is \(y_{it}\),
\[\pi_{t y_{it}} = \Pr(Y_i = 1 \mid y_{it}) \sim \text{Beta}(\alpha_{t {y_it}}, \beta_{t y_{it}})\]
We use the set cardinality operator \(\mid \ldots \mid\) to obtain the posterior distributions of \(\alpha_t\) and \(\beta_t\) as:
\[\alpha_{t0} = \alpha_{\mu 0} + \left| Y_i = 1, y_{it} = 0 \right| \] \[\alpha_{t1} = \alpha_{\mu 1} + \left| Y_i = 0, y_{it} = 0 \right| \] \[\beta_{t0} = \beta_{\mu 0} + \left| Y_i = 1, y_{it} = 1 \right| \] \[\beta_{t1} = \beta_{\mu 1} + \left| Y_i = 0, y_{it} = 1 \right| \]
i.e. a prior value \((\alpha_{\mu 0}, \alpha_{\mu 1}, \beta_{\mu 0}, \beta_{\mu 1})\) plus the number of subjects for which the final response is known to be 1 for \(\alpha_{tx}\) (or 0 for \(\beta_{tx}\)) and the response at time \(t\) is \(x\).
The \(\alpha_{tx}\) and \(\beta_{tx}\) parameters are independently estimated using only patients in their model instance, and may or not have identical priors \(\alpha_{\mu *}\) and \(\beta_{\mu *}\) depending on the Model Priors selection in FACTS. A common non-informative prior for the \(\pi_{t0}\) and \(\pi_{t1}\) parameters is \(\text{Beta}(1,1)\).
Logistic regression
The Logistic regression longitudinal model works similarly to the Beta Binomial imputation model. The difference is in the method of modeling the probability of a transition from an interim visit to the final visit \(\Pr(Y_i = 1 \mid y_{it})\). Like the Beta Binomial model, the logistic regression model imputes a patient’s final endpoint given their most recent observed response.
The final endpoint response \(Y_i\) is modeled as:
\[Y_i \sim \text{Bernoulli}(\pi_{t y_{it}})\]
where \(\pi_{t y_{it}}\) is the probability of a response at the final endpoint time given that its early observed endpoint at time \(t\)$ is \(y_{it}\). Then, we define the parameter
\[\theta_{ty_{it}} = \text{logit}\left( \pi_{ty_{it}} \right) = \log\left( \frac{\pi_{ty_{it}}}{1 - \pi_{ty_{it}}} \right)\].
The priors on \(\theta_{t0}\) and \(\theta{t1}\) are:
\[\theta_{t0} \sim \text{N}(\mu_0, \sigma_0^2)\] \[\theta_{t1} \sim \text{N}(\mu_1, \sigma_1^2)\]
The model computes the posterior distribution of \(\theta_{t0}\) and \(\theta_{t1}\) using all patients on arms belonging to the model instance that have observed endpoint values at time \(t\) and the final endpoint time \(T\).
The priors on \(\theta_{t0}\) and \(\theta_{t1}\) may be shared across model instances and/or visits depending on the selection made in the Model Priors section of the FACTS Logistic regression Longitudinal model page.
A possible starting place for non-informative priors in this model would be: \(\mu=0\), \(\sigma=2\). A weakly informative set of priors that an early response makes a final response more likely could be \[\theta_{t0} \sim \text{N}(-.75, 1.25^2)\] and \[\theta_{t1} \sim \text{N}(0.75, 1.25^2)\]
Restricted Markov Model (Absorbing Markov Chain)
The restricted markov model is special in the sense that it can be used if and only if the “Use longitudinal modeling” check box is checked, the “Enable Special Longitudinal Options” check box is checked, and “Use restriced Markov model” is selected. When these conditions are met the Virtual Subject Response tab changes and the Design > Longitudinal tab only has the Restricted Markov option.
The Absorbing Markov Chain model assumes that at each visit a subject is in one of three states – responder (1), stable (S), or failure (0). The responder and failure states are absorbing, meaning that once a patient has entered one of those states they must remain in that state for the remainder of the visits. Patients in the stable state may move to a responder or a failure in subsequent visits.
Unlike most other longitudinal models in FACTS, the Restricted Markov Model estimates the probability of a result at a visit based on the visit right before it, rather than predicting directly to the final endpoint from the early visit.
\[\Pr(y_{it} = n \mid y_{i, t-1} = S) \sim \text{Dirichlet}(\{\alpha_{0,t}, \alpha_{1,t}\, \alpha_{S,t}\}) \text{ for } t\ge 2\]
Where n can be 0, 1, or S, denoting the probability of going to the Fail state, the Responder state, or the Stable state at visit \(t\) from the Stable state at visit \(t-1\). \(t\) must be greater than or equal to \(2\), because we do not impute the first visit – a subject missing visit 1 and all visits after does not contribute to the longitudinal model or dose response model.
The priors for the \(\alpha\) parameters are specified in terms of the prior number of transitions from Stable at \(t-1\) to each different state at time \(t\). For example, if the prior value for the parameter \(\alpha_{1,3}\) is \(2\), we are putting apriori information into the Dirichlet distribution suggesting that \(2\) patients transitioned from Stable at visit 2 to Responders at visit 3. Specified priors can either be common or different across model instances based on user specification.
The parameters defining the posterior distribution of the state probabilities are available in closed form as:
\[\alpha_{0,t} = \gamma_{0,t} + \left|y_{it}=0, y_{i, t-1} = S\right|\] \[\alpha_{S,t} = \gamma_{S,t} + \left|y_{it}=S, y_{i, t-1} = S\right|\] \[\alpha_{1,t} = \gamma_{1,t} + \left|y_{it}=1, y_{i, t-1} = S\right|\]
To create a dichotomous endpoint, the user specifies in the Study > Study Info > Design Options section whether patients remaining in a stable state at the final visit should be dichotomized as responders or failures.
Dichotomous Endpoint: Dichotomized Continuous Longitudinal Model
The user may select (on the Study tab) to assume that the dichotomous final endpoint is generated by observing continuous longitudinal data and then dichotomizing the final endpoint based on whether it is greater than or less than a provided threshold. If the user selects this option, then they may select any of the continuous longitudinal models specified in the Continuous Longitudinal Models section. The engine will impute incomplete subjects according to the continuous model, resulting in a continuous imputed final endpoint. The imputed dichotomous final endpoint is then simply the dichotomized version of the continuous imputation.
All priors and methods are identical to the continuous longitudinal models mentioned above.