In this case, the hazard parameter is similar to the harvest parameter used in models for exploited populations [27], [45].
Rather than being unspecified, as in basic Cox proportional hazards models, the baseline hazard was set to be proportional to a proxy of influenza activity based on community surveillance data, with the constant of proportionality estimated as a parameter in the model.
Therefore, we can write where λ0 t) is the baseline hazard function, β is the p-dimensional parameter vector corresponding to Z i, and η i is the indicator η i = 1 if individual i eventually experiences the event (uncured) and η i = 0 if individual i never experiences the event, with the cured incidence c i = Pr(η i = 0 | X i ).
where h 1 is the hazard rate and h 0 is the unknown baseline hazard rate, X is a vector of covariates, and β is a vector of parameters.
The Cox regression model (Cox, 1972) for the hazard of death at time t can be expressed as: (11) where is the baseline hazard function, is the vector of parameters and is the vector of risk factor variables with corresponding sample value of for the i-th sample.
The hazard of failure at time t for patient i is modelled as where h0 t) is either a parametric or semi-parametric baseline hazard function, α is a parameter measuring the association between the observed longitudinal measurement and the hazard of failure at time t, and x i is a vector of further explanatory variables with regression parameters β.
(5)The baseline hazard h0 t) is defined by one or more estimated parameters, and x1, x2, … xn represent a set of predictors (e.g. one of the metrics in Tables 4 that were related to OS).
From an interpretability standpoint, the two-piece models were more appealing as they yielded parameter estimates that directly corresponded to changes in hazard at the specified time point, which may be useful in targeting interventions (while the parameter estimates for a Weibull model describe the shape of the baseline hazard, which is not directly meaningful).
The estimation procedures are semiparametric in that a baseline hazard function is nonparametrically specified.
The Cox proportional hazards model specifies the hazard function as λ t | x t = λ 0 t exp x t ′ β, where λ0 is the baseline hazard, x t) is the vector of observed covariate values at time t and β is the vector of unknown regression parameters.
