This function specifies the probability that an incident will end

This function specifies the probability that an incident will end before transpired time t. F(t) is also known as the failure function. Another basic function in hazard-based modeling is the survivor Bcl-xL apoptosis function S(t), which is expressed as follows: St=Pr⁡T≥t=1−Pr⁡T

H(t) = −ln S(t). Based on the log cumulative hazard scale, with a covariates vector z, the proportional hazards model can be expressed as follows: ln⁡H(t ∣ zi)=ln⁡H0(t)+βTzi. (3) Given H(t) = −ln S(t), (3) can be rewritten in the following equivalent form [37]: ln⁡−ln⁡S(t ∣ zi)=ln⁡−ln⁡S0t+βTzi, (4) where S0(t) = S(t∣0) is the baseline survival function and βT is a vector of parameters to be estimated for covariates z. Equation (4) can be generalized to [36] gθS(t ∣ zi)=s(x,γ)+βTzi, (5) where gθ(·) is a monotonic increasing function depending on a parameter θ, x = ln t and γ is an adjustable parameter vector. Royston and Parmar [36] took gθ(·) to be Aranda-Ordaz’s function: gθs=ln⁡s−θ−1θ, (6) where θ > 0. The limit of gθ(s) as θ tends

toward 0 is ln (−ln s), so that when θ = 0, the proportional hazards model can be expressed as gθS(t∣z) = ln (−ln (S(t∣z))). When θ = 1, the proportional odds model can be expressed as gθS(t∣z) = ln (S(t∣z)−1 − 1). When gθ(·) is defined as an inverse normal cumulative distribution function, the probity model can be expressed as gθS(t∣z) = −Φ−1(S(t∣z)), where Φ−1() is the inverse normal distribution function. As flexible mathematical functions, splines are defined by piecewise polynomials, but with some constraints to ensure that the overall curve is smooth;

the split points at which the polynomials join are known as knots [41]. Cubic splines are the most commonly used splines in practice. Restricted cubic splines [42] are used in this study with the restriction that the fitted function is forced to be linear before the first knot and after the final knot. Restricted cubic splines offer greater flexibility than standard parametric models in terms of the shape of the hazard function [37]. Restricted cubic splines with m distinct internal knots, k1,…, km, and two boundary knots, kmin and kmax , can be fit by creating m + 1 derived variables. A restricted cubic Anacetrapib spline function is defined as follows: sx,γ=γ0+γ1x+γ2v1x+⋯+γm+1vmx. (7) The derived variables vj(x) (also known as the basis function) can be calculated as follows: vjx=x−kj+3−λjx−kmin⁡+3−1−λjx−kmax⁡+3, (8) where for j = 1,…, mλj = (kmax − kj)/(kmax − kmin ) and (x − a)+ = max (0, x − a). The baseline distribution is Weibull or log-logistic with m = 0, meaning that no internal and no boundary knots are specified; that is, s(x, γ) = γ0 + γ1x [36].

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