**5. Spatial survival models**

Generally, survival analysis models are specified through their hazard function, *h t*ð Þ, whose intuitive interpretation is that *h t*ð Þ*δt* is the conditional probability that a patient will die in the interval ð Þ *t*, *t* þ *δt* , given tat they have survived until time *t*. The most widely used approach to modeling *h t*ð Þ, at least in medical applications, is to use a semi-parametric formulation [44]. In this approach, the hazard for the *i*-th patient is modeled as

$$h(t\_i) = h\_0(t\_i) \exp\left(\mathbf{x}\_i^\prime \boldsymbol{\beta}\right),\tag{15}$$

where *xi* is a vector of explanatory variables for patient *i* and *h*0ð Þ*t* is an unspecified baseline hazard function. This is known as a proportional hazards (PH) model, because for any two patients *i* and *j*, *h t*ð Þ*<sup>i</sup> =h tj* does not change over time [1].

Another key idea in survival analysis is frailty, this corresponds to the random effects term used; time-to-event data will be group into strata, such as clinical sites, geographic regions, etc. This gives rise to mixed models, which include a random effect (the frailty) that correspond to a stratum's overall health status [30]. To illustrate, let *tij* be the time to death or censoring for subject *j* in stratum *i*, *j* ¼ 1, … , *ni*, *i* ¼ 1, … , *m*. Let **x***ij* be a vector of individual specific covariates, then

$$h\left(\mathbf{t}\_{ij}, \mathbf{x}\_{\vec{\eta}}\right) = h\_0\left(\mathbf{t}\_{\vec{\eta}}\right) \exp\left(\mathbf{x}\_{ij}'\boldsymbol{\beta} + \mathcal{W}\_i\right),\tag{16}$$

where *Wi* es the stratum-specific frailty term, designed to capture differences among strata; strata are typically denoted by *si*, *i* ¼ 1, … , *m*, so *si* denotes the location of the *i*-th patient and *Wi* ¼ *W s*ð Þ*<sup>i</sup>* . It can be assumed that the *Wi* are independent identical distribution (iid), i.e.

$$\mathcal{W}\_i \sim \mathcal{N}\left(\mathbf{0}, \sigma^2\right). \tag{17}$$

But it can also be assumed that *Wi* arises from a Gaussian process, i.e. if **W** ¼ ð Þ *W*1, … , *Wm* , then

$$\mathbf{W} \sim \mathcal{N} \mathcal{M} \left( \mathbf{0}, \sigma^2 \mathbf{R}(\phi) \right). \tag{18}$$

This way, suppose subjects are observed at *m* distinct spatial locations *s*1, … , *sm* ∈ *A*. Let *tij* be a random event time associated with the *j*-th subject in *si*, assume the survival time *tij* lies in the interval *aij*, *bij* , *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>m</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, … , *ni*; and **<sup>x</sup>***ij* be a related pdimensional vector of covariates, then are defined proportional hazard (PH) frailty

*Spatial Modeling in Epidemiology DOI: http://dx.doi.org/10.5772/intechopen.104693*

models, accelerated failure time (AFT) frailty models and proportional odds (PO) frailty models.

PH frailty models are the extensions of the population hazards model which is best known as the Cox model [44] a widely pursued model in survival analysis. PH frailty models extends the Cox model such that the hazard of an individual depends in addition on an unobserved random variable *W*, then introducing an additive frailty term *Wi* for each individual in the exponent of the hazard function as follows

$$h(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}) = h\_0(t\_{\vec{\eta}}) e^{\mathbf{x}\_{\vec{\eta}}^\prime \beta + W\_i}. \tag{19}$$

The corresponding survival function and the density are given by

$$\begin{split} \mathcal{S}\left(\mathbf{t}\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}\right) &= \mathcal{S}\_{0}\left(\mathbf{t}\_{\vec{\eta}}\right)^{\mathbf{x}\_{\vec{\eta}}^{\prime\beta+W\_{i}}}, \\ f\left(\mathbf{t}\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}\right) &= \mathbf{e}^{\mathbf{x}\_{\vec{\eta}}^{\prime}\beta+W\_{i}} \mathcal{S}\_{0}\left(\mathbf{t}\_{\vec{\eta}}\right)^{\mathbf{x}\_{\vec{\eta}}^{\prime\beta+W\_{i}}-1} f\_{0}\left(\mathbf{t}\_{\vec{\eta}}\right), \end{split} \tag{20}$$

where *S*0ð Þ� , *f* <sup>0</sup>ð Þ� and *h*0ð Þ� are the baseline survival function, baseline density and baseline hazard function assumed to be unique for all individual in the study population respectively.

Accelerated failure time frailty model extends the AFT model such that the hazard of an individual depends in addition on an unobserved random variable *W* [45–47]. Introducing an additive frailty term *Wi* for each individual in the exponent of the hazard function it becomes:

$$h\left(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}\right) = h\_0\left(e^{\mathbf{x}\_{\vec{\eta}}^{\prime}\beta + W\_i} t\_{\vec{\eta}}\right) e^{\mathbf{x}\_{\vec{\eta}}^{\prime}\beta + W\_i}.\tag{21}$$

The survival function and density are given by

$$\begin{aligned} \mathcal{S}(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}) &= \mathcal{S}\_0 \Big( e^{\mathbf{x}\_{\vec{\eta}}^\prime \beta + W\_i} t\_{\vec{\eta}} \Big), \\ \mathcal{f}\left(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}\right) &= e^{\mathbf{x}\_{\vec{\eta}}^\prime \beta + W\_i} f\_0 \Big( e^{\mathbf{x}\_{\vec{\eta}}^\prime \beta + W\_i} t\_{\vec{\eta}} \Big), \end{aligned} \tag{22}$$

where *S*0ð Þ� , *f* <sup>0</sup>ð Þ� and *h*0ð Þ� are the baseline survival function, baseline density and baseline hazard function assumed to be unique for all individual in the study population respectively.

Finally, proportional odds frailty model is given by

$$h(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}) = h\_0 \frac{1}{\mathbf{1} + \left[e^{-\mathbf{x}\_{\vec{\eta}}^\prime \beta - \mathcal{W}\_i} - \mathbf{1}\right] \mathbb{S}\_0(t\_{\vec{\eta}})} \,. \tag{23}$$

The survival function and density are given by

$$\begin{split} S(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}) &= \frac{S\_0(t\_{\vec{\eta}})e^{-\mathbf{x}\_{\vec{\eta}}^\prime \beta - W\_i}}{1 + \left(e^{-\mathbf{x}\_{\vec{\eta}}^\prime \beta - W\_i} - \mathbf{1}\right)S\_0(t\_{\vec{\eta}})}, \\ f\left(t\_{\vec{\eta}}, \mathbf{x}\_{\vec{\eta}}\right) &= \frac{f\_0\left(t\_{\vec{\eta}}\right)e^{-\mathbf{x}\_{\vec{\eta}}^\prime \beta - W\_i}}{\left[1 + \left(e^{-\mathbf{x}\_{\vec{\eta}}^\prime \beta - W\_i} - \mathbf{1}\right)S\_0\left(t\_{\vec{\eta}}\right)\right]^2}, \end{split} \tag{24}$$

where *S*0ð Þ� , *f* <sup>0</sup>ð Þ� and *h*0ð Þ� are the baseline survival function, baseline density and baseline hazard function assumed to be unique for all individual in the study population respectively.

In the frailty models, it is possible to deal with left, right and interval censoring of the data. Among the packages that exist in the R statistical software to perform spatial survival analysis is the *spBayesSurv* package [48]; the function *survregbayes* estimates the parameters of the PH, AFT and PO spatial models under the classical and Bayesian approach; also reports the posterior mean and median of the regression coefficients and of the parameters of the covariance function of the Gaussian process, *σ*<sup>2</sup> and *ϕ*, with their 95% credible intervals. The *spBayesSurv* package uses the powered exponential function (Table 0) to model the spatial correlation of the data.

Also in R, there is the *spatsurv* package [49], which implements the function *survspat* that fits parametric PH spatial survival models. This function reports the estimates and posterior median of the parameters *β*, *σ*<sup>2</sup> and *ϕ* with the respective credibility intervals.
