### Abstract

Statistical Appendix: Cox survival analysis using Gaussian process priors In this supplementary file, we describe in a detail how to apply the Gaussian processes (GP) in Cox survival analyses using the proportional hazards model. This statistical methodology is applied in the paper “Stratification of the risk for gastrointestinal stromal tumour recurrence after surgery: a combined analysis of ten populationbased cohorts”. For the individual i, where i =1,..., n, we have observed survival time yi (possibly right censored) with a censoring indicator δi, where δi =0if the ith observation is uncensored and δi =1if the observation is right censored. The traditional approach to analyze continuous time-to-event data is to assume the Cox proportional hazard function 1 hi(t) =h0(t) exp(x T i β), (1) where h0 is the unspecified baseline hazard rate, xi is the d × 1 vector of covariates for the ith patient and β is the vector of regression coefficients. The matrix X =[x1,..., xn] T of size n × d includes all covariate observations. The Cox model with a linear predictor can be extended to more general form to enable, for example, additive and non-linear effects of covariates. 2,3 We extend the proportional hazards model by hi(t) = exp(log(h0(t)) + ηi(xi)), (2) where the linear predictor is replaced with the latent predictor ηi depending on the covariates xi. By assuming a Gaussian process prior 4 over η =(η1,..., ηn) T, smooth nonlinear effects of continuous covariates are possible, and if there are dependencies between covariates, the GP can model these interactions implicitly. A zero-mean GP prior is set for η, which results in the zero-mean multivariate Gaussian distribution p(η|X) =N (0,C(X, X)), (3) where C(X, X) is the n × n covariance matrix whose elements are given by the covariance function of the GP. The covariance function defines the smoothness and scale properties of the latent function, and we choose a sum of constant and non-stationary neural network covariance function 5 c(xi, xj) =σc + 2 π sin−1

### Keyphrases

gaussian process covariance function proportional hazard model linear predictor paper stratification covariate observation latent predictor supplementary file non-stationary neural network covariance function ith patient gaussian process prior cox survival analysis traditional approach zero-mean gp prior cox proportional hazard function non-linear effect continuous covariates latent function regression coefficient scale property cox survival general form ith observation gastrointestinal stromal tumour recurrence covariance matrix matrix x1 survival time yi zero-mean multivariate gaussian distribution statistical appendix smooth nonlinear effect cox model statistical methodology unspecified baseline hazard rate continuous time-to-event data combined analysis