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Thus, we are forced to work with discrete times. Furthermore, discrete times are used when the latent scale of the response times is discrete. High dimensional discrete survival data Assume there are n independent subjects (i = l, 2, 3,��, n) and p features per subject, where p >> n. Because this design matrix will be singular, traditional statistical methods (eg, OLS) are not applicable. The data are often presented as follows: Let Yi represent the discrete survival time response variable that takes on the values (j = 1, 2��, K), where K is the largest value of Y observed. To facilitate SERCA the formation of the likelihood, we define an n �� K response matrix as follows: yij={1ifyi=j0otherwise A p �� 1 vector of covariates, xi, is observed for each subject. The forward CR model with a complementary log-log link function With discrete survival data, we are generally interested in modeling the discrete hazard rate defined as ��ij=��j(xi)=P(Yi=j|Yi��j,xi). This is also the form of a probability modeled by a forward CR model. Furthermore, if it is reasonable to assume that the data were generated by a continuous-time proportional hazards model, then Rigosertib purchase we use the complementary log-log (cloglog) link function,5 log[?log(1?��ij)]=��j+xi�� Here ��j represents the intercept, or threshold, for the jth class. Notice that ��j is the only component of the model that depends on time. Thus, the functions for the K time points are parallel, Wortmannin in vitro which implies we are assuming proportional hazards. Likelihood We define the likelihood as a product of n conditionally independent Bernoulli random variables,6 where ��ij is the discrete hazard rate and (1 ? ��ij) is the conditional complement of ��ij given by P(Yi>j|Yi��j,xi) for the forward CR model. L=��i=1n��j=1K?1��ijyij(1?��ij)��Kk=jyik?yij Now define ��j=(��1j, ��2j,��, ��nj). When using the cloglog link, the derivative of the log-likelihood is then given by ��logL�Ħ�p=��j=1K?1[xPTexp?exp��j+X��+��j+X��[yi��j?��k=jKyk?yj1?��j]] We use the generalized monotone incremental forward stagewise algorithm to solve for the penalized solution: ��^=argmax��(log[L(��,��|y,X)]?�ˡ�P=1P|��P|) The tuning parameter, ��, controls the amount of shrinkage. As �� increases, the number of parameter estimates that will be shrunk to zero also increases. Using these coefficient estimates and the estimates for the ����s (described later), we can recursively estimate the probability that subject i belongs to class j where P(Yi=j|xi)=��ij*P(Yi��j|xi)={��ijfor j=1��ij*[1?��i=1j?1P(Yi=j|xi)]for?1