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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 [http://en.wikipedia.org/wiki/SERCA 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 [http://www.selleckchem.com/products/ON-01910.html 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, [http://www.selleckchem.com/products/wortmannin.html 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 | |||
Revision as of 17:20, 1 December 2016
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