Yi Yj ao 3 Zi ao 3 Zj Y Yj 3 Zi 3 Zj

which does not involves the intercept. Therefore, there is no need to estimate the intercept in the regression model for our purpose. While the traditional leave-one-out cross validation evaluates model performance in predicting the mean response and involves the estimation of the intercept, our proposed leave-two-out cross validation procedure estimates the error in predicting response difference and leaves the intercept as a nuisance parameter.

The leave-two-out cross validation uses each pair of observations as a validation sample, with the remaining data serving as a training sample. In the training sample that excludes subjects i and j, let <f—(i j) (•), 3—(ij) Zi and 3-(iij) Zj be the partial least squares estimates of ¥>(•), 3oZi and 3oZj with k latent variables, respectively. If these estimates are recomputed for all n(n — 1)/2 possible pairs (i,j), the mean-squared prediction error for the response difference between two cases with k latent variables can be estimated by C(k) =

2{n(n—l)}-1 £ { 0k_(ij)(YO)—^-(ij)(YO) — (3k:Z);Zi —3-^Zj) }2 •

For any k, C(k) requires O(n2) partial least squares model estimates, so we recommend using a stochastic estimate C*(k) of C(k). The most natural estimate is the observed mean-squared prediction error over randomly selected training samples. The number of training samples should be chosen so that the estimated standard error for C*(k) is no larger than 10% of C*(k). The number of latent variables K is selected to minimize C*(k).

In the linear regression model, the response of a future subject with a set of covariates is usually predicted by its conditional expectation given the covari-ates. However, in right-censored data, there is generally no unbiased estimator of the conditional mean of the response. Often the conditional median of the response can be well estimated if the censoring proportion is not too large. In those cases, the conditional median can be used to predict the response of a future subject in the accelerated failure time model, which corresponds to minimizing the mean absolute difference loss function. Another advantage of using the conditional median is that the median of T, the response variable of our primary interest, can be obtained easily from the monotone transformation function T = g—1(Y). This method is similar to that of [YWL92] for predicting the response of future subjects.

For a future subject with a covariate vector Zf, the predicted response based on the estimated conditional median of Y is given by / Zf + S—1(0.5), where / is the partial least squares parameter estimate from the observed data set {(Yi, Si, Zi)} and S£(-) is the Kaplan-Meier estimator of the survival function of the residuals using the empirical residuals {(Yi — / Zi,Si)}. The corresponding prediction for T is g—1(3 Zf + S—1(0.5)).

The next section shows the simulation studies exploring the predictive power of the accelerated failure time model using partial least squares and the Buckley-James fitting algorithm.

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