D variance .For the response approach with left-censoring, a possibly distinct set of covariates with measurement errors could also be associated for the skew-t (ST) distribution element exactly where we assume that the outcome variable follows an ST distribution [18, 22, 23] so as to incorporate skewness. Hence, the response variable yij for the ith topic at the jth occasion is formulated by(3)exactly where xij is definitely an s1 ?1 vector of covariates, g(? is really a nonlinear known function, d(? is definitely an s1dimensional vector-valued linear function, j is definitely an s1 ?1 individual-specific time-dependent parameter vector, ?is an s2 ?1 population parameter vector, bi = (bi1, …, bis3)T is definitely an s3 ?1 vector of random-effects possessing a multivariate typical distribution with variance b, ei = (ei1, …, eini)T follows a multivariate ST distribution with degrees of freedom , scale parameter 2 and an ni ?ni skewness diagonal matrix ? i) = diag( i1, …, in ) with ni ?1 e e e , then skewness parameter vector i = ( i1, …, in )T. In certain, if e e e i ? i) = Ini and i = 1ni with 1ni = (1, …, 1)T, implying that our interest is an general e e e e skewness measure. In the model (3), we assume that the individual-specific parameters j rely on the accurate (but unobservable) covariate z* (tij) as opposed to the observed covariate z(tij), which may be measured with errors; we discuss a covariate approach model subsequent.iStat Med. Author manuscript; available in PMC 2014 September 30.Dagne and HuangPage2.3. Covariate models In this paper, we consider covariate models for modeling measurement errors in timedependent covariates [7, 24, 25, 26]. We adopt a versatile empirical nonparametric mixedeffects model having a typical distribution to quantify the covariate process as follows.(four)NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscriptwhere w(tij) and hi(tij) are unknown nonparametric smooth fixed-effects and random-effects functions, respectively, and = ( 1, …, ni)T follows a multivariate regular distribution with scale parameter two. Let z* (tij) = w(tij) + hi(tij) be the correct but unobserved covariate values at time tij. Note that the fixed smooth function w(t) represents population typical in the covariate approach, when the random smooth function hi(t) measures inter-individual variation within the covariate process. We assume that hi(t) is a realization of a zero-mean stochastic procedure. To fit model (4), we apply a regression spline method to w(t) and hi(t). The key concept of regression spline is to approximate w(t) and hi(t) by using a linear combination of spline basis functions (for more particulars see [6, 27]).4-bromo-2,6-dimethylpyridine site As an illustration, w(t) and hi(t) is often approximated by a linear mixture of basis functions p(t) = 0(t), 1(t), .53103-03-0 supplier .PMID:28440459 ., p-1(t)T and ?q(t) = ?(t), ?(t), …, -1(t)T, respectively. Which is,(five)exactly where ( , …, -1)T can be a p ?1 vector of fixed-effects and ai = (ai0, …, ai,q-1)T (q p in = 0 p order to limit the dimension of random-effects) is really a q ?1 vector of random-effects possessing a multivariate normal distribution with imply zero variance-covariance matrix a. For our model, we take into account natural cubic spline bases with the percentile-based knots. To choose an optimal degree of regression spline and numbers of knots, i.e., optimal sizes of p and q, the Akaike info criterion (AIC) or the Bayesian data criterion (BIC) is normally applied [6, 27]. Replacing w(t) and hi(t) by their approximations wp(t) and hiq(t), we are able to approximate model (four) by.