Time-dependent covariates

Covariates affected by treatment allocation. Variables measured after randomisation (e.g. compliance, duration of treatment) should not be used as covariates in a model for evaluation of the treatment effect as these may be influenced by the treatment received. A similar issue concerns late baselines , that is covariate measures that are based on data captured after randomisation. The term time-dependent covariate is sometimes used in relation to each of the examples above. 

Platt RW, Joseph KS, Ananth CV, Grondines J, Abrahamowicz M, Kramer MS (2004) A proportional hazards model with time-dependent covariates and time-varying effects for analysis of fetal and infant death. Am J Epidemiol, 160(3) 199-206. 

Higgins, K.M., Davidian, M. and Giltinan, D.M. A two-step approach to measurement error in time-dependent covariates in nonlinear mixed-effects models, with application to IGF-1 pharmacokinetics. Journal of the American Statistical Association 1997 93 436-448. 

Wu, L. and Wu, H. Missing time-dependent covariates in human immunodeficiency virus dynamic models. Applied Statistics 2002b 51 297-318. 

When each of these components is accounted for in the model, then a stationary time-series model is obtained and the outcome (or dependent variable) can now be treated like a normal distribution. In other words, the mean, variances, and correlations would no longer (for analysis purposes) change over the time sequence and any trends reported would be statistically meaningful and could also be used for future descriptors. The overall fit of the model can also be improved with the addition of other known explanatory factors that can account for intervention effects, and other time-dependent covariates. 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

The distinctive feature of the dynamic case is the time evolution of the estimate and its error covariance matrix. Their time dependence is given by... 

Similarly, in equation (6.9) for the covariances the matrices A and B are now time-dependent. We define the interaction representation by setting... 

They determine the variances and the covariance of the fluctuations of nx and nY around the values of the macroscopic solution of (5.5). Rather than study the general time-dependent state, however, we concentrate on the stationary state. 

Concentration-time curves for three individuals with different kidney functions CLCr are shown in Figure 1. The broken lines CLcr represent the time dependent CLcr as a measure of the kidney function. For the subject shown in the center panel, the CLCr decreases at 3.5 days causing a steep increase in the drug concentration. Dots are observations CONC (observed concentration), full lines PRED (model predictions for the population with p = 0) correspond to the model predictions for a typical individual with a specific set of mean covariates CLCr, WT and SEX (fixed effects). The broken lines IPRE (model predictions for the... 

It should also be recognized that regression-based methods, whether they are linear models or GAMs, do not take into account whether a covariate is time dependent nor do they reflect the potentially correlated behavior of the parameters. These methods compare the value of a single dependent variable against the value of a single independent variable. If the independent variable varies with time then some summary of the independent variable must be used. For example, the baseline, the median value, or the value at the midpoint of the time interval on which the variable was collected are sometimes used. 

Data splitting is fairly straightforward and covered in detail in the next section on validation. It simply implies that data to be modeled are partitioned based on differences in sampling (i.e., windows where suspect 0 are believed to be constant). The most common data splits to explore pharmacokinetic time dependencies would be single-dose, chronic non-steady-state, and steady-state conditions. Data subsets are modeled individually with all parameters and variability estimates along with any relevant covariate expressions compared in a manner similar to a validation procedure (see next section). Data can be combined in a leave-one-out strategy (see cross-validation description) to examine the uniformity of data windows. ... 

The local PCA is the eigenvalue problem for the time-dependent variance-covariance matrix C(t) of the distribution function, whose elements are defined by c,y(t) = ((x/t) -- (Xj))), where X is the tth coordinate and the average is taken between... 

Using simulation experiments, Bertrand finds that an increase in a or a decrease in CLL increases the standard deviation of quoted flow times. Similarly, an increase in a increases the standard deviation of actual flow times. The covariance of actual and quoted flow times increase in a and decrease in CLL. He also finds that using time-phased workload information improves the standard deviation of lateness (cri) however, the improvement decreases for larger values of a. The parameter CLL does not seem to have much impact on aL, however, it has a significant impact on the mean lateness (E[L]) and its best value depends on a. For decreasing CLL values, the sensitivity of mean quoted flow time to mean flow time increases. These results indicate that for the objective of minimizing (1) DDM policies which use time-phased work-... 

Classical electrodynamics, i.e.. Maxwell s unquantized theory for time-dependent electric and magnetic fields is inherently a covariant relativistic theory— in the sense of Einstein and Lorentz not Newton and Galilei — fitting perfectly well to the theory of special relativity as we shall understand in chapter 3. In this section, only those basic aspects of elementary electrodynamics will be... 

Eq. (7.8) is the most general covariant form of the inhomogeneous Maxwell equations, which immediately imply the continuity equation dy.j = + div = 0 of section 5.2.3, and Eq. (7.9) is the covariant time-dependent Dirac equation in the presence of external electric and magnetic fields. The homogeneous Maxwell equations are automatically satisfied by the sole existence of... 

Other samphng methods do not rely on the time-dependent evolution of a stmc-ture, but try to cover protein flexibility in collective variables. Collective variables describe the global motiorts within a system and can be derived fiom an MD simulation or from network models. From an MD simulation a pritrcipal comporrent arraly-sis of the covariance matrix of the N atom positiorts yields a set of N eigenvectors. The first ten to twenty eigenvectors contribute significantly to the overall atorttic fluctuations and are therefore called esserrtial modes or soft modes [63]. 

Figure 2 (A) A schematic drawing of an array of one-dimensional spectra of a homo-nuclear three-spin system with incremented evolution time q. We suppose that the magnetizations of the spins labeled with i, j, and k change with time as depicted in (B). Since the way that the magnetizations of / and k change with is somewhat correlated, a covariance cross-peak appears between / and k, as drawn in (Q, whereas the time dependence of the j magnetization is quite different from others, resulting in no appreciable covariance cross-peaks.

Figure 10 Mixing-time dependence of covariance spectra in C-labeled samples of Ap42 fibril mixed with curcumin. (A) and (B) were obtained with a mixing time of 50 ms in Ap42 labeled at the 17-21 residues and at the N-terminus (see Fig. 8). (C) and (D) were obtained with a mixing time of 500 ms. Reprinted with permission from Ref. [76], Copyright 2011 Elsevier.

For structural analysis, one often needs to perform correlation experiments for various mixing times, since the mixing-time dependence of the cross-peak intensities provides important distance restrictions. Unfortunately, the cross-peaks of the DTD spectrum build up with the mixing time in a somewhat different way from those in the FT spectrum. To retain the quantitative feature of the buildup curves, Kaiser et al. proposed a template DTD scheme [93]. In template DTD, covariance processing is performed only on the data acquired with the longest mixing time. Then, the individual FT spectra obtained with various mixing times are multiplied by this common covariance spectrum (Fig. 21). As demonstrated in Fig. 22, the buildup curves of the template DTD spectra show similar behavior to those of the FT spectra. 

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