Survival function for the grades (1: Well-differentiated; 2: Moderately differentiated; 3: Poorly differentiated or undifferentiated) using all complete data (n = 233,125).

Survival function for the grades (1: Well-differentiated; 2: Moderately differentiated; 3: Poorly differentiated or undifferentiated) using all complete data (n = 233,125).</p

Variables in the Equation using all the data (n = 233,125).

Variables in the Equation using all the data (n = 233,125).</p

Evaluating (<i>α</i> = 0.05) and the power of testing <i>H</i>_<i>o</i>: <i>β</i>₂ = 0 vs <i>H</i>_<i>a</i>: <i>β</i>₂ ≠ 0 adjusting for the auxiliary variable (Z) in the model.

(Continuous risk factor) {Censoring variable = ci = U(0,1)*1.5}.</p

Evaluating the (<i>α</i> = 0.05) and the power of testing <i>H</i>_<i>o</i>: <i>β</i>₂ = 0 vs <i>H</i>_<i>a</i>: <i>β</i>₂ ≠ 0 adjusting for (Z) in the model.

(Binary risk factor) {Censoring variable = ci = U(0,1)*1.5}.</p

Estimating 95% confidence Interval length and coverage probability (CP) of the Hazard Ratio (HR) (Binary risk factor) {Censoring variable = ci = U(0,1)*1.5}.

Estimating 95% confidence Interval length and coverage probability (CP) of the Hazard Ratio (HR) (Binary risk factor) {Censoring variable = ci = U(0,1)*1.5}.</p

Estimating 95% confidence interval length and coverage probability (CP) of the Hazard Ratio (HR) (Continuous risk factor) {Censoring variable = ci = U(0,1)*1.5}.

Estimating 95% confidence interval length and coverage probability (CP) of the Hazard Ratio (HR) (Continuous risk factor) {Censoring variable = ci = U(0,1)*1.5}.</p

sj-docx-1-smm-10.1177_09622802231168248 - Supplemental material for Misclassification simulation extrapolation method for a Weibull accelerated failure time model

Supplemental material, sj-docx-1-smm-10.1177_09622802231168248 for Misclassification simulation extrapolation method for a Weibull accelerated failure time model by Varadan Sevilimedu, Lili Yu and Hani Samawi in Statistical Methods in Medical Research</p

Estimating Hazard ratio (HR) estimation and their MSE (Continuous risk factor) {Censoring variable = ci = U(0,1)*1.5}.

Estimating Hazard ratio (HR) estimation and their MSE (Continuous risk factor) {Censoring variable = ci = U(0,1)*1.5}.</p

Estimation of Hazard ratio (HR) estimation and their MSE (Binary risk factor) {Censoring variable = ci = U(0,1)*1.5}.

Estimation of Hazard ratio (HR) estimation and their MSE (Binary risk factor) {Censoring variable = ci = U(0,1)*1.5}.</p

Computing Protein–Protein Association Affinity with Hybrid Steered Molecular Dynamics

Computing protein–protein association affinities is one of the fundamental challenges in computational biophysics/biochemistry. The overwhelming amount of statistics in the phase space of very high dimensions cannot be sufficiently sampled even with today’s high-performance computing power. In this article, we extend a potential of mean force (PMF)-based approach, the hybrid steered molecular dynamics (hSMD) approach we developed for ligand–protein binding, to protein–protein association problems. For a protein complex consisting of two protomers, P1 and P2, we choose <i>m</i> (≥3) segments of P1 whose <i>m</i> centers of mass are to be steered in a chosen direction and <i>n</i> (≥3) segments of P2 whose <i>n</i> centers of mass are to be steered in the opposite direction. The coordinates of these <i>m</i> + <i>n</i> centers constitute a phase space of 3­(<i>m</i> + <i>n</i>) dimensions (3­(<i>m</i> + <i>n</i>)­D). All other degrees of freedom of the proteins, ligands, solvents, and solutes are freely subject to the stochastic dynamics of the all-atom model system. Conducting SMD along a line in this phase space, we obtain the 3­(<i>m</i> + <i>n</i>)­D PMF difference between two chosen states: one single state in the associated state ensemble and one single state in the dissociated state ensemble. This PMF difference is the first of four contributors to the protein–protein association energy. The second contributor is the 3­(<i>m</i> + <i>n</i> – 1)­D partial partition in the associated state accounting for the rotations and fluctuations of the (<i>m</i> + <i>n</i> – 1) centers while fixing one of the <i>m</i> + <i>n</i> centers of the P1–P2 complex. The two other contributors are the 3­(<i>m</i> – 1)­D partial partition of P1 and the 3­(<i>n</i> – 1)­D partial partition of P2 accounting for the rotations and fluctuations of their <i>m</i> – 1 or <i>n</i> – 1 centers while fixing one of the <i>m</i>/<i>n</i> centers of P1/P2 in the dissociated state. Each of these three partial partitions can be factored exactly into a 6D partial partition in multiplication with a remaining factor accounting for the small fluctuations while fixing three of the centers of P1, P2, or the P1–P2 complex, respectively. These small fluctuations can be well-approximated as Gaussian, and every 6D partition can be reduced in an exact manner to three problems of 1D sampling, counting the rotations and fluctuations around one of the centers as being fixed. We implement this hSMD approach to the Ras–RalGDS complex, choosing three centers on RalGDS and three on Ras (<i>m</i> = <i>n</i> = 3). At a computing cost of about 71.6 wall-clock hours using 400 computing cores in parallel, we obtained the association energy, −9.2 ± 1.9 kcal/mol on the basis of CHARMM 36 parameters, which well agrees with the experimental data, −8.4 ± 0.2 kcal/mol